# Statistics

## Mean, Median and Mode

### Learning Objectives

After completing this section, you should be able to:
1. Calculate the mode of a dataset.
2. Calculate the median of a dataset.
3. Calculate the mean of a dataset.
4. Contrast measures of central tendency to identify the most representative average.
5. Solve application problems involving mean, median, and mode.
What exactly do we mean when we describe something as "average"? Is the height of an average person the height that more people share than any other? What if we line up every person in the world, in order from shortest to tallest, and find the person right in the middle: Is that person’s height the average? Or maybe it’s something more complicated.
Imagine a game where you and a friend are trying to guess the typical person’s height. Once the guesses are made, you bring in every person and measure their height. You and your friend figure out how far off each of your guesses were from the actual value, then square that number. The result is the number of points you earn for that person. After we check every height and award points accordingly, the person with the lower score wins (because a lower score means that person’s guess was, overall, closer to the actual values). Could we define the average height to be the number that you should guess to give you the smallest possible score?
Each of these three methods of determining the “average” is commonly used. They are all methods of measuring centrality (or central tendency). Centrality is just a word that describes the middle of a set of data. All give potentially different results, and all are useful for different reasons. In this section, we’ll explore each of these methods of finding the “average.”

### The Mode

In our discussion of average heights, the first possible definition we offered was the height that more people share than any other. This is the mode, or the value that appears most often. If there are two modes, the data are bimodal.
Let’s look at some examples.
When we have a complete list of the data or a stem-and-leaf plot, it’s pretty straightforward to find the mode; we just need to find the number that appears most often. If we’re given a frequency distribution instead, the technique is different (but just as straightforward): we’re looking for the number with the highest frequency.
What happens if there is no number in the data that appears more than once? In that case, by our definition, every data value is a mode. But according to some other definitions, the data would have no mode. In practice, though, it doesn’t really matter; if no data value appears more than once, then the mode is not helpful at all as a measure of centrality.

### The Median

Let's revisit our example of trying to identify the height of the “average” person. If we lined everyone up in order by height and found the person right in the middle, that person’s height is called the median, or the value that is greater than no more than half and less than no more than half of the values.
Let’s look at a really simple example. Consider the following list of numbers: 11, 12, 13, 13, 14. Is the first number on the list, 11, the median? There are no values less than 11 (that’s 0%), and there are four values greater than 11 (that’s 80%). Since more than 50% of the data are greater than 11, the definition is violated; it’s not the median. Here’s a chart with the rest of the data, with red shading to show where the definition is violated:

Only 13 has no violations, so it’s the median according to the definition. In practice, we find the median just like we described in the average height example: by lining up all the data values in order from smallest to largest and picking the value in the middle. For our easy example (with data values 11, 12, 13, 13, 14), that first 13 is right in the middle; there are two values to the left and two values to the right. If there’s not one value right in the middle, we pick the two closest, then choose the number exactly between them. For example, let’s say we have the data 41, 44, 46, 53. Since there are an even number of data values in our list, we can’t pick the one right in the middle. The two closest to the middle are 44 and 46, so we’ll choose the number halfway between those to be the median: 45. As this example shows, the median (unlike the mode) doesn’t have to be a number in our original set of data.
In the examples we’ve looked at so far, it’s been pretty easy to identify which number is right in the middle. If we had a very large dataset, though, it might be harder. Fortunately, we have some formulas to help us with that.
Let’s put those formulas to work in an example.
Now, let’s tackle an example with an even number of values.

### The Mean

Recall our example of ways we could identify the “average” height of an individual. The last method we discussed was also the most complicated. It involved a game where the player guesses a height, then figures out how far off that guess is from every single person’s height. Those differences get squared and added together to get a score. Our next measure of centrality gives the lowest possible score: No other guess would beat it in the game. Given a dataset containing n total values, the mean of the dataset is the sum of all the data values, divided by n.
This is a computation you have likely done before. In many places, including spreadsheet programs like Microsoft Excel and Google Sheets, this number is called the average. For statisticians, though, the word average has too many possible meanings, so they prefer the one we’ll use: mean.
As the number of data values we are considering grows, the computation for the mean gets more and more complicated. That’s why people generally trust technology to perform that computation.
Note that a recent update to Google Sheets introduced a new function called “MODE.MULT,” which will find every mode (not just the first one on the list).

### Which Is Better: Mode, Median, or Mean?

If the mode, median, and mean all purport to measure the same thing (centrality), why do we need all three? The answer is complicated, as each measure has its own strengths and weaknesses. The mode is simple to compute, but there may be more than one. Further, if no data value appears more than once, the mode is entirely unhelpful. As for the mean and median, the main difference between these two measures is how each is affected by extreme values.
Consider this example: the mean and median of 1, 2, 3, 4, 5 are both 3. But what if the dataset is instead 1, 2, 3, 4, 10? The median is still 3, but the mean is now 4. What this example shows is that the mean is sensitive to extreme values, while the median isn’t. This knowledge can help us decide which of the two is more relevant for a given dataset. If it is important that the really high or really low values are reflected in the measure of centrality, then the mean is the better option. If very high or low values are not important, however, then we should stick with the median.
The decision between mean and median only really matters if the data are skewed. If the data are symmetric, then the mean and median are going to be approximately equal, and the distinction between them is irrelevant. If the data are skewed, the mean gets pulled in the direction of the skew (i.e., if the data are right-skewed, then the mean will be bigger than the median; if the data are left-skewed, the opposite relation is true).

### Check Your Understanding

### Key Terms

1. mode
2. bimodal
3. median
4. mean

### Key Concepts

1. The mode of a dataset is the value that appears the most frequently. The median is a value that is greater than or equal to no more than 50% of the data and less than or equal to no more than 50% of the data. The mean is the sum of all the data values, divided by the number of units in the dataset.
2. The median of a dataset is not affected by outliers, but the mean will be biased toward outliers. This distinction might affect which measure of centrality is used to summarize a dataset.

### Formulas

Suppose we have a set of data with  values, ordered from smallest to largest. If  is odd, then the median is the data value at position . If  is even, then we find the values at positions  and . If those values are named  and , then the median is defined to be .

### Videos

1. Compute Measures of Centrality Using Google Sheets


# Statistics

## Range and Standard Deviation

### Learning Objectives

After completing this section, you should be able to:
1. Calculate the range of a dataset
2. Calculate the standard deviation of a dataset
Measures of centrality like the mean can give us only part of the picture that a dataset paints. For example, let’s say you’ve just gotten the results of a standardized test back, and your score was 138. The mean score on the test is 120. So, your score is above average! But how good is it really? If all the scores were between 100 and 140, then you know your score must be among the best. But if the scores ranged from 0 to 200, then maybe 140 is good, but not great (though still above average). Knowing information about how the data are spread out can help us put a particular data value in better context. In this section, we’ll look at two numbers that help us describe the spread in the data: the range and the standard deviation. These numbers are called measures of dispersion.

###  The Range

Our first measure of dispersion is the range, or the difference between the maximum and minimum values in the set. It’s the measure we used in the standardized test example above.
Let’s look at a couple of examples.
For large datasets, finding the maximum and minimum values can be daunting. There are two ways to do it in a spreadsheet. First, you can ask the spreadsheet program to sort the data from smallest to largest, then find the first and last numbers on the sorted list. The second method uses built-in functions to find the minimum and maximum.
In either method, once you’ve found the maximum and minimum, all you have to do is subtract to find the range.
The range is very easy to compute, but it depends only on two of the data values in the entire set. If there happens to be just one unusually high or low data value, then the range might give a distorted measure of dispersion. Our next measure takes every single data value into account, making it more reliable.

###  The Standard Deviation

The standard deviation is a measure of dispersion that can be interpreted as approximately the average distance of every data value from the mean. (This distance from the mean is the “deviation” in “standard deviation.”)
To compute the standard deviation using the formula, we follow the steps below:
1. Compute the mean of all the data values.
2. Subtract the mean from each data value.
3. Square those differences.
4. Add up the results in step 3.
5. Divide the result in step 4 by
6. Take the square root of the result in step 5.
Let’s see that process in action.
The computation for the standard deviation is complicated, even for just a small dataset. We’d never want to compute it without technology for a large dataset! Luckily, technology makes this calculation easy.

### Check Your Understanding

### Key Terms

1. range
2. standard deviation

### Key Concepts

1. The range of a dataset is the difference between its largest and smallest values. The standard deviation is approximately the mean difference (in absolute value) that individual units fall from the mean of the dataset.

### Formulas

Here, s is the standard deviation,  represents each data value,  is the mean of the data values,  is the number of data values, and the capital sigma () indicates that we take a sum.

### Videos

1. Find the Minimum and Maximum Using Google Sheets
2. Find the Standard Deviation Using Google Sheets


# Statistics

## Percentiles

### Learning Objectives

After completing this section, you should be able to:
1. Compute percentiles.
2. Solve application problems involving percentiles.
A college admissions officer is comparing two students. The first, Anna, finished 12th in her class of 235 people. The second, Brian, finished 10th in his class of 170 people. Which of these outcomes is better? Certainly 10 is less than 12, which favors Brian, but Anna’s class was much bigger. In fact, Anna beat out 223 of her classmates, which is  of her classmates, while Brian bested 160 out of 170 people, or 94%. Comparing the proportions of the data values that are below a given number can help us evaluate differences between individuals in separate populations. These proportions are called percentiles. If  of the values in a dataset are less than a number , then we say that  is at the th percentile.

### Finding Percentiles

There are some other terms that are related to "percentile" with meanings you may infer from their roots. Remember that the word percent means “per hundred.” This reflects that percentiles divide our data into 100 pieces. The word quartile has a root that means “four.” So, if a data value is at the first quantile of a dataset, that means that if you break the data into four parts (because of the quart-), this data value comes after the first of those four parts. In other words, it’s greater than 25% of the data, placing it at the 25th percentile. Quintile has a root meaning “five,” so a data value at the third quintile is greater than three-fifths of the data in the set. That would put it at the 60th percentile. The general term for these is quantiles (the root quant– means “number”).
In Mean, Median, and Mode, we defined the median as a number that is greater than no more than half of the data in a dataset and is less than no more than half of the data in the dataset. With our new term, we can more easily define it: The median is the value at the 50th percentile (or second quartile).
Let’s look at some examples.
In each of the examples above, the computations were made easier by the fact that the we were looking for percentiles that “came out evenly” with respect to the number of values in our dataset. Things don’t always work out so cleanly. Further, different sources will define the term percentile in different ways. In fact, Google Sheets has three built-in functions for finding percentiles, none of which uses our definition. Some of the definitions you’ll see differ in the inequality that is used. Ours uses “less than or equal to,” while others use “less than” (these correspond roughly to Google Sheets’ ‘PERCENTILE.INC’ and ‘PERCENTILE.EXC’). Some of them use different methods for interpolating values. (This is what we did when we first computed medians without technology; if there were an even number of data values in our dataset, found the mean of the two values in the middle. This is an example of interpolation. Most computerized methods use this technique.) Other definitions don’t interpolate at all, but instead choose the closest actual data value to the theoretical value. Fortunately, for large datasets, the differences among the different techniques become very small.
So, with all these different possible definitions in play, what will we use? For small datasets, if you’re asked to compute something involving percentiles  without technology
  , use the technique we used in the previous example. In all other cases, we’ll keep things simple by using the built-in ‘PERCENTILE’ and ‘PERCENTRANK’ functions in Google Sheets (which do the same thing as the ‘PERCENTILE.INC’ and ‘PERCENTRANK.INC’ functions; they’re “inclusive, interpolating” definitions).

### Check Your Understanding

### Key Terms

1. percentile
2. quartile
3. quintile
4. quantile

### Key Concepts

1. The percentile rank of a data value is the percentage of all values in the dataset that are less than or equal to the given value.

### Videos

1. Using RANK, PERCENTRANK, and PERCENTILE in Google Sheets


# Statistics

## The Normal Distribution

### Learning Objectives

After completing this section, you should be able to:
1. Describe the characteristics of the normal distribution.
2. Apply the 68-95-99.7 percent groups to normal distribution datasets.
3. Use the normal distribution to calculate a -score.
4. Find and interpret percentiles and quartiles.
Many datasets that result from natural phenomena tend to have histograms that are symmetric and bell-shaped. Imagine finding yourself with a whole lot of time on your hands, and nothing to keep you entertained but a coin, a pencil, and paper. You decide to flip that coin 100 times and record the number of heads. With nothing else to do, you repeat the experiment ten times total. Using a computer to simulate this series of experiments, here’s a sample for the number of heads in each trial:
54, 51, 40, 42, 53, 50, 52, 52, 47, 54
It makes sense that we’d get somewhere around 50 heads when we flip the coin 100 times, and it makes sense that the result won’t always be exactly 50 heads. In our results, we can see numbers that were generally near 50 and not always 50, like we thought.

### Moving Toward Normality

Let’s take a look at a histogram for the dataset in our section opener:
This is interesting, but the data seem pretty sparse. There were no trials where you saw between 43 and 47 heads, for example. Those results don’t seem impossible; we just didn’t flip enough times to give them a chance to pop up. So, let’s do it again, but this time we'll perform 100 coin flips 100 times. Rather than review all 100 results, which could be overwhelming, let's instead visualize the resulting histogram.

From the histogram, we see that most of the trials resulted in between, say, 44 and 56 heads. There were some more unusual results: one trial resulted in 70 heads, which seems really unlikely (though still possible!). But we’re starting to maybe get a sense of the distribution. More data would help, though. Let’s simulate another 900 trials and add them to the histogram!

We can still see that 70 is a really unusual observation, though we came close in another trial (one that had 68 heads). Now, the distribution is coming more into focus: It looks quite symmetric and bell-shaped. Let’s just go ahead and take this thought experiment to an extreme conclusion: 10,000 trials.
The distribution is pretty clear now. Distributions that are symmetric and bell-shaped like this pop up in all sorts of natural phenomena, such as the heights of people in a population, the circumferences of eggs of a particular bird species, and the numbers of leaves on mature trees of a particular species. All of these have bell-shaped distributions. Additionally, the results of many types of repeated experiments generally follow this same pattern, as we saw with the coin-flipping example; this fact is the basis for much of the work done by statisticians. It’s a fact that’s important enough to have its own name: the Central Limit Theorem.

###  The Normal Distribution

In the coin flipping example above, the distribution of the number of heads for 10,000 trials was close to perfectly symmetric and bell-shaped:

Because distributions with this shape appear so often, we have a special name for them: normal distributions. Normal distributions can be completely described using two numbers we’ve seen before: the mean of the data and the standard deviation of the data. You may remember that we described the mean as a measure of centrality; for a normal distribution, the mean tells us exactly where the center of the distribution falls. The peak of the distribution happens at the mean (and, because the distribution is symmetric, it’s also the median). The standard deviation is a measure of dispersion; for a normal distribution, it tells us how spread out the histogram looks. To illustrate these points, let’s look at some examples.
Let’s put it all together to identify a completely unknown normal distribution.

### Properties of Normal Distributions: The 68-95-99.7 Rule

The most important property of normal distributions is tied to its standard deviation. If a dataset is perfectly normally distributed, then 68% of the data values will fall within one standard deviation of the mean. For example, suppose we have a set of data that follows the normal distribution with mean 400 and standard deviation 100. This means 68% of the data would fall between the values of 300 (one standard deviation below the mean: ) and 500 (one standard deviation above the mean: ). Looking at the histogram below, the shaded area represents 68% of the total area under the graph and above the axis:
Since 68% of the area is in the shaded region, this means that  of the area is found in the unshaded regions. We know that the distribution is symmetric, so that 32% must be divided evenly into the two unshaded tails: 16% in each.
Of course, datasets in the real world are never perfect; when dealing with actual data that seem to follow a symmetric, bell-shaped distribution, we’ll give ourselves a little bit of wiggle room and say that approximately 68% of the data fall within one standard deviation of the mean.
The rule for one standard deviation can be extended to two standard deviations. Approximately 95% of a normally distributed dataset will fall within 2 standard deviations of the mean. If the mean is 400 and the standard deviation is 100, that means 95% calculation describes the way we compute standardized scores. (two standard deviations below the mean: ) and 600 (two standard deviations above the mean: ). We can visualize this in the following histogram:

As before, since 95% of the data are in the shaded area, that leaves 5% of the data to go into the unshaded tails. Since the histogram is symmetric, half of the 5% (that’s 2.5%) is in each.
We can even take this one step further: 99.7% of normally distributed data fall within 3 standard deviations of the mean. In this example, we’d see 99.7% of the data between 100 (calculated as ) and 700 (calculated as ). We can see this in the histogram below, although you may need to squint to find the unshaded bits in the tails!

This observation is formally known as the 68-95-99.7 Rule.
There are more problems we can solve using the 68-95-99.7 Rule. but first we must understand what the rule implies. Remember, the rule says that 68% of the data falls within one standard deviation of the mean. Thus, with normally distributed data with mean 100 and standard deviation 10, we have this distribution:

Since we know that 68% of the data lie within one standard deviation of the mean, the implication is that 32% of the data must fall beyond one standard deviation away from the mean. Since the histogram is symmetric, we can conclude that half of the 32% (or 16%) is more than one standard deviation above the mean and the other half is more than one standard deviation below the mean:

Further, we know that the middle 68% can be split in half at the peak of the histogram, leaving 34% on either side:

So, just the “68” part of the 68-95-99.7 Rule gives us four other proportions in addition to the 68% in the rule. Similarly, the “95” and “99.7” parts each give us four more proportions:

We can put all these together to find even more complicated proportions. For example, since the proportion between 100 and 120 is 47.5% and the proportion between 100 and 110 is 34%, we can subtract to find that the proportion between 110 and 120 is :

### Standardized Scores

 When we want to apply the 68-95-99.7 Rule, we must first figure out how many standard deviations above or below the mean our data fall. This calculation is common enough that it has its own name: the standardized score. Values above the mean have positive standardized scores, while those below the mean have negative standardized scores. Since it's common to use the letter  to represent a standard score, this value is also often referred to as a -score.
So far, we’ve only really considered -scores that are whole numbers, but in general they can be any number at all. For example, if we have data that are normally distributed with mean 80 and standard deviation 6, the value 85 is five units above the mean, which is less than one standard deviation. Dividing by the standard deviation, we get . Since 85 is  of one standard deviation above the mean, we’d say that the standardized score for 85 is  (which is positive, since ). This calculation describes the way we compute standardized scores.

###  Using Google Sheets to Find Normal Percentiles

The 68-95-99.7 Rule is great when we’re dealing with whole-number -scores. However, if the -score is not a whole number, the Rule isn’t going to help us. Luckily, we can use technology to help us out. We’ll talk here about the built-in functions in Google Sheets, but other tools work similarly.
Let’s say we’re working with normally distributed data with mean 40 and standard deviation 7, and we want to know at what percentile a data value of 50 would fall. That corresponds to finding the proportion of the data that are less than 50. If we create our histogram and mark off whole-number multiples of the standard deviation like we did before, we’ll see why the 68-95-99.7 Rule isn’t going to help:
Since 50 doesn’t line up with one of our lines, the 68-95-99.7 Rule fails us. Looking back at  and , the best we can say is that 50 is between the 84th and 99.5th percentiles, but that’s a pretty wide range. Google Sheets has a function that can help; it’s called NORM.DIST. Here’s how to use it:
1. Click in an empty cell in your worksheet.
2. Type “=NORM.DIST(“
3. Inside the parentheses, we must enter a list of four things, separated by commas: the data value, the mean, the standard deviation, and the word “TRUE”. These have to be entered in this order!
4. Close the parentheses, and hit Enter. The result is then displayed in the cell; convert it to a percent to get the percentile.
So, for our example, we should type “=NORM.DIST(50, 40, 7, TRUE)” into an empty cell, and hit Enter. The result is 0.9234362745; converting to a percent and rounding, we can conclude that 50 is at the 92nd percentile. Let’s walk through a few more examples.
Google Sheets can also help us go the other direction: If we want to find the data value that corresponds to a given percentile, we can use the NORM.INV function. For example, if we have normally distributed data with mean 150 and standard deviation 25, we can find the data value at the 30th percentile as follows:
1. Click on an empty cell in your worksheet.
2. Type “=NORM.INV(“
3. Inside the parentheses, we’ll enter a list of three numbers, separated by commas: the percentile in question expressed as a decimal, the mean, and the standard deviation. These must be entered in this order!
4. Close the parentheses and hit Enter. The desired data value will be in the cell!
In our example, we want the 30th percentile; converting 30% to a decimal gives us 0.3. So, we’ll type “=NORM.INV(0.3, 150, 25)” to get 136.8899872; let’s round that off to 137.

### Check Your Understanding

1. normal distribution
2. 68-95-99.7 Rule
3. standardized score (-score)

### Key Concepts

1. Normally distributed data follow a bell-shaped, symmetrical distribution.
2. The mean of normally distributed data falls at the peak of the distribution. The standard deviation of normally-distributed data is the distance from the peak to either of the inflection points.
3. Data that are normally distributed follow the 68-95-99.7 Rule, which says that approximately 68% of the data fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations.
4. The -score for a data value is the number of standard deviations that value falls above (or below, if the -score is negative) the mean.
5. We can use the normal distribution to estimate percentiles.

### Formulas

If  is a member of a normally distributed dataset with mean  and standard deviation , then the standardized score for  is
If you know a -score but not the original data value , you can find it by solving the previous equation for :


# Statistics

## Applications of the Normal Distribution

### Learning Objectives

After completing this section, you should be able to:
1. Apply the normal distribution to real-world scenarios.
As we saw in The Normal Distribution, the word “standardized” is closely associated with the normal distribution. This is why tests like college entrance exams, state achievement tests for K–12 students, and Advanced Placement tests are often called “standardized tests”: scores are assigned in a way that forces them to follow a normal distribution, with a mean and standard deviation that are consistent from year to year. Standardization also allows people like college admissions officers to directly compare an applicant who took the ACT (a college entrance exam) to an applicant who instead chose to take the SAT (a different college entrance exam). Standardization allows us to compare individuals from different groups; this is among the most important applications of the normal distribution. We’ll explore this and other real-world uses of the normal distribution in this section.

### College Entrance Exams

There are two good ways to compare two data values from different groups: using -scores and using percentiles. The two methods will always give consistent results (meaning that we won’t find, for example, that the first value is better using -scores but the second value is better using percentiles), so use whichever method is more comfortable for you.

### Coin flipping

In the opening of The Normal Distribution, we saw that the number of heads we get when we flip a coin 100 times is distributed normally. It can be shown that if  is the number of flips, then the mean of that distribution is  and the standard deviation is  (as long as ). So, for 100 flips, the mean of the distribution is 50 and the standard deviation is 5.
          In that opening example, one of our early runs gave us 70 heads in 100 flips, which we noted seemed unusual. Using the normal distribution, we can identify exactly how unusual that really is. Using Google Sheets, the formula “=NORM.DIST(70, 50, 5, TRUE)” gives us 0.999968, which is the 99.997th percentile! How is that useful? Suppose you need to test whether a coin is fair, and so you flip it 100 times. While we might be suspicious if we get 70 heads out of the 100 flips, we now have a numerical measure for how unusual that is: If the coin were fair, we would expect to see 70 heads (or more) only  of the time. That’s really unlikely! Analysis like this is related to hypothesis testing, an important application of statistics in the sciences and social sciences.

### Analyzing Data That Are Normally Distributed

Whenever we’re working with a dataset that has a distribution that looks symmetric and bell-shaped, we can use techniques associated with the normal distribution to analyze the data.

### Check Your Understanding

### Key Concepts

1. We can use -scores to compare data values from different datasets.


# Statistics

## Scatter Plots, Correlation, and Regression Lines

### Learning Objectives

After completing this section, you should be able to:
1. Construct a scatter plot for a dataset.
2. Interpret a scatter plot.
3. Distinguish among positive, negative and no correlation.
4. Compute the correlation coefficient.
5. Estimate and interpret regression lines.
One of the most powerful tools statistics gives us is the ability to explore relationships between two datasets containing quantitative values, and then use that relationship to make predictions. For example, a student who wants to know how well they can expect to score on an upcoming final exam may consider reviewing the data on midterm and final exam scores for students who have previously taken the class. It seems reasonable to expect that there is a relationship between those two datasets: If a student did well on the midterm, they were probably more likely to do well on the final than the average student. Similarly, if a student did poorly on the midterm, they probably also did poorly on the final exam.
Of course, that relationship isn’t set in stone; a student’s performance on a midterm exam doesn’t cement their performance on the final! A student might use a poor result on the midterm as motivation to study more for the final. A student with a really good grade on the midterm might be overconfident going into the final, and as a result doesn’t prepare adequately.
The statistical method of regression can find a formula that does the best job of predicting a score on the final exam based on the student’s score on the midterm, as well as give a measure of the confidence of that prediction! In this section, we’ll discover how to use regression to make these predictions. First, though, we need to lay some graphical groundwork.

### Relationships Between Quantitative Datasets

Before we can evaluate a relationship between two datasets, we must first decide if we feel that one might depend on the other. In our exam example, it is appropriate to say that the score on the final depends on the score on the midterm, rather than the other way around: if the midterm depended on the final, then we’d need to know the final score first, which doesn’t make sense.
Here’s another example: if we collected data on home purchases in a certain area, and noted both the sale price of the house and the annual household income of the purchaser, we might expect a relationship between those two. Which depends on the other? In this case, sale price depends on income: people who have a higher income can afford a more expensive house. If it were the other way around, people could buy a new, more expensive house and then expect a raise! (This is very bad advice.)
It's worth noting that not every pair of related datasets has clear dependence. For example, consider the percent of a country’s budget devoted to the military and the percent earmarked for public health. These datasets are generally related: as one goes up, the other goes down. However, in this case, there’s not a preferred choice for dependence, as each could be seen as depending on the other. When exploring the relationship between two datasets, if one set seems to depend on the other, we’ll say that dataset contains values of the response variable (or dependent variable). The dataset that the response variable depends on contains values of what we call the explanatory variable (or independent variable). If no dependence relationship can be identified, then we can assign either dataset to either role.
Once we’ve assigned roles to our two datasets, we can take the first step in visualizing the relationship between them: creating a scatter plot.

### Creating Scatter Plots

A scatter plot is a visualization of the relationship between two quantitative sets of data. The scatter plot is created by turning the datasets into ordered pairs: the first coordinate contains data values from the explanatory dataset, and the second coordinate contains the corresponding data values from the response dataset. These ordered pairs are then plotted in the -plane. Let's return to our exam example to put this into practice.
For large datasets, it’s impractical to create scatter plots manually. Luckily, Google Sheets automates this process for us.

### Reading and Interpreting Scatter Plots

Scatter plots give us information about the existence and strength of a relationship between two datasets. To break that information down, there are a series of questions we might ask to help us. First: Is there a curved pattern in the data? If the answer is “yes,” then we can stop; none of the linear regression techniques from here to the end of this section are appropriate.  and  show several examples of scatter plots that can help us identify these curved patterns.
Once we have confirmed that there is no curved pattern in our data, we can move to the next question: Is there a linear relationship? To answer this, we must look at different values of the explanatory variable and determine whether the corresponding response values are different, on average. It's important to look at the values “on average” because, in general, our scatter plots won’t include just one corresponding response point for each value of the explanatory variable (i.e., there may be multiple response values for each explanatory value). So, we try to look for the center of those points. Let’s look again at , but consider some different values for the explanatory variable. Let’s highlight the points whose -values are around 50 and those that are around 80:
Now, we can estimate the middle of each group of points. Let's add our estimated averages to the plot as starred points:
Since those two starred points occur at different heights, we can conclude that there’s likely a relationship worth exploring.
Here’s another example using a different set of data:
Let’s look again at the points near 50 and near 80, and estimate the middles of those clusters:
Notice that there’s not much vertical distance between our two starred points. This tells us that there’s not a strong relationship between these two datasets.

### Positive and Negative Linear Relationships

Another way to assess whether there is a relationship between two datasets in a scatter plot is to see if the points seem to be clustered around a line (specifically, a line that’s not horizontal). The stronger the clustering around that line is, the stronger the relationship.
Once we’ve established that there’s a relationship worth exploring, it’s time to start quantifying that relationship. Two datasets have a positive linear relationship if the values of the response tend to increase, on average, as the values of the explanatory variable increase. If the values of the response decrease with increasing values of the explanatory variable, then there is a negative linear relationship between the two datasets. The strength of the relationship is determined by how closely the scatter plot follows a single straight line: the closer the points are to that line, the stronger the relationship. The scatter plots
      in  to  depict varying strengths and directions of linear relationships.
The strength and direction (positive or negative) of a linear relationship can also be measured with a statistic called the correlation coefficient (denoted ). Positive values of  indicate a positive relationship, while negative values of  indicate a negative relationship. Values of  close to 0 indicate a weak relationship, while values close to  correspond to a very strong relationship. Looking again at  to , the correlation coefficients for each, in sequential order, are: ‒1, ‒0.97, ‒0.55, ‒0.03, 0.61, 0.97, and 1.
There’s no firm rule that establishes a cutoff value of  to divide strong relationships from weak ones, but  is often given as the dividing line (i.e., if  or  the relationship is strong, and if  the relationship is weak).
The formula for computing  is very complicated; it’s almost never done without technology. Google Sheets will do the computation for you using the CORREL function. The syntax works like this: if your explanatory values are in cells A2 to A50 and the corresponding response values are in B2 to B50, then you can find the correlation coefficient by entering “=CORREL(A2:A50, B2:B50)”. (Note that the order doesn’t matter for correlation coefficients; “=CORREL(B2:B50, A2:A50)” will give the same result.)
Let’s put all of this together in an example.

### Linear Regression

The final step in our analysis of the relationship between two datasets is to find and use the equation of the regression line. For a given set of explanatory and response data, the regression line (also called the least-squares line or line of best fit) is the line that does the best job of approximating the data.
What does it mean to say that a particular line does the “best job” of approximating the data? The way that statisticians characterize this “best line” is rather technical, but we’ll include it for the sake of satisfying your curiosity (and backing up the claim of "best"). Imagine drawing a line that looks like it does a pretty good job of approximating the data. Most of the points in the scatter plot will probably not fall exactly on the line; the distance above or below the line a given point falls is called that point’s residual. We could compute the residuals for every point in the scatter plot. If you take all those residuals and square them, then add the results together, you get a statistic called the sum of squared errors for the line (the name tells you what it is: “sum” because we’re adding, “squared” because we’re squaring, and “errors” is another word for “residuals”). The line that we choose to be the “best” is the one that has the smallest possible sum of squared errors. The implied minimization (“smallest”) is where the “least” in “least squares” comes from; the “squares” comes from the fact that we’re minimizing the sum of squared errors. This is very similar to the process we outlined in the "game" that we used to introduce the mean.  Both the regression line and the mean are designed to minimize a sum of squared errors. Here ends the super technical part.

### Finding the Equation of the Regression Line

So, how do we find the equation of the regression line? Recall the point-slope form of the equation of a line:
The regression line has two properties that we can use to find its equation. First, it always passes through the point of means. If  and  are the means of the explanatory and response datasets, respectively, then the point of means is . We’ll use that as the point in the point-slope form of the equation. Second, if  and  are the standard deviations of the explanatory and response datasets, respectively, and if  is the correlation coefficient, then the slope is . Putting all that together with the point-slope formula gives us this:
Let's walk through an example.
As you can see, finding the equation of the regression line involves a lot of steps if you have to find all of the values of the needed quantities yourself. But, as usual, technology comes to our rescue. This video (which you actually watched earlier when learning how to create scatter plots) covers the regression line at around the 3:30 mark. Note that Google Sheets calls it the "trendline."
Let's put this into practice.

### Using the Equation of the Regression Line

Once we’ve found the equation of the regression line, what do we do with it? We’ll look at two possible applications: making predictions and interpreting the slope.
We can use the equation of the regression line to predict the response value  for a given explanatory value . All we have to do is plug that explanatory value into the formula and see what response value results. This is useful in two ways: first, it can be used to make a guess about an unknown data value (like one that hasn’t been observed yet). Second, it can be used to evaluate performance (meaning, we can predict an outcome given a particular event). In , we created a scatter plot of final exam scores vs. midterm exam scores using this data:
The equation of the regression line is , where  is the final exam score and  is the midterm exam score. If Frank scored 85 on the midterm, then our prediction for his final exam score is . To use the regression line to evaluate performance, we use a data value we’ve already observed. For example, Allison scored 88 on the midterm. The regression line predicts that someone who scores an 88 on the midterm will get  on the final. Allison actually scored 84 on the final, meaning she underperformed expectations by almost 4 points .
The second application of the equation of the regression line is interpreting the slope of the line to describe the relationship between the explanatory and response datasets. For the exam data in the previous paragraph, the slope of the regression line is 0.687. Recall that the slope of a line can be computed by finding two points on the line and dividing the difference in the -values of those points by the difference in the -values. Keeping that in mind, we can interpret our slope as  . Multiplying both sides of that equation by the denominator of the fraction, we get . Thus, a one-point increase in the midterm score would result in a predicted increase in the final score of 0.687 points. A ten-point drop in the midterm score would give us a decrease in the predicted final score of 6.87 points. In general, the slope gives us the predicted change in the response that corresponds to a one unit increase in the explanatory variable.

### Extrapolation

A very common misuse of regression techniques involves extrapolation, which involves making a prediction about something that doesn't belong in the dataset.

### Correlation Does Not Imply Causation

One of the most common fallacies about statistics has to do with the relationship between two datasets. In the dataset “Public”, we find that the correlation coefficient between the 75th percentile math SAT score and the 75th percentile verbal SAT score is 0.92, which is really strong. The slope of the regression line that predicts the verbal score from the math score is 0.729, which we might interpret as follows: “If the 75th percentile math SAT score goes up by 10 points, we’d expect the corresponding verbal SAT score to go up by just over 7 points.”
Does the increasing math score cause the increase in the verbal score? Probably not. What’s really going on is that there’s a third variable that’s affecting them both: To raise the SAT math score by 10 points, a school will recruit students who do better on the SAT in general; these students will also naturally have higher SAT verbal scores. This third variable is sometimes called a lurking variable or a confounding variable. Unless all possible lurking variables are ruled out, we cannot conclude that one thing causes another.

### Check Your Understanding

### Key Terms

1. response variable (dependent variable)
2. explanatory variable (independent variable)
3. scatter plot
4. positive linear relationship
5. negative linear relationship
6. correlation coefficient
7. regression line (least-squares line, line of best fit)

### Key Concepts

1. If one variable affects the value of another variable, we say the first is an explanatory variable and the second is a response variable.
2. Scatter plots place a point in the -plane for each unit in the dataset. The -value is the value of the explanatory variable, and the -value is the value of the response variable.
3. The correlation coefficient  gives us information about the strength and direction of the relationship between two variables. If  is positive, the relationship is positive: an increase in the value of the explanatory variable tends to correspond to an increase in the value of the response variable. If  is negative, the relationship is negative: an increase in the value of the explanatory variable tends to correspond to a decrease in the value of the response variable. Values of  that are close to 0 indicate weak relationships, while values close to –1 or indicate strong relationships.
4. The regression line for a relationship between two variables is the line that best represents the data. It can be used to predict values of the response variable for a given value of the explanatory variable.

### Formulas

If a line has slope  and passes through a point , then the point-slope form of the equation of the line is:
Suppose  and  are explanatory and response datasets that have a linear relationship. If their means are  and  respectively, their standard deviations are  and  respectively, and their correlation coefficient is , then the equation of the regression line is:

### Videos

1. Making Scatter Plots in Google Sheets

### Projects

1. Browse through some news websites to find five stories that report on data and include data visualizations. Can you tell from the report how the data were collected? Was randomization used? Are the visualizations appropriate for the data? Are the visualizations presented in a way that might bias the reader?
2. We discussed three measures of centrality in this chapter: the mode, the median, and the mean. In a broader context, the mean as we discussed it is more properly called the arithmetic mean, to distinguish it from other types of means. Examples of these include the geometric mean, harmonic mean, truncated mean, and weighted mean. How are these computed? How do they compare to the arithmetic mean? In what situations would each of these be preferred to the arithmetic mean?
3. Simpson’s Paradox is a statistical phenomenon that can sometimes appear when we observe a relationship within several subgroups of a population, but when the data for all thegroups are analyzed all together, the opposite relationship appears. Find some examples of Simpson’s Paradox in real-world situations, and write a paragraph or two that would explain the concept to someone who had never studied statistics before.

### Chapter Review

### Gathering and Organizing Data

### Visualizing Data

### Mean, Median and Mode

### Range and Standard Deviation

### Percentiles

### The Normal Distribution

### Applications of the Normal Distribution

### Scatter Plots, Correlation, and Regression Lines

### Chapter Test


# Metric Measurement

## Introduction

You are planning a road trip from your home state to the sunny beaches of Mexico and need to prepare a budget. While in the United States, gasoline is sold in gallons and distances are measured in miles, but in almost any other country you will find that gasoline is sold in liters and distance is measured in kilometers.
Whether you’re traveling, baking, watching an international sporting event, working with machine tools, or using scientific equipment, it's important to understand the metric system, or the International System of Units (SI). The metric system is a decimal measuring system that uses meters, liters, and grams to quantify length, capacity, and mass. It is used in all but three countries in the world, including the United States.


# Metric Measurement

## The Metric System

### Learning Objectives

After completing this section, you should be able to:
1. Identify units of measurement in the metric system and their uses.
2. Order the six common prefixes of the metric system.
3. Convert between like unit values.
Even if you don’t travel outside of the United States, many specialty grocery stores utilize the metric system. For example, if you want to make authentic tamales you might visit the nearest Hispanic grocery store. While shopping, you discover that there are two brands of masa for the same price, but one bag is marked 1,200 g and the other 1 kg. Which one is the better deal? Understanding the metric system allows you to understand that 1,200 grams is equivalent to 1.2 kg, so the 1,200 g bag is the better deal.

### Units of Measurement in the Metric System

Units of measurement provide common standards so that regardless of where or when an object or substance is measured, the results are consistent. When measuring distance, the units of measure might be feet, meters, or miles. Weight might be expressed in terms of pounds or grams. Volume or capacity might be measured in gallons, or liters. Understanding how metric units of measure relate to each other is essential to understanding the metric system:
1. The metric unit for distance is the meter (m). A person’s height might be written as 1.8 meters (1.8 m). A meter slightly longer than a yard (3 feet), while a centimeter is slightly less than half an inch.
2. The metric unit for area is the square meter (m2). The area of a professional soccer field is 7,140 square meters (7,140 m2).
3. The metric base unit for volume is the cubic meter (m3). However, the liter (L), which is a metric unit of capacity, is used to describe the volume of liquids. Soda is often sold in 2-liter (2 L) bottles.
4. The gram (g) is a metric unit of mass but is commonly used to express weight. The weight of a paper clip is approximately 1 gram (1 g).
5. The metric unit for temperature is degrees Celsius (°C). The temperature on a warm summer day might be 24 °C.
While the U.S. Customary System of Measurement uses ounces and pounds to distinguish between weight units of different sizes, in the metric system a base unit is combined with a prefix, such as kilo– in kilogram, to identify the relationship between smaller or larger units.
While there are other base units in the metric system, our discussions in this chapter will be limited to units used to express length, area, volume, weight, and temperature.

### Metric Prefixes

Unlike the U.S. Customary System of Measurement in which 12 inches is equal to 1 foot and 3 feet are equal to 1 yard, the metric system is structured so that the units within the system get larger or smaller by a power of 10. For example, a centimeter is , or 100 times smaller than a meter, while the kilometer is 103, or 1,000 times larger than a meter.
The metric system combines base units and unit prefixes reasonable to the size of a measured object or substance. The most used prefixes are shown in . An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died From Drinking Chocolate Milk.

### Converting Metric Units of Measure

Imagine you order a textbook online and the shipping detail indicates the weight of the book is 1 kg. By attaching the letter “k” to the base unit of gram (g), the unit used to express the measure is or 1,000 times greater than a gram. One kilogram is equivalent to 1,000 grams.
The tip of a highlighter measures approximately 1 cm. The letter “c” attached to the base unit of meter (m) means the unit used to express the measure is  of a meter. One meter is equivalent to 100 centimeters.
A conversion factor is used to convert from smaller metric units to bigger metric units and vice versa. It is a number that when used with multiplication or division converts from one metric unit to another, both having the same base unit. In the metric system, these conversion factors are directly related to the powers of 10. The most common used conversion factors are shown in .

### Check Your Understanding

### Key Terms

1. metric system
2. meter (m)
3. gram (g)
4. liter (L)
5. square meter (m2)
6. cubic meter (m3)
7. degrees Celsius (°C)
8. conversion factor

### Key Concepts

1. The metric system is decimal system of weights and measures based on base units of meter, liter, and gram. The system was first proposed in 1670 and has since been adopted as the International System of Units and used in nearly every country in the world.
2. Each successive unit on the metric scale is 10 times larger than the previous one. To convert between units with the same base unit, you must either multiply or divide by a power of 10.
3. The most used prefixes are listed in . An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died from Drinking Chocolate Milk.

### Videos

1. U.S. Office of Education: Metric Education
2. Neil deGrasse Tyson Explains the Metric System

### Formulas

You can convert between unit sizes with the same base unit using the conversion factors shown in .


# Metric Measurement

## Measuring Area

### Learning Objectives

After completing this section, you should be able to:
1. Identify reasonable values for area applications.
2. Convert units of measures of area.
3. Solve application problems involving area.
Area is the size of a surface. It could be a piece of land, a rug, a wall, or any other two-dimensional surface with attributes that can be measured in the metric unit for distance-meters. Determining the area of a surface is important to many everyday activities. For example, when purchasing paint, you’ll need to know how many square units of surface area need to be painted to determine how much paint to buy.
Square units indicate that two measures in the same units have been multiplied together. For example, to find the area of a rectangle, multiply the length units and the width units to determine the area in square units.
Note that to accurately calculate area, each of the measures being multiplied must be of the same units. For example, to find an area in square centimeters, both length measures (length and width) must be in centimeters.

### Reasonable Values for Area

Because area is determined by multiplying two lengths, the magnitude of difference between different square units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a square meter  is  times greater in area than a square centimeter . The relationships between benchmark metric area units are shown in the following table.
An essential understanding of metric area is to identify reasonable values for area. When testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you determine whether the given area is reasonable for the situation.

### Converting Units of Measures for Area

Just like converting units of measure for distance, you can convert units of measure for area. However, the conversion factor, or the number used to multiply or divide to convert from one area unit to another, is not the same as the conversion factor for metric distance units. Recall that the conversion factor for area is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are shown in .

### Solving Application Problems Involving Area

While it may seem that solving area problems is as simple as multiplying two numbers, often determining area requires more complex calculations. For example, when measuring the area of surfaces, you may need to account for portions of the surface that are not relevant to your calculation.
When calculating area, you must ensure that both distance measurements are expressed in terms of the same distance units. Sometimes you must convert one measurement before using the area formula.
When calculating area, you may need to use multiple steps, such as converting units and subtracting areas that are not relevant.

### Check Your Understanding

### Key Terms

1. area
2. square units

### Key Concepts

1. Area describes the size of a two-dimensional surface. It is the amount of space contained within the lines of a two-dimensional space.
2. Area is measured in square meters units; in the metric system the base unit for area is square meters ().

### Videos

1. Converting Metric Units of Area
2. Why the Metric System Matters

### Formulas

To determine the area of rectangular-shaped objects:

You can convert between metric area units using the conversion factors shown in .


# Metric Measurement

## Measuring Volume

### Learning Objectives

After completing this section, you should be able to:
1. Identify reasonable values for volume applications.
2. Convert between like units of measures of volume.
3. Convert between different unit values.
4. Solve application problems involving volume.
Volume is a measure of the space contained within or occupied by three-dimensional objects. It could be a box, a pool, a storage unit, or any other three-dimensional object with attributes that can be measured in the metric unit for distance–meters. For example, when purchasing an SUV, you may want to compare how many cubic units of cargo the SUV can hold.
Cubic units indicate that three measures in the same units have been multiplied together. For example, to find the volume of a rectangular prism, you would multiply the length units by the width units and the height units to determine the volume in square units:
Note that to accurately calculate volume, each of the measures being multiplied must be of the same units. For example, to find a volume in cubic centimeters, each of the measures must be in centimeters.

### Reasonable Values for Volume

Because volume is determined by multiplying three lengths, the magnitude of difference between different cubic units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a cubic meter, m3, is  times greater in area than a cubic centimeter, cm3. This relationship between benchmark metric volume units is shown in the following table.
To have an essential understanding of metric volume, you must be able to identify reasonable values for volume. When testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you determine whether the given volume is reasonable for the situation.

### Converting Like Units of Measures for Volume

Just like converting units of measure for distance, you can convert units of measure for volume. However, the conversion factor, the number used to multiply or divide to convert from one volume unit to another, is different from the conversion factor for metric distance units. Recall that the conversion factor for volume is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are illustrated in .

### Understanding Other Metric Units of Volume

When was the last time you purchased a bottle of soda? Was the volume of the bottle expressed in cubic centimeters or liters? The liter (L) is a metric unit of capacity often used to express the volume of liquids. A liter is equal in volume to 1 cubic decimeter. A milliliter is equal in volume to 1 cubic centimeter. So, when a doctor orders 10 cc (cubic centimeters) of saline to be administered to a patient, they are referring to 10 mL of saline.
The most frequently used factors for converting from cubic meters to liters are listed in .

### Solving Application Problems Involving Volume

Knowing the volume of an object lets you know just how much that object can hold. When making a bowl of punch you might want to know the total amount of liquid a punch bowl can hold. Knowing how many liters of gasoline a car’s tank can hold helps determine how many miles a car can drive on a full tank. Regardless of the application, understanding volume is essential to many every day and professional tasks.

### Check Your Understanding

### Key Terms

1. volume
2. cubic units

### Key Concepts

1. Volume is a measure of the space contained within or occupied by three-dimensional objects.
2. Volume is measured in cubic units; the base unit for volume in the metric system is cubic meters (m3)
3. The liter (L) is a metric unit of capacity but is often used to express the volume of liquids. One liter is equivalent to one cubic decimeter, which is the volume of a cube measuring .

### Formulas

To determine the volume of a rectangular prism:
You can convert between metric volume units and metric capacity units using the relationships shown in .

### Videos

1. How to Convert Cubic Centimeters to Cubic Meters
2. Converting Metric Units of Volume


# Metric Measurement

## Measuring Weight

### Learning Objectives

After completing this section, you should be able to:
1. Identify reasonable values for weight applications.
2. Convert units of measures of weight.
3. Solve application problems involving weight.
In the metric system, weight is expressed in terms of grams or kilograms, with a kilogram being equal to 1,000 grams. A paper clip weighs about 1 gram. A liter of water weighs about 1 kilogram. In fact, in the same way that 1 liter is equal in volume to 1 cubic decimeter, the kilogram was originally defined as the mass of 1 liter of water. In some cases, particularly in scientific or medical settings where small amounts of materials are used, the milligram is used to express weight. At the other end of the scale is the metric ton (mt), which is equivalent to 1,000 kilograms. The average car weighs about 2 metric tons.
Any discussion about metric weight must also include a conversation about mass. Scientifically, mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The amount of mass of an object remains constant no matter where the object is. Identical objects located on Earth and on the moon will have the same mass, but the weight of the objects will differ because the moon has a weaker gravitational force than Earth. So, objects with the same mass will weigh less on the moon than on Earth.
Since there is no easy way to measure mass, and since gravity is just about the same no matter where on Earth you go, people in countries that use the metric system often use the words mass and weight interchangeably. While scientifically the kilogram is only a unit of mass, in everyday life it is often used as a unit of weight as well.

### Reasonable Values for Weight

To have an essential understanding of metric weight, you must be able to identify reasonable values for weight. When testing for reasonableness, you should assess both the unit and the unit value. Only by examining both can you determine whether the given weight is reasonable for the situation.

### Converting Like Units of Measures for Weight

Just like converting units of measure for distance, you can convert units of measure for weight. The most frequently used conversion factors for metric weight are illustrated in .

### Solving Application Problems Involving Weight

From children’s safety to properly cooking a pie, knowing how to solve problems involving weight is vital to everyday life. Let’s review some ways that knowing how to work with metric weight can facilitate important decisions and delicious eating.

### Check Your Understanding

### Key Terms

1. mass

### Key Concepts

1. Mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The mass of an object never changes; the weight of an object changes depending on the force of gravity. An object with the same mass would weigh less on the moon than on Earth because the moon’s gravity is less than that of Earth.
2. In the metric system, weight and mass are often used interchangeably and are expressed in terms in grams or kilograms.

### Videos

1. Metric System: Units of Weight
2. Using Drones to Weigh Whales?
3. Metric Units of Mass: Convert mg, g, and kg

### Formulas

You can convert between metric weight units using the conversion factors shown in .


# Metric Measurement

## Measuring Temperature

### Learning Objectives

After completing this section, you should be able to:
1. Convert between Fahrenheit and Celsius.
2. Identify reasonable values for temperature applications.
3. Solve application problems involving temperature.
When you touch something and it feels warm or cold, what is that really telling you about that substance? Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature.
In the metric system, temperature is measured using the Celsius (°C) scale. Because temperature is a condition of the physical properties of a substance, the Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor).

### Converting Between Fahrenheit and Celsius Temperatures

Understanding how to convert between Fahrenheit and Celsius temperatures is an essential skill in understanding metric temperatures. You likely know that below 32 °F means freezing temperatures and perhaps that the same holds true for 0 °C. While it may be difficult to recall that water boils at 212 °F, knowing that it boils at 100 °C is a fairly easy thing to remember.
But what about all the temperatures in between? What is the temperature in degrees Celsisus on a scorching summer day? What about a cool autumn afternoon? If a recipe instructs you to preheat the oven to 350 °F, what Celsius temperature do you set the oven at?
 lists common temperatures on both scales, because we don’t use Celsius temperatures daily it’s difficult to remember them. Fortunately, we don’t have to. Instead, we can convert temperatures from Fahrenheit to Celsius and from Celsius to Fahrenheit using a simple algebraic expression.

### Reasonable Values for Temperature

While knowing the exact temperature is important in most cases, sometimes an approximation will do. When trying to assess the reasonableness of values for temperature, there is a quicker way to convert temperatures for an approximation using mental math. These simpler formulas are listed in .

### Solving Application Problems Involving Temperature

Whether traveling abroad or working in a clinical laboratory, knowing how to solve problems involving temperature is an important skill to have. Many food labels express sizes in both ounces and grams. Most rulers and tape measures are two-sided with one side marked in inches and feet and the other in centimeters and meters. And while many thermometers have both Fahrenheit and Celsius scales, it really isn’t practical to pull out a thermometer when cooking a recipe that uses metric units. Let’s review at few instances where knowing how to fluently use the Celsius scale helps solve problems.

### Check Your Understanding

### Key Terms

1. temperature

### Key Concepts

1. Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature.
2. In the metric system, temperature is measured using the Celsius (°C) scale.
3. The Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor).

### Videos

1. Misconceptions About Temperature
2. Temperature Conversion Trick
3. Learn the Metric System in 5 Minutes

### Formulas

 To convert temperature from Fahrenheit to Celsius:

 To convert temperature from Celsius to Fahrenheit:

 To estimate temperature from Fahrenheit to Celsius:

 To estimate temperature from Celsius to Fahrenheit:

### Projects

### Cooking

1. Take a favorite recipe that uses customary measures and convert the measures and cooking temperature to the metric system.
2. Find a recipe that uses metric measures and convert the measures and cooking temperature to the U.S. Customary System of Measurement, using cups, tablespoons, or teaspoons as required.
3. What did you observe? Was it easier to convert from one system to another? Which system allows for more precise measurements? What kitchen tools would you need in your kitchen if you used the metric system?

### Shopping

1. Compare the average gas price in California to the average gas price in Puerto Rico.
2. What conversions did you need to make to do the comparison?
3. Do you think that the price of the gasoline is affected by the units in which it is sold?

### Sports

1. What system of measurement is used for track and field events? Why do you think this system is used?
2. What system of measurement is used for football? Why do you think this system is used?
3. Research various sports records. Which units of measurement is used? What do you think influenced the unit of measure used?

### Chapter Review

### The Metric System

### Measuring Area

### Measuring Volume

### Measuring Weight

### Measuring Temperature

### Chapter Test


# Geometry

## Introduction

The painting The School of Athens presents great figures in history such as Plato, Aristotle, Socrates, Euclid, Archimedes, and Pythagoras. Other scientists are also represented in the painting.
To the ancient Greeks, the study of mathematics meant the study of geometry above all other subjects. The Greeks looked for the beauty in geometry and did not allow their geometrical constructions to be “polluted” by the use of anything as practical as a ruler. They permitted the use of only two tools—a compass for drawing circles and arcs, and an unmarked straightedge to draw line segments. They would mark off units as needed. However, they never could be sure of what the units meant. For instance, how long is an inch? These mathematicians defined many concepts.
The Greeks absorbed much from the Egyptians and the Babylonians (around 3000 BCE), including knowledge about congruence and similarity, area and volume, angles and triangles, and made it their task to introduce proofs for everything they learned. All of this historical wisdom culminated with Euclid in 300 BCE.
Euclid (325–265 BC) is known as the father of geometry, and his most famous work is the 13-volume collection known as The Elements, which are said to be “the most studied books apart from the Bible.” Euclid brought together everything offered by the Babylonians, the Egyptians, and the more refined contributions by the Greeks, and set out, successfully, to organize and prove these concepts as he methodically developed formal theorems.
This chapter begins with a discussion of the most basic geometric tools: the point, the line, and the plane. All other topics flow from there. Throughout the eight sections, we will talk about how to determine angle measurement and learn how to recognize properties of special angles, such as right angles and supplementary angles. We will look at the relationship of angles formed by a transversal, a line running through a set of parallel lines. We will explore the concepts of area and perimeter, surface area and volume, and transformational geometry as used in the patterns and rigid motions of tessellations. Finally, we will introduce right-angle trigonometry and explore the Pythagorean Theorem.


# Geometry

## Points, Lines, and Planes

### Learning Objectives

After completing this section, you should be able to:
1. Identify and describe points, lines, and planes.
2. Express points and lines using proper notation.
3. Determine union and intersection of sets.
In this section, we will begin our exploration of geometry by looking at the basic definitions as defined by Euclid. These definitions form the foundation of the geometric theories that are applied in everyday life.
In The Elements, Euclid summarized the geometric principles discovered earlier and created an axiomatic system, a system composed of postulates. A postulate is another term for axiom, which is a statement that is accepted as truth without the need for proof or verification. There were no formal geometric definitions before Euclid, and when terms could not be defined, they could be described. In order to write his postulates, Euclid had to describe the terms he needed and he called the descriptions “definitions.” Ultimately, we will work with theorems, which are statements that have been proved and can be proved.

### Points and Lines

The first definition Euclid wrote was that of a point. He defined a point as “that which has no part.” It was later expanded to “an indivisible location which has no width, length, or breadth.” Here are the first two of the five postulates, as they are applicable to this first topic:
1. Postulate 1: A straight line segment can be drawn joining any two points.
2. Postulate 2: Any straight line segment can be extended indefinitely in a straight line.
Before we go further, we will define some of the symbols used in geometry in :
From , we see the variations in lines, such as line segments, rays, or half-lines. What is consistent is that two collinear points (points that lie on the same line) are required to form a line. Notice that a line segment is defined by its two endpoints showing that there is a definite beginning and end to a line segment. A ray is defined by two points on the line; the first point is where the ray begins, and the second point gives the line direction. A half-line is defined by two points, one where the line starts and the other to give direction, but an open circle at the starting point indicates that the starting point is not part of the half-line. A regular line is defined by any two points on the line and extends infinitely in both directions. Regular lines are typically drawn with arrows on each end.
There are numerous applications of line segments in daily life. For example, airlines working out routes between cities, where each city’s airport is a point, and the points are connected by line segments. Another example is a city map. Think about the intersection of roads, such that the center of each intersection is a point, and the points are connected by line segments representing the roads. See .

### Parallel Lines

Parallel lines are lines that lie in the same plane and move in the same direction, but never intersect. To indicate that the line  and the line  are parallel we often use the symbol  The distance  between parallel lines remains constant as the lines extend infinitely in both directions. See .

### Perpendicular Lines

Two lines that intersect at a  angle are perpendicular lines and are symbolized by . If  and  are perpendicular, we write  When two lines form a right angle, a  angle, we symbolize it with a little square  See .

### Defining Union and Intersection of Sets

Union and intersection of sets is a topic from set theory that is often associated with points and lines. So, it seems appropriate to introduce a mini-version of set theory here. First, a set is a collection of objects joined by some common criteria. We usually name sets with capital letters. For example, the set of odd integers between 0 and 10 looks like this:  When it involves sets of lines, line segments, or points, we are usually referring to the union or intersection of set.
The union of two or more sets contains all the elements in either one of the sets or elements in all the sets referenced, and is written by placing this symbol  in between each of the sets. For example, let set  and let set  Then, the union of sets A and B is
The intersection of two or more sets contains only the elements that are common to each set, and we place this symbol  in between each of the sets referenced. For example, let’s say that set  and let set  Then, the intersection of sets  and  is

### Planes

A plane, as defined by Euclid, is a “surface which lies evenly with the straight lines on itself.” A plane is a two-dimensional surface with infinite length and width, and no thickness. We also identify a plane by three noncollinear points, or points that do not lie on the same line. Think of a piece of paper, but one that has infinite length, infinite width, and no thickness. However, not all planes must extend infinitely. Sometimes a plane has a limited area.
We usually label planes with a single capital letter, such as Plane , as shown in , or by all points that determine the edges of a plane. In the following figure, Plane  contains points  and , which are on the same line, and point , which is not on that line. By definition,  is a plane. We can move laterally in any direction on a plane.
One way to think of a plane is the Cartesian coordinate system with the -axis marked off in horizontal units, and -axis marked off in vertical units. In the Cartesian plane, we can identify the different types of lines as they are positioned in the system, as well as their locations. See .
This plane contains points , , and . Points  and  are colinear and form a line segment. Point  is not on that line segment. Therefore, this represents a plane.
To give the location of a point on the Cartesian plane, remember that the first number in the ordered pair is the horizontal move and the second number is the vertical move. Point  is located at  point  is located at  and point  is located at  We can also identify the line segment as
Two other concepts to note: Parallel planes do not intersect and the intersection of two planes is a straight line. The equation of that line of intersection is left to a study of three-dimensional space. See .
To summarize, some of the properties of planes include:
1. Three points including at least one noncollinear point determine a plane.
2. A line and a point not on the line determine a plane.
3. The intersection of two distinct planes is a straight line.

### Check Your Understanding

### Key Terms

1. line segment
2. plane
3. union
4. intersection
5. parallel
6. perpendicular

### Key Concepts

1. Modern-day geometry began in approximately 300 BCE with Euclid’s Elements, where he defined the principles associated with the line, the point, and the plane.
2. Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.
3. The union of two sets,  and , contains all points that are in both sets and is symbolized as
4. The intersection of two sets  and  includes only the points common to both sets and is symbolized as


# Geometry

## Angles

### Learning Objectives

After completing this section, you should be able to:
1. Identify and express angles using proper notation.
2. Classify angles by their measurement.
3. Solve application problems involving angles.
4. Compute angles formed by transversals to parallel lines.
5. Solve application problems involving angles formed by parallel lines.
Unusual perspectives on architecture can reveal some extremely creative images. For example, aerial views of cities reveal some exciting and unexpected angles. Add reflections on glass or steel, lighting, and impressive textures, and the structure is a work of art. Understanding angles is critical to many fields, including engineering, architecture, landscaping, space planning, and so on. This is the topic of this section.
We begin our study of angles with a description of how angles are formed and how they are classified. An angle is the joining of two rays, which sweep out as the sides of the angle, with a common endpoint. The common endpoint is called the vertex. We will often need to refer to more than one vertex, so you will want to know the plural of vertex, which is vertices.
In , let the ray  stay put. Rotate the second ray  in a counterclockwise direction to the size of the angle you want. The angle is formed by the amount of rotation of the second ray. When the ray  continues to rotate in a counterclockwise direction back to its original position coinciding with ray  the ray will have swept out  We call the rays the “sides” of the angle.

### Classifying Angles

Angles are measured in radians or degrees. For example, an angle that measures  radians, or 3.14159 radians, is equal to the angle measuring  An angle measuring  radians, or 1.570796 radians, measures  To translate degrees to radians, we multiply the angle measure in degrees by  For example, to write  in radians, we have

To translate radians to degrees, we multiply by  For example, to write  radians in degrees, we have

Another example of translating radians to degrees and degrees to radians is  To write in degrees, we have  To write  in radians, we have . However, we will use degrees throughout this chapter.
Several angles are referred to so often that they have been given special names. A straight angle measures ; a right angle measures  an acute angle is any angle whose measure is less than  and an obtuse angle is any angle whose measure is between  and  See .
An easy way to measure angles is with a protractor (). A protractor is a very handy little tool, usually made of transparent plastic, like the one shown here.
With a protractor, you line up the straight bottom with the horizontal straight line of the angle. Be sure to have the center hole lined up with the vertex of the angle. Then, look for the mark on the protractor where the second ray lines up. As you can see from the image, the degrees are marked off. Where the second ray lines up is the measurement of the angle.

### Notation

Naming angles can be done in couple of ways. We can name the angle by three points, one point on each of the sides and the vertex point in the middle, or we can name it by the vertex point alone. Also, we can use the symbols  or  before the points. When we are referring to the measure of the angle, we use the symbol . See .
We can name this angle , or , or

### Adjacent Angles

Two angles with the same starting point or vertex and one common side are called adjacent angles. In , angle  is adjacent to . Notice that the way we designate an angle is with a point on each of its two sides and the vertex in the middle.

### Supplementary Angles

Two angles are supplementary if the sum of their measures equals  In , we are given that  so what is  These are supplementary angles. Therefore, because , and as  we have

### Complementary Angles

Two angles are complementary if the sum of their measures equals  In , we have  and  What is the  These are complementary angles. Therefore, because  the

### Vertical Angles

When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example,  shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well.

### Transversals

When two parallel lines are crossed by a straight line or transversal, eight angles are formed, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. See . Angles 1, 2, 7, and 8 are called exterior angles, and angles 3, 4, 5, and 6 are called interior angles.

### Alternate Interior Angles

Alternate interior angles are the interior angles on opposite sides of the transversal. These two angles have the same measure. For example,  and  are alternate interior angles and have equal measure;  and  are alternate interior angles and have equal measure as well. See .

### Alternate Exterior Angles

Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example, in ,  and  are alternate exterior angles and have equal measures;  and  are alternate exterior angles and have equal measures as well.

### Corresponding Angles

Corresponding angles refer to one exterior angle and one interior angle on the same side as the transversal, which have equal measures. In ,  and  are corresponding angles and have equal measures;  and  are corresponding angles and have equal measures;  and  are corresponding angles and have equal measures;  and  are corresponding angles and have equal measures as well.

### Check Your Understanding

### Key Terms

1. vertex
2. right angle
3. acute angle
4. obtuse angle
5. straight angle
6. complementary
7. supplementary

### Key Concepts

1. Angles are classified as acute if they measure less than  obtuse if they measure greater than  and less than  right if they measure exactly  and straight if they measure exactly
2. If the sum of angles equals , they are complimentary angles. If the sum of angles equals , they are supplementary.
3. A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior angles, vertical angles, and corresponding angles

### Formula

To translate an angle measured in degrees to radians, multiply by
To translate an angle measured in radians to degrees, multiply by


# Geometry

## Triangles

### Learning Objectives

After completing this section, you should be able to:
1. Identify triangles by their sides.
2. Identify triangles by their angles.
3. Determine if triangles are congruent.
4. Determine if triangles are similar.
5. Find the missing side of similar triangles.
How were the ancient Greeks able to calculate the radius of Earth? How did soldiers gauge their target? How was it possible centuries ago to estimate the height of a sail at sea? Triangles have always played a significant role in how we find heights of objects too high to measure or distances between objects too far away to calculate. In particular, the concept of similar triangles has countless applications in the real world, and we shall explore some of those applications in this section.
Technology has given us instruments that allow us to find measurements of distant objects with little effort. However, it is all based on the properties of triangles discovered centuries ago. In this section, we will explore the various types of triangles and their special properties, as well as how to measure interior and exterior angles. We will also explore congruence theorems and similarity.

### Identifying Triangles

Joining any three noncollinear points with line segments produces a triangle. For example, given points , , and , connected by the line segments  and  we have a triangle, as shown in .
Triangles are classified by their angles and their sides. All angles in an acute triangle measure  One of the angles in a right triangle measures  symbolized by □. One angle in an obtuse triangle measures between  and  Sides that have equal length are indicated by the same hash marks.  illustrates the shapes of the basic triangles, their names, and their properties.
A few other facts to remember as we move forward:
1. The points where the line segments meet are called the vertices (plural for vertex).
2. We often refer to sides of a triangle by the angle they are opposite. In other words, side  is opposite angle , side  is opposite angle , and side  is opposite angle .
We want to add a special note about right triangles here, as they are referred to more than any other triangle. The side opposite the right angle is its longest side and is called the hypotenuse, and the sides adjacent to the right angle are called the legs.
One of the most important properties of triangles is that the sum of the interior angles equals  Euclid discovered and proved this property using parallel lines. The completed sketch is shown in .
This is how the proof goes:
Step 1: Start with a straight line  and a point  not on the line.
Step 2: Draw a line through point  parallel to the line .
Step 3: Construct two transversals (a line crossing the parallel lines), one angled to the right and one angled to the left, to intersect the parallel lines.
Step 4: Because of the property that alternate interior angles inside parallel lines are equal, we have that

Step 5: Notice that  by the straight angle property.
Step 6: Therefore, by substitution,  and  we have that

Therefore, the sum of the interior angles of a

### Congruence

If two triangles have equal angles and their sides lengths are equal, the triangles are congruent. In other words, if you can pick up one triangle and place it on top of the other triangle and they coincide, even if you have to rotate one, they are congruent.

### The Congruence Theorems

The following theorems are tools you can use to prove that two triangles are congruent. We use the symbol  to define congruence. For example, .
Side-Side-Side (SSS). If three sides of one triangle are equal to the corresponding sides of the second triangle, then the triangles are congruent. See .
We have that ,  and  then
Side-Angle-Side (SAS). If two sides of a triangle and the angle between them are equal to the corresponding two sides and included angle of the second triangle, then the triangles are congruent. See . We see that  and , , then .
Angle-Side-Angle (ASA). If two angles and the side between them in one triangle are congruent to the two corresponding angles and the side between them in a second triangle, then the two triangles are congruent. See . Notice that , and , , then
Angle-Angle-Side (AAS). If two angles and a nonincluded side of one triangle are congruent to two angles and the nonincluded corresponding side of a second triangle, then the triangles are congruent.
See . We see that , , and , then .

### Similarity

If two triangles have the same angle measurements and are the same shape but differ in size, the two triangles are similar. The lengths of the sides of one triangle will be proportional to the corresponding sides of the second triangle. Note that a single fraction  is called a ratio, but two fractions equal to each other is called a proportion, such as
This rule of similarity applies to all shapes as well as triangles. Another way to view similarity is by applying a scaling factor, which is the ratio of corresponding measurements between an object or representation of the object, to an image that produces the second, similar image.
For example, why are the two images in  are similar? These two images have the same proportions between elements. Therefore, they are similar.

### Check Your Understanding

### Key Terms

1. acute
2. obtuse
3. isosceles
4. equilateral
5. hypotenuse
6. congruence
7. similarity
8. scaling factor

### Key Concepts

1. The sum of the interior angles of a triangle equals
2. Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal.
3. The congruence theorems include the following: SAS, two sides and the included angle of one triangle are congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same in a second triangle; AAS, two angles and the non-included side of one triangle are congruent to the same in a second triangle.
4. Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are equal, regardless of size.