CHIRAG GOYAL — Published On January 5, 2022

This article was published as a part of the Data Science Blogathon.

## Introduction

Statistical Moments plays a crucial role while we specify our probability distribution to work with since, with the help of moments, we can describe the properties of statistical distribution. Therefore, they are helpful to describe the distribution.

In Statistical Estimation and Testing of Hypothesis, which all are based on the numerical values arrived for each distribution, we required the statistical moments.

So, In this article, we will be discussing primary statistical moments in a detailed manner.

1. What is the Moment in Statistics?

2. Understanding Four Statistical Moments

– The Expected Value or Mean
– Variance and Standard Deviation
– Skewness
– Kurtosis

3. Different Types of Moments

– Raw Moments
– Centered Moments
– Standardized Moments

## What is the Moment in Statistics?

In Statistics, Moments are popularly used to describe the characteristic of a distribution. Let’s say the random variable of our interest is X then, moments are defined as the X’s expected values.

For Example, E(X), E(X²), E(X³), E(X⁴),…, etc.

What is the use of Moments?

– These are very useful in statistics because they tell you much about your data.

– The four commonly used moments in statistics are- the mean, variance, skewness, and kurtosis.

To be ready to compare different data sets we will describe them using the primary four statistical moments.

Let’s discuss each of the moments in an exceedingly detailed manner:

## The First Moment

– The first central moment is the expected value, known also as an expectation, mathematical expectation, mean, or average.

– It measures the location of the central point.

Case-1: When all outcomes have the same probability of occurrence

It is defined as the sum of all the values the variable can take times the probability of that
value occurring. Intuitively, we can understand this as the arithmetic mean.

Case-2: When all outcomes don’t have the same probability of occurrence

This is the more general equation that includes the probability of each outcome and is defined as the summation of all the variables multiplied by the corresponding probability.

Conclusion

For equally probable events, the expected value is exactly the same as the Arithmetic Mean. This is one of the most popular measures of central tendency, which we also called Averages. But, there are some other common measures also like, Median and Mode.

– Median — The middle value

– Mode — The most likely value.

## The Second Moment

– The second central moment is Variance”.

– It measures the spread of values in the distribution OR how far from the normal.

– Variance represents how a set of data points are spread out around their mean value.

For Example, for a sample dataset, you can find the variance as mentioned below: Standard deviation

Standard deviation is just a square root of the variance and is commonly used since the unit
of random variable X and Standard deviation is the same, so interpretation is easier.

For Example, For a normal distribution: Image Source:

– Within 1 st Standard Deviation: 68.27% of the data points lie

– Within 2 nd Standard Deviation: 95.45% of the data points lie

– Within 3 rd Standard Deviation: 99.73% of the data points lie

Now. let’s understand the answer to the given questions:

“Why Variance is preferred over Mean Absolute Deviation(MAD)?”

Variance is preferred over MAD due to the following reasons:

Mathematical properties: The function of variance in both Continuous and differentiable.

For a Population, the Standard Deviation of a sample is a more consistent estimate: If we picked the repeated samples from a normally distributed population, then the standard deviations of samples are less spread out as compared to mean
absolute deviations.

## The Third Moment

– The third statistical moment is “Skewness”.

– It measures how asymmetric the distribution is about its mean.

We can differentiate three types of distribution with respect to its skewness:

Symmetrical distribution

If both tails of a distribution are symmetrical, and the skewness is equal to zero, then that distribution is symmetrical.

Positively Skewed

In these types of distributions, the right tail (with larger values) is longer. So, this also tells us about ‘outliers’ that have values higher than the mean. Sometimes, this is also referred to as:

– Right-skewed

– Right-tailed

– Skewed to the Right

Negatively skewed

In these types of distributions, the left tail (with small values) is longer. So, this also tells us about ‘outliers’ that have values lower than the mean. Sometimes, this is also referred to as:

– Left-skewed

– Left-tailed

– Skewed to the Left Image Source:

For Example, For a Normal Distribution, which is Symmetric, the value of Skewness equals 0 and that distribution is symmetrical.

In general, Skewness will impact the relationship of mean, median, and mode in the  described manner:

For a Symmetrical distribution: Mean = Median = Mode

For a positively skewed distribution: Mode < Median < Mean (large tail of high values)

For a negatively skewed distribution: Mean < Median < Mode (large tail of small values)

But the above generalization is not true for all possible distributions.

For Example, if one tail is long, but the other is heavy, this may not work. The best way to explore your data is to first compute all three estimators and then try to draw conclusions based on the results, rather than just focusing on the general rules.

Other Formula of Calculating Skewness:

Skewness = (Mean-Mode)/SD = 3*(Mean-Median)/SD

Since, (Mode = 3*Median-2*Mean)

Some transformations to make the distribution normal:

For Positively skewed (right): Square root, log, inverse, etc.

For Negatively skewed (left): Reflect and square[sqrt(constant-x)], reflect and log, reflect
and inverse, etc.

## The Fourth Moment

– The fourth statistical moment is “kurtosis.

– It measures the amount in the tails and outliers.

– It focuses on the tails of the distribution and explains whether the distribution is flat or rather with a high peak. This measure informs us whether our distribution is richer in extreme values than the normal distribution.

For Example, For a normal distribution, the value of Kurtosis equals 3

For Kurtosis not equal to 3, there are the following cases:

– Kurtosis<3 [Lighter tails]: Negative kurtosis indicates a broad flat distribution.

– Kurtosis>3 [Heavier tails]: Positive kurtosis indicates a thin pointed distribution.

In general, we can differentiate three types of distributions based on the Kurtosis:

Mesokurtic

These types of distributions are having the kurtosis of 3 or excess kurtosis of 0. This category includes the normal distribution and some specific binomial distributions.

Leptokurtic

These types of distributions are having a kurtosis greater than 3, or excess kurtosis greater than 0. This is the distribution with fatter tails and a more narrow peak.

Platykurtic

These types of distributions are having the kurtosis smaller than 3 or excess kurtosis less than 0(negative). This is a distribution with very thin tails compared to the normal distribution.

Now, let’s define what is Excess Kurtosis:

Excess Kurtosis = Kurtosis – 3

Understanding of Kurtosis related to Outliers:

– Kurtosis is defined as the average of the standardized data raised to the fourth power. Any standardized values less than |1| (i.e. data within one standard deviation of the mean) will contribute petty to kurtosis.

– The standardized values that will contribute immensely are the outliers.

– Therefore, the high value of Kurtosis alerts about the presence of outliers.

## Different Types of Moments

Let’s discuss three different types of moments:

Raw Moments

The raw moment or the n-th moment about zero of a probability density function f(x) is the expected value of X^n. It is also known as the Crude moment.

Centered Moments

A central moment is a moment of a probability distribution of a random variable defined about the mean of the random variable’s i.e, it is the expected value of a specified integer power of the deviation of the random variable from the mean.

Standardized Moments

A standardized moment of a probability distribution is a moment that is normally a higher degree central moment, but it is normalized typically by dividing the standard deviation which renders the moment scale-invariant.

This ends our discussion!

## Endnotes

I hope that you have enjoyed the article. If you like it, share it with your friends also. Something not mentioned or want to share your thoughts? Feel free to comment below And I’ll get back to you. 😉

If you want to read my previous blogs, you can read from here. 