Descriptive Statistics:

6 min readJan 26, 2026

Descriptive Statistics is a way to describe and summarize data. It helps us understand large amounts of data easily. Instead of looking at many numbers, descriptive statistics shows important information using simple values like average, middle value, and most common value. It also tells us how much the data is spread out. Descriptive statistics uses tables, charts, and graphs to present data clearly. It does not predict future results; it only explains what the data shows. This makes data easy to read, understand, and compare.

Key Types

There are four types

Measure of central tendency (mean,median,mode)
Measure of dispersion(range,variance,standard deviation)
Measure of shape(skewness,kurtosis)
Measure of relationship(correlation,covariance,rankcorrelation)

1. Measures of central tendency

Numerical values that represents the middle value of a dataset.

a) Mean(Average):

sum of all data values divided by no of values.

Mean = sum of all values/ no of values

Example:
Data: 2, 4, 6, 8
Mean = (2 + 4 + 6 + 8) ÷ 4 = 5

b) Mediun

Middle value of a dataset.when all the values are arranged in ascending/descending order. 1. arrange the data in order. 2. if the no of the observation is odd. or if the no of observation is even.

If odd number of values → middle value
If even number of values → average of two middle values

Example (odd):
Data: 3, 5, 7
Median = 5

Example (even):
Data: 2, 4, 6, 8
Median = (4 + 6) ÷ 2 = 5

c) Mode

Value that occurs must frequently in a dataset

Example:
Data: 1, 2, 2, 3, 4
Mode = 2

One mode → Unimodal
Two modes → Bimodal
No repeated values → No mode

Use: Useful for categorical data (like colors, sizes).

2. Measure of dispersion

Describe how spread out of the data values are from the central value

a) Range:

Difference between maximum value and minimum value in a data set.

Formula:
Range = Maximum − Minimum

Example:
Data: 10, 20, 30, 40
Range = 40 − 10 = 30

b) Variance:

How far data values from the mean

values are close to the mean => Low variance

values are far the mean => High variance

Example:

Variance=1/n∑(x−xˉ)2

Data: 2, 4, 6
Mean = 4

Deviations:
(2−4)² = 4
(4−4)² = 0
(6−4)² = 4

Variance = (4 + 0 + 4) ÷ 3 = 2.67

c) Standard Deviation

Standard Deviation (SD) is a measure of dispersion that shows how much data values deviate from the mean.

Simple Example

Data: 2, 4, 6
Mean = 4

Squared deviations:
(2−4)² = 4
(4−4)² = 0
(6−4)² = 4

Variance = (4 + 0 + 4) ÷ 3 = 2.67

Standard Deviation = √2.67 ≈ 1.63

d) Quartiles

Divides dataset into fourequal parts after the arranging in ascending order

Types of Quartiles

Q1 (First Quartile)
→ The value below which 25% of the data lies
→ Also called the lower quartile
Q2 (Second Quartile)
→ The median of the data
→ 50% data is below and 50% is above
Q3 (Third Quartile)
→ The value below which 75% of the data lies
→ Also called the upper quartile

Example (Small Data Set)

Data:
2, 4, 6, 8, 10, 12, 14, 16

(Already in ascending order)

Step 1: Find Q2 (Median)

Middle of data = average of 8 and 10
Q2 = (8 + 10) / 2 = 9

Step 2: Find Q1

Lower half → 2, 4, 6, 8
Median of lower half = (4 + 6) / 2 = 5
Q1 = 5

Step 3: Find Q3

Upper half → 10, 12, 14, 16
Median of upper half = (12 + 14) / 2 = 13
Q3 = 13

Q1 = 5
Q2 = 9
Q3 = 13

3. Measure of shape(skewness,kurtosis)

It shows whether data is balanced (or) tilted

a) Skewness

tells us shape of data,shape of data means balanced or pulled more to one side.

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If the data is evenly spread on both sides => Not skewed (symmetric)

if the data has long tail on the rightside.=> Positive skewed (right skewness)

if the data has long tail on the leftside.=> Negative skewed (left skewness)

Skewness tells us how data is tilted (leaning) to the left or right instead of being perfectly balanced.

1. Positive Skewness (Right Skew)

In a positively skewed distribution, the right tail is longer than the left which means most data points are on the left with a few large values pulling the distribution to the right.

Relationship:

Mean > Median > Mode
Examples: Income distribution, exam scores and stock market returns.

2. Negative Skewness (Left Skew)

In a negatively skewed distribution, the left tail is longer which means most data points are on the right with a few smaller values pulling the distribution to the left.

Relationship:

Mean < Median < Mode

Examples: Test scores on easy exams, age at retirement and gestational age at birth.

3. Zero Skewness (Symmetrical Distribution)

Zero skewness shows a perfectly symmetrical distribution where the mean, median and mode are equal. In a symmetrical distribution, the data points are evenly distributed around the central point.

Relationship:

Mean = Median = Mode

b) Kurtosis

Kurtosis is a statistical measure that describes the shape of a data distribution especially how heavy or light the tails are. It tells us whether a dataset has outliers than a normal distribution or if most data points stay closer to the average.

High Kurtosis (Leptokurtic): Data has extreme outliers, peaks sharply at the mean.
Low Kurtosis (Platykurtic): Data is more evenly spread, flatter peak and lighter tails.
Normal Kurtosis (Mesokurtic): Distribution resembles the normal curve, standard probability of extreme events.

Calculation of Kurtosis

To calculate kurtosis in statistics, you can follow these steps:

1. Compute the Mean (μ): Calculate the arithmetic mean of the dataset.

2. Compute the Variance (σ2): Calculate the variance of the dataset, which is the average of the squared differences from the mean.

3. Compute the Standard Deviation (σ): Take the square root of the variance to find the standard deviation.

4. Compute the Fourth Moment (μ4): Calculate the fourth moment of the dataset, which is the average of the fourth power of the differences from the mean.

5. Compute Kurtosis: The formula for calculating kurtosis is: