Introduction to Statistics — Part 2
By: Saiteja Kura
Tags: Mathematics, Data Science, Descriptive Statistics, Statistics, Data Visualization

Measures of Centrality:

A measure of centrality is a measure that attempts to describe the whole dataset with a single value that represents the centre of its distribution or a typical value in the distribution.

To demonstrate the measures of centrality let us take an example. Consider the scores of Virat Kohli in the last 10 ODI matches. The scores look like 21, 28, 29, 26, 22, 22, 24, 16, 38, 198.

The mean is calculated by adding all the scores and dividing by the total matches played. The mean of the above scores is 42.5. It actually means that Virat Kohli’s scores were around 42 in the last 10 matches he played.

The mode is the most frequent value appearing in the data. The mode of the above scores is 22. There may be a case where a data set can have 2 modes too.

The median is calculated by sorting all the scores and picking up the score which is actually located at the center of the data(there is a difference depending on the number of data points). The scores after sorting look like — 16, 21, 22, 22, 24, 26, 28, 29, 38, 198. The median of the above scores is 25. This means that 50% of the scores were less than 25 in the last 10 matches played by Virat Kohli.

Sensitivity to outliers:

Let us extend our first example in order to illustrate the sensitivity of various measures of centrality to outliers. Let us consider scores of 2 batsmen in the last 10 ODIs.

Virat Kohli: 1, 8, 9, 16, 24, 12, 12, 16, 38, 264.

Rohit Sharma: 3, 9, 14, 26, 52, 52, 54, 46, 58, 68

Q) Who performed better in the series?

If we consider the total runs scored, Virat Kohli leads the charts by 400 runs when compared to Rohit Sharma’s 382. If we calculate the mean, Virat Kohli’s mean which is 40 exceeds Rohit Sharma’s mean of 38.2. But is that conclusive enough? Let us calculate the median, Virat Kohli’s median of 14 is very much lesser than Rohit Sharma’s 49. It means that Virat Kohli scored less than 14 runs in the 50 % of matches he’s played which is abysmal whereas Rohit Sharma Scored less than 49 in 50% of matches he’s played which seems to be decent. Why are the mean and medium showing two completely opposite stats? This is where outliers come into the picture.

Outliers are values which are far off from the other points in the data.

In the above example we can clearly see there is an outlier(264) in Virat Kohli’s stats. Let us remove the outliers(We must remove n outliers from the end and from the beginning of the data) and calculate the new mean which is also known as trimmed mean. In the same way let us calculate the trimmed median.

The removal of outliers shows a clear picture.

Now as you can see the trimmed mean shows the actual interpretation which is needed and also you can notice that the mean varied a lot but the median did not, after removing the outliers.

The mean is very sensitive to the outliers whereas median and mode aren’t.

Measures of Centrality for different distributions:

For Symmetric distributions:

Mean=Median=Mode(for Unimodal distributions)

Mean=Median and there exist 2 Modes (for Bimodal distributions)

For Left Skewed Distributions:

Mean<Median<Mode(Mostly true!)

But if heavy tail exists in between then Median<Mean<Mode

For Right Skewed Distributions:

Mean>Median>Mode(Mostly true!)

But if heavy tail exists in between then Median>Mean>Mode