Introduction to Statistics for Data Science
By: Super Albert
Tags: Programming, Statistics, Data Science, Education, Machine Learning

Normal Distribution

The Normal Distribution is one of the most important concepts in Statistics, since the majority of tests require normally distributed data. Normal distribution describes how equally your data points are distributed along a given scale and where the majority of data accumulates towards the center. This distribution is also known as Gaussian curve.

A Normal Distribution exists if your data is symmetrical, bell-shaped, centered and unimodal.

Measure of Central Tendency

The measure of central tendency refers to the idea there’s one number that best summarizes the entire set. The most popular are mean, median and mode.

Mean

This is considered the most reliable measure of the measure of central tendency for making assumptions about a population from a single sample. The μ symbol is used to described the population value whereas the x̅ to describe the sample mean.

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We can find the mean by summing all the components and then dividing the sum by the number of components. As already said, it’s the most common measure of central tendency, but it has the downside of being easily affected by outliers. Sometimes, due to outliers, the mean might not be enough to make conclusions.

Wikipedia

Median

The median is the midpoint or the “middle” value in your orderly ascending dataset. It is also known as the 50th percentile. In order to avoid the error provoked in the mean by outliers, it is usually a good idea to also calculate the median.

Finding the median in sets of data with an odd and even number of values — Wikipedia

But what about the representation that most values give?

Mode

The mode shows us the value that occurs most often. It can be used for numerical as well as categorical variables. If there are not a single value does not appear more than once you say there is no mode.

W hich is the best measure?

The measures should be used together instead of independently. There is no best and using only one is not advisable. Moreover, in a normal distribution, these measures all fall at the same midline point. This means that the mean, mode and median are all equal!

Measures of variability

The measure of variability refers to the idea of measuring the dispersion in our data according to the mean value. The most known measures of variability are the range, interquartile range (IQR), variance and standard deviation.

Range

The range is the most obvious measure of dispersion and describes the difference between the largest and the smallest points in your data.

Range is 99–12 = 87

Interquartile range (IQR)

The IQR is a measure of variability between the upper (75th) and lower (25th) quartiles. The data is sorted into ascending order and divided into four quarters.

While the range measures the range of values in which our dataset is distributed, the interquartile range measures the interval of values where the majority of values lies in.

Variance

The variance as well as the standard deviation are more complex forms of measuring how much the data disperse from the mean value of the dataset.

The variance is found by computing the difference between every data point and the mean, squaring that value and summing for all available data points. In the end, the variance is calculated by dividing the sum by the total number of available points.

Squaring the difference has two main purposes : Dispersion is non-negative, by powerering the subtraction by 2 we ensure we do not have negative values and thus there is not the chance of them canceling out.

Amplifies the effect of large differences

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The problem with Variance is that because of the squaring, it is not in the same unit of measurement as the original data. This is why the Standard Deviation is used more often because it is in the original unit. Squared dollars means nothing in statistics.

Standard Deviation

Usually standard deviation is much more meaningful than variance. It is the preferred measure of variability as it is directly interpretable.

Standard deviation is basically the square root of our variance.

Standard deviation is best used when data presents a unimodal shape. In a normal distribution, approximately 34% of data points fall one standard deviation away from the mean. Since a normal distribution is symmetrical, we have 68.2% of data points one standard deviation away from the mean. Around 95% of points fall between two standard deviation from the mean whereas 99.7% fall under three standard deviation.

With the Z-Score, you can check how many standard deviations below (or above) the mean, a specific data point is.