Aspects of Quantitative Data
By: Ruchit Sherathiya
Tags: Data, Quantitative Analysis, Data Science, Statistics

So now let’s discuss these aspects in detail.

1. Centre

The Centre of the distribution is also known as the middle of the distribution. The centre has three main measures. They are

A. Mean

Mean means the average value of the set of values. The formula for finding the mean is as follows:

MEAN = Sum of all values / The number of values

Example:

Mean example

Here the sum of all 5 values is 161 and hence as per the formula the mean of this list is 32.20

B. MEDIAN

Median means the middle value of the given list of elements. It divides the list of elements into two different parts.

If the number of elements in the list is odd then the median will be the middle element of the list.

Median odd example

The median of this odd list is 50 as 50 is the middle element in the list.

If the number of elements in the list is even then the median will be the sum of both the middle elements.

Median even example

The median of this even list is (50+22)/2 i.e. 36.

C. Mode

The mode of the data is calculated as the most frequent element in the list. Many times there are all unique values in the list. If there are multiple values with the same frequency then such data is known as bimodal. Similarly, if n values are having the same frequency then it is known as multimodal.

Example:

Example of Mode.

Here, the mode of the data is 50 because the value 50 is repeated for a couple of times.

2. Spread

Spread refers to the variation of the dataset. In layman language, one can say that how far are the points from one another. Spread has four measures. They are:

A. Range

The range is the simplest method of statistics. It can be calculated as the difference between the last and first elements of the data.

Example of range

Here the difference between the last and first element is 50–12 =38 which is the range of the given data.

B. Interquartile Range (IQR)

The IQR is the difference between the third quartile and the first quartile of the given data.

The formula for the IQR is: IQR = Q3-Q1

The median of the given data is the point from where the data is divided into two parts i.e the lower half and the upper half.

Example of IQR with even sample size

Here the data is divided into two parts with the help of median i.e. 71. Then Quartile 1 is taken as 64 which the median for the lower half and Quartile 3 is taken as 77 which is the median for the upper half. Hence, the difference between the third quartile and the first quartile is 77–64 = 13 which is the interquartile range for the given data.

Similarly

Example of IQR with an odd sample size

Here the data is divided into two parts with the help of median i.e. 72. The median is not included in any of the halves. Then Quartile 1 is taken as 64 which the median for the lower half and Quartile 3 is taken as 79 which is the median for the upper half. Hence, the difference between the third quartile and the first quartile is 79–64 = 15 which is the interquartile range for the given data.

C. Standard Deviation

Standard Deviation is the measurement of the average distance between each element and the mean. Whenever the standard deviation is low, we can say that data points are close to the mean of the dataset, while a high standard deviation states that the spread of the elements is more.

The formula for Standard Deviation.

For finding the standard deviation we need to follow the following steps

FIRST: Find the mean of the data

SECOND: For each point, find the square of its distance to the mean.

THIRD: Sum up all the points from the second step

FOURTH: Divide the sum by the total number of values/records and then take the square root of it.

Example:

SD example

Here the value of the mean is 32.20 and the sum of each point’s distance to mean is 2012.8 which when further divided by 5 gives 402.56 and then taking the square root of it we will get 20.06 as standard deviation.

NOTE: This formula is for the population data only. If you are calculating for the sample data then instead of n you need to take n-1.

D. Variance

Variance is a square of the average distance between each quantity and mean. i.e. it is square of standard deviation.

Variance = (S.D.) * (S.D.)

From the above example of standard deviation, we can say that the variance of the data is 20.06 * 20.06 = 402.56.

3. Shape

The shape attribute helps in identifying the actual distribution or the pattern of the data. Now, as we know that the quantitative data deals with numbers, so for analyzing the trend or knowing the distribution of quantitative data we generally use the histogram chart.

Histogram is generally in the four following pattern.

A. Uniform

Uniform histogram shows that there is consistency present in the data. The data which founds to be uniform may be bimodal. In this case, the frequency of each class remains almost similar to that of the other classes.

Uniform Histogram (Source:pqsystems.com)

B. Symmetric

Symmetric histogram shows that the values occurs at regular frequencies. It is often known as bell shaped curve because it forms the shape in the similar manner. It is also known as the normal distribution of the data. Mean and Standard deviation is preferred for the calculations.

Here, MEAN = MEDIAN = MODE.

Symmetric Histogram (Source:pqsystems.com)

C. Right Skewed

Generally, when the data is found to be sorted in descending order then it is known as Right Skewed Histogram. Here, the mean is pulled higher and the median and mode are closer to each other. Right skewed means that the tail is towards the right side of the data.

Here, MEAN > MEDIAN > MODE.

Right skewed Histogram (Source:pqsystems.com)

D. Left Skewed

Generally, when the data is found to be sorted in ascending order then it is known as left skewed data. Here, the mean is pulled to the lower side. Left skewed means that the tail of the data is towards the left side of the data.

Here, MEAN < MEDIAN < MODE.

4. Outliers

Outliers are defined as the data points that fall very far away from the rest of the values in our dataset. Generally, outliers does not help in analysis. They can either be extremely high or extremely low values. For detecting the outliers generally scatter plot or box plot is used.

The Steps to detect and deal with the outliers is as follows.

A. Firstly, one needs to notice that the outliers exists in the data. One needs to realize the impact that the outliers have on summary stats.

B. If there is typing error(typo) we need to either remove them or fix them.

C. One needs to understand the impact on the questions that he/she is trying to answer about the data.

D. Using Five Number Summary is better rather than using mean, standard deviation or variance when we have outliers.

E. Be careful while reporting and one should know how to ask the right question.