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B. Com. II Practical . Moments, skewness, and kurtosis

Moment, Skewness and Kurtosis

Moment, Skewness and Kurtosis

Introduction

In statistics, we often use measures like mean, median, and mode to find the center or typical value of data. We also use range, variance, and standard deviation to understand how spread out the data is.

However, these measures do not tell us everything about the shape of the data. To fully understand the data’s distribution, we need to look at its shape, skewness, and kurtosis.

Moments help describe various aspects of the data distribution beyond just the center and spread.

Skewness tells us if the data is asymmetric, meaning it leans more to one side — left or right.

Kurtosis describes how peaked or flat the data distribution is and shows the presence of extreme values or outliers.

By studying moments, skewness, and kurtosis, we get a better idea of how the data behaves, which helps in choosing the right statistical tests, making better visualizations, and understanding any unusual values in the data.

Moments

There are three types of moments:

  • Raw moment
  • Central moment
  • Moment about any point 'a'

Raw moment

If the rth moment is measured from zero (0 or origin) then the rth moment of X is called the rth raw moment. It is denoted by μ'r.

μ'r = (Σ xr) / n     where r = 1, 2, 3, 4, …

Here, x = each observation in the data set, and n = total number of observations.

To calculate the rth raw moment, raise each data value to the power r, then find the average of these values. Since it is measured from zero, it is also called the moment about the origin.

Central moment

If the rth moment is measured from the mean (x̄) then the rth moment of X is called the rth central moment. It is denoted by μr.

μr = (Σ (x - x̄)r) / n     where r = 1, 2, 3, 4, …

The 1st central moment is always zero because the sum of deviations taken from its mean is zero, i.e. μ1 = 0.

Moment about any point 'a'

If the rth moment is measured from any arbitrary point a then the rth moment of X is called the rth moment about point 'a' and denoted by μr(a).

μr(a) = (Σ (x - a)r) / n     where r = 1, 2, 3, 4, …

If we replace a = 0, we get the raw moment. If we replace a = mean, we get the central moment.

Relation Between Central Moments and Raw Moments (about Origin)

Central moments can be expressed in terms of raw moments (moments about origin) using the following relations. Let μ'r denote rth raw moment and μr denote rth central moment.

  • First central moment: μ1 = 0
  • Second central moment: μ2 = μ'2 - (μ'1)2
  • Third central moment: μ3 = μ'3 - 3 μ'2 μ'1 + 2 (μ'1)3
  • Fourth central moment: μ4 = μ'4 - 4 μ'3 μ'1 + 6 μ'2 (μ'1)2 - 3 (μ'1)4

Skewness

Definition: Skewness is a statistical measure that tells us about the asymmetry of a data distribution. It shows whether the data is symmetrically distributed around the mean or leans more toward one side — left or right.

Types of Skewness

  • Symmetrical Distribution
    Left and right sides of the graph are roughly the same. Mean = Median = Mode. Skewness = 0.
  • Positively Skewed (Right Skewed)
    Tail is longer on the right side. Mean > Median > Mode. Skewness > 0.
  • Negatively Skewed (Left Skewed)
    Tail is longer on the left side. Mean < Median < Mode. Skewness < 0.

Formula for Skewness (Using Central Moments)

Karl Pearson's coefficient of skewness: Sk = (Mean − Mode) / SD

Moment-based skewness (common):

Skewness = μ3 / (μ2)3/2

Interpretation: If Sk < 0 then distribution is negatively skewed; if Sk = 0 then symmetric; if Sk > 0 then positively skewed.

Why Skewness Matters

  • It helps decide whether to use mean or median as a better measure of center.
  • It helps in choosing parametric or non-parametric tests.
  • In modeling and machine learning, correcting skewness can improve model accuracy.

Kurtosis

Definition: Kurtosis is a statistical measure that describes the "peakedness" or flatness of a data distribution compared to a normal distribution. It indicates how sharply the data peaks at the center and how heavy or light the tails are.

Formula for Kurtosis

β₂ = μ4 / (μ2)2
Excess kurtosis: γ₂ = β₂ − 3

Types of Kurtosis

TypeDescriptionβ₂ Valueγ₂ Value
MesokurticNormal peak (like normal distribution)= 3= 0
LeptokurticSharper peak, heavy tails (more outliers)> 3> 0
PlatykurticFlatter peak, light tails (fewer outliers)< 3< 0

Interpretation and Importance

Kurtosis helps detect outliers, is useful in risk analysis (finance), and affects the performance of statistical models — high kurtosis may suggest need for data transformation.

Example / Explanation

Kurtosis is used to describe how concentrated the data is around the mean. A sharp, tall peak (leptokurtic) suggests many values close to the mean but with heavy tails (outliers). A flat curve (platykurtic) suggests the data is more evenly spread with fewer extreme values. A normal bell curve is mesokurtic.

Practical Exercises (From the Uploaded Word File)

ShikshanPrasarakSantha’s
PADMABHUSHAN VASANTRAODADA PATIL MAHAVIDHYALAYA
KAVATHE MAHANKAL
DEPARTMENT OF STATISTICS
B.Com. II (OE): Basic Statistics Practical - III

Expt. No. 1      Date:   /  / 2025
Title: Boxplot

Q.1 Draw a Boxplot for following data.

55, 60, 62, 65, 67, 68, 70, 72, 76, 77, 80, 75, 90, 88, 85, 92, 88, 64, 75, 70.

Q2. The statistics test score of students in class A and class B are given below:

Class A: 55, 60, 62, 65, 68, 70, 72, 65, 67, 73, 75, 80, 85, 88, 90.

Class B: 50, 52, 61, 60, 55, 66, 67, 70, 71, 79, 77, 68, 55, 69, 72.

Q.3 Draw Boxplot for following data set.

165, 189, 110, 148, 150, 198, 180, 175, 168, 188, 154, 156, 164, 155, 140, 159, 160, 158, 149, 197, 150, 120.

Q.4 Draw Boxplot for following data set.

40, 65, 10, 49, 55, 87, 78, 21, 45, 48, 59, 67, 75, 86, 99, 65, 45, 45, 47, 58, 99.

Central Moments, Skewness, and Kurtosis

Q.1: Calculate first four Central Moments and discuss Skewness and Kurtosis

Given Data:

Age (xi) 5 10 15 20 25 30 35
No. of Students (fi) 1 7 14 24 15 13 4

Step 1: Compute Mean (First Central Moment)

Mean:   μ1 = μ = Σ fixi / Σ fi

Σ fi = 1+7+14+24+15+13+4 = 78
Σ fixi = 1*5 + 7*10 + 14*15 + 24*20 + 15*25 + 13*30 + 4*35 = 1670

Mean: μ = 1670 / 78 ≈ 21.41 years

Step 2: Deviations and Central Moments

Age xi xi-μ (xi-μ)² (xi-μ)³ (xi-μ)⁴ fi(xi-μ)² fi(xi-μ)³ fi(xi-μ)⁴
5-16.41269.34-4420.172493.2269.34-4420.172493.2
10-11.41130.2-1485.716942.6911.4-10399.9118798.2
15-6.4141.1-263.71689.3574.8-3691.823620.2
20-1.411.988-2.83.9547.7-67.294.8
253.5912.946.4166.2193.56962493
308.5973.8634.45448959.48247.223724
3513.59184.72508.234078738.810032.8136312

Step 3: First Four Central Moments

μ1 = 0
μ2 = Σ fi(xi-μ)² / Σ fi = 3694.6 / 78 ≈ 47.37
μ3 = Σ fi(xi-μ)³ / Σ fi = 1397.8 / 78 ≈ 17.91
μ4 = Σ fi(xi-μ)⁴ / Σ fi = 394535.4 / 78 ≈ 5055.57

Step 4: Skewness and Kurtosis

Skewness (γ1) = μ3 / μ23/2

μ23/2 ≈ 47.37 * √47.37 ≈ 47.37 * 6.884 ≈ 326.2
Skewness ≈ 17.91 / 326.2 ≈ 0.055

Interpretation: Almost symmetric distribution.

Kurtosis (β2) = μ4 / μ2²

Kurtosis ≈ 5055.57 / 2244.7 ≈ 2.25
Interpretation: Platykurtic (flatter than normal distribution)

Step 5: Conclusion

  • Mean age = 21.41 years
  • Distribution is slightly positively skewed (almost symmetric)
  • Distribution is platykurtic (flatter than normal)
Central Moments, Skewness, and Kurtosis

Calculate First Four Central Moments and Discuss Skewness and Kurtosis

Given Data:

Age Group 0–5 5–10 10–15 15–20 20–25 25–30
No. of Students (fi) 1 12 15 25 11 4

Step 1: Find Class Midpoints (xi)

Age Group 0–5 5–10 10–15 15–20 20–25 25–30
Midpoint xi 2.5 7.5 12.5 17.5 22.5 27.5

Step 2: Compute Mean (μ)

Mean: μ = Σ fixi / Σ fi

Σ fi = 1+12+15+25+11+4 = 68
Σ fixi = 1*2.5 + 12*7.5 + 15*12.5 + 25*17.5 + 11*22.5 + 4*27.5 = 1075

Mean: μ = 1075 / 68 ≈ 15.81

Step 3: Deviations and Central Moments

xi xi-μ (xi-μ)² (xi-μ)³ (xi-μ)⁴ fi(xi-μ)² fi(xi-μ)³ fi(xi-μ)⁴
2.5-13.31177.2-2357.131354.7177.2-2357.131354.7
7.5-8.3169.1-574.94773.7829.2-6898.857284.4
12.5-3.3110.96-36.3120.1164.4-544.51801.5
17.51.692.864.828.171.5120.5202.5
22.56.6944.8300.12009.2492.83301.122101.2
27.511.69136.71597.118668.4546.86388.474673.6

Step 4: First Four Central Moments

μ1 = 0
μ2 = Σfi(xi-μ)² / Σfi = 2281.9 / 68 ≈ 33.55
μ3 = Σfi(xi-μ)³ / Σfi = 10.4 / 68 ≈ 0.153
μ4 = Σfi(xi-μ)⁴ / Σfi = 187417.9 / 68 ≈ 2756.7

Step 5: Skewness and Kurtosis

Skewness (γ1) = μ3 / μ23/2

μ23/2 ≈ 33.55 * √33.55 ≈ 33.55 * 5.79 ≈ 194.3
Skewness ≈ 0.153 / 194.3 ≈ 0.00079

Interpretation: Nearly symmetric distribution.

Kurtosis (β2) = μ4 / μ2²

Kurtosis ≈ 2756.7 / 33.55² ≈ 2756.7 / 1125.7 ≈ 2.45
Interpretation: Platykurtic (flatter than normal distribution)

Step 6: Conclusion

  • Mean age ≈ 15.81 years
  • Distribution is nearly symmetric (skewness ≈ 0)
  • Distribution is platykurtic (flatter than normal)
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