B. Com. II Practical . Moments, 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 r^th moment is measured from zero (0 or origin) then the r^th moment of X is called the r^th raw moment. It is denoted by μ'_r.

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

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

To calculate the r^th 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 r^th moment is measured from the mean (x̄) then the r^th moment of X is called the r^th central moment. It is denoted by μ_r.

μ_r = (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 r^th moment is measured from any arbitrary point a then the r^th moment of X is called the r^th moment about point 'a' and denoted by μ_r(a).

μ_r(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 r^th raw moment and μ_r denote r^th 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

Type	Description	β₂ Value	γ₂ Value
Mesokurtic	Normal peak (like normal distribution)	= 3	= 0
Leptokurtic	Sharper peak, heavy tails (more outliers)	> 3	> 0
Platykurtic	Flatter 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.

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 f_ix_i / Σ f_i

Σ f_i = 1+7+14+24+15+13+4 = 78
Σ f_ix_i = 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	-16.41	269.34	-4420.1	72493.2	269.34	-4420.1	72493.2
10	-11.41	130.2	-1485.7	16942.6	911.4	-10399.9	118798.2
15	-6.41	41.1	-263.7	1689.3	574.8	-3691.8	23620.2
20	-1.41	1.988	-2.8	3.95	47.7	-67.2	94.8
25	3.59	12.9	46.4	166.2	193.5	696	2493
30	8.59	73.8	634.4	5448	959.4	8247.2	23724
35	13.59	184.7	2508.2	34078	738.8	10032.8	136312

Step 3: First Four Central Moments

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

Step 4: Skewness and Kurtosis

Skewness (γ₁) = μ₃ / μ₂^3/2

μ₂^3/2 ≈ 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)

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 (x_i)

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 f_ix_i / Σ f_i

Σ f_i = 1+12+15+25+11+4 = 68
Σ f_ix_i = 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	-13.31	177.2	-2357.1	31354.7	177.2	-2357.1	31354.7
7.5	-8.31	69.1	-574.9	4773.7	829.2	-6898.8	57284.4
12.5	-3.31	10.96	-36.3	120.1	164.4	-544.5	1801.5
17.5	1.69	2.86	4.82	8.1	71.5	120.5	202.5
22.5	6.69	44.8	300.1	2009.2	492.8	3301.1	22101.2
27.5	11.69	136.7	1597.1	18668.4	546.8	6388.4	74673.6

Step 4: First Four Central Moments

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

Step 5: Skewness and Kurtosis

Skewness (γ₁) = μ₃ / μ₂^3/2

μ₂^3/2 ≈ 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)