Moments in Statistics: A Comprehensive Guide

# Moments in Statistics: A Comprehensive Guide

Vikram Singh
Assistant Manager - Content
Updated on Apr 30, 2024 09:03 IST

Data can come in all shapes and sizes, just like collections of toys! Statistics help us understand these collections. Moments are like special tools used to describe how the data is spread out. The most common moments, like mean (average) and variance (how spread out the data is), are like understanding how many toys you have and how close they are to each other. This article will explain what moments are in statistics, using fun examples to show you how they help us get a feel for our data!

Moments in statistics are quantitative measure that describes the specific characteristics of a probability distribution. It helps to understand the data set’s shape, spread, and central tendency. This article will discuss what moments in statistics are and four key moments: Mean, Variance & Standard Deviation, Skewness, and Kurtosis.

Let’s start the article with the formal definition of Moments in Statistics.

Table of Content

## What are Moments in Statistics?

Moments in statistics are quantitative measures (or a set of statistical parameters) that describe the specific characteristics of a probability distribution. In simple terms, the moment measures how spread out or concentrated the number in a dataset is around the central value, such as the mean.

Note:

• Mathematically, for any random variable X, the moments of X are defined as the expected values of X.
• Example: E[X], E[X^2], E[X^3],….
• Moments in statistics are closely related to the concepts of the moment in Physics.
• If a function represents the physical density of the body, then the
• zeroth moment is the total mass.
• the first moment divided by the total mass is the centre of mass.
• the second moment is rotational inertia.
• In statistics, the first four moments are Mean, Variance, Skewness, and Kurtosis.
• Each of these four moments gives information about the dataset, such as central location, dispersion, asymmetry, and outliers.

Now, let’s discuss the first four moments in detail. Each of these four moments

### Four Keys of Moments in Statistics

#### First Moment – Mean

The first moment is referred to as mean. Mean is a measure of central tendency that represents the average value of the dataset. It is defined as the ratio of the sum of all observations to the total number of observations.

i.e.,

Mean (µ) = Σxᵢ / n

where,

Σxᵢ: Sum of all data points (xᵢ)

n: Number of data points

Example: Let the heights of 5 students in a class be 150cm, 155cm, 160m, 165cm, and 180cm. Find the class mean height.

Solution

Sum of height of all students = 150 + 155 + 160 + 165 + 180 = 810

Total number of students = 5

=> Mean =  810 / 5 = 162.

Hence, the mean height of the class is 162 cm.

Note:  Mean is highly sensitive to skewness and outliers.

Also Read: How to Calculate Mean using Mean Formula.

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#### Second Moment – Variance

Variance

Variance is the measure of dispersion, referred to as the second moment in statistics. It is the average squared difference of data points from the mean. The higher the variance value, the greater the variability (i.e., data points are far away from the mean) within the dataset, and vice versa.

Variance measures how far each data point in the dataset is from the mean.

Formula of Variance

Variance (σ²) = Σ(xᵢ – µ)²/n

where:

Σ(xᵢ – µ)²: Sum of the squared difference between each data point (xᵢ) and the mean (µ)

n: Number of data points

Standard Deviation

It is defined as the square root of variance. It compares the spread of two different datasets with approximately the same mean.

Formula of Standard Deviation

Standard Deviation = (variance)1/2

#### Third Moment – Skewness

Skewness is the third moment, which measures the deviation of the given distribution of a random variable from a symmetric distribution. In simple terms, skewness means the lack of straightness or symmetry.

Formula of Skewness

Skewness (γ) = Σ(xᵢ – µ)³/(n * σ³)

where:

Σ(xᵢ – µ)³: Sum of the cubed difference between each data point (xᵢ) and the mean (µ)
n: Number of data points
σ: Standard deviation

There are two types of skewness:

Positive Skewness: In positive skewness, the extreme data values are greater, which, in turn, increases the mean value of the dataset.

In positive skewness: Mode < Median < Mean.

Negative Skewness: In negative skewness, the extreme data values are smaller, which, in turn, decreases the mean value of the dataset.

In negative skewness: Mean < Median < Mode.

Note: When there is no skewness in the dataset, it is referred to as Symmetrical Distribution.

In Symmetrical Distribution: Mean = Median = Mode.

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#### Fourth Moment – Kurtosis

Kurtosis is the fourth moment, which measures the presence of outliers in the distribution. It gives the graph as either heavily-tailed or lightly-tailed due to the presence of outliers. In simple terms, kurtosis measures the peakedness or flatness of a distribution.

• If the graph has a shorter tail and a flat top, then Kurtosis is said to be high.
• If the graph has a higher peak and lower tail, then the kurtosis is said to be low.

There are three types of Kurtosis:

Mesokurtic: This is the same as Normal distribution, i.e., a type of distribution in which the extreme ends of the graph are similar.

Lepotokurtic: This distribution indicates that a more significant percentage of data is present near the tail, which implies the longer tail. Lepotokurtic has a greater value of kurtosis than Mesokurtic.

Platykurtic: This distribution indicates that there is less data in the tail portion, which implies a shorter tail. Platykurtic has a lesser value of kurtosis than Mesokurtic.