Descriptive Statistics

Descriptive statistics helps us summarize and understand data. It does not predict anything. It only explains what the data looks like.

Descriptive statistics is mainly divided into four measures:

1. Measure of Central Tendency
2. Measure of Dispersion
3. Measure of Shape
4. Measures of Relationship

1. Measures of Central Tendency

(Finding the Center of Data)

The center of data means a value that represents the whole dataset.

Many people say "average," but average can be misleading because there are different types of averages.

Mean

Mean is the average value of the data.

The mean is called the balance point of data. If all values were placed on a scale, the mean is where the scale balances.

Formula:

Mean=∑x/n

​ OR

Mean = Number of values/Sum of all values​

Where:

• ∑x = sum of all values
• n = number of values

Solved Example (Mean)

Data: 10, 20, 30, 40

Mean=10+20+30+40/4=100/4=25

Problem with Mean

One extreme value can change the mean a lot.

Example: Salaries = 20k, 25k, 30k, 200,000

The mean becomes very high, but most people earn much less.

When Mean Is Useful

• Marks
• Sensor data
• Balanced data

When Mean Is Misleading

• Salaries
• Income
• House prices

Median

The median is the middle value after sorting the data.

Case 1: Odd Number of Values

Median = middle value

Example: Data: 10, 20, 30

Median = 20

Case 2: Even Number of Values

Median = average of two middle values

Formula:

Median = (Middle value 1 + Middle value 2)/2 ​

Solved Example: Data: 10, 20, 30, 40

Median=20+30/2=25

Why Median Is Important

• Not affected by extreme values
• Shows what a typical person earns

Used for:

• Salaries
• Income
• Property prices

Mode

Mode is the value that appears most frequently.

Solved Example: Data: 2, 3, 3, 5, 7

Mode = 3

Mode is useful for categorical data:

• Brands
• Sizes
• Payment methods

A dataset can have no mode, one mode, or more than one mode.

Why Businesses Prefer Median Over Mean

In uneven data:

• Mean changes easily
• Median stays stable

That is why companies report median salary, not mean salary.

2. Measures of Dispersion

(Understanding Data Spread)

The center alone is not enough.

Two datasets can have the same average but behave very differently.

Dispersion shows how spread out the data is.

Range

Range is simple but unreliable because it uses only two values.

Formula:

Range=Maximum value−Minimum value

Solved Example: Data: 10, 20, 50

Range = 50 − 10 = 40

Variance

Measures how far data values are from the mean.

1. Population Variance

Formula:

Where:

• σ² = population variance
• x = each data value
• xˉ = mean of the population
• n = total number of values

Used when: You have data for the entire population.

2. Sample Variance

Formula:

Where:

• s² = sample variance
• x = each data value
• xˉ = sample mean
• n = number of values in the sample

Used when: You have only a sample from the population.

Simple Difference

• Population variance → divide by N
• Sample variance → divide by n − 1

Solved Example (Variance)

Data: 10, 20, 30

Step 1: Mean

xˉ=10+20+30/3=20

Step 2: Differences from mean

• (10 − 20)² = 100
• (20 − 20)² = 0
• (30 − 20)² = 100

Step 3: Variance

100+0+100/3=66.67

Standard Deviation

Standard deviation is the square root of variance.

Meaning:

• Low standard deviation → stable and consistent data
• High standard deviation → high uncertainty and risk

Formula:

For a population (N):

For Sample (n):

Solved Example:

sqrt{66.67} = 8.16

Quartiles and Interquartile Range (IQR)

Quartiles divide the data into four equal parts after sorting the data.

Each part represents 25% of the data.

• Q1 → 25% 25% of the data lies below Q1.
• Q2 → Median (50%) 50% of the data lies below Q2. Q2 is the median of the data.
• Q3 → 75% 75% of the data lies below Q3.
• Q4 → 100% (Maximum value) All data lies below Q4. Q4 is the maximum value in the dataset

Formula:

IQR=Q3−Q1

IQR shows the spread of the middle 50% of the data.

Solved Example: Q1 = 20, Q3 = 40

IQR = 40 − 20 = 20

3. Measures of Shape

(Understanding Data Pattern)

Data is not always symmetrical.

Shape tells us how data is distributed.

Skewness

Skewness shows which side of the data is longer.

There are three types of skewness.

1. Right Skewness (Positive Skewness)

• Long tail on the right side
• Few very large values
• Mean > Median > Mode

Common examples:

• Income
• Salaries
• Delivery time

Numerical example:

Data: 10, 12, 14, 15, 100

• Median = 14
• Mean = (10 + 12 + 14 + 15 + 100) / 5 = 30.2

Since mean > median > mode, the data is right-skewed.

2. Left Skewness (Negative Skewness)

• Long tail on the left side
• A few very small values
• Mean < Median < Mode

Common examples:

• Exam scores
• Product ratings (mostly high)

Numerical example:

Data: 10, 80, 85, 90, 95

• Median = 85
• Mean = (10 + 80 + 85 + 90 + 95) / 5 = 72

Since mean < median < mode, the data is left-skewed.

3. Zero Skewness (Symmetrical Data)

• Data is balanced on both sides
• No long tail
• Mean = Median = Mode

Common examples:

• Heights of people
• Controlled measurements

Numerical example:

Data: 40, 45, 50, 55, 60

• Mean = 50
• Median = 50
• Mode = 50

This data is symmetrical (zero skewness).

Kurtosis

Kurtosis describes how extreme values behave in data.

• High kurtosis → many extreme values
• Low kurtosis → fewer extreme values

There are three types of kurtosis.

1. Mesokurtic (Normal data)

• Normal amount of extreme values

Numerical example: Data: 45, 50, 52, 55, 60

Values are close together. Extreme values are normal. This data is mesokurtic.

2. Leptokurtic (High kurtosis)

• Many extreme values

Numerical example: Data: 10, 12, 14, 15, 100

One very large value exists. This data has high kurtosis (leptokurtic).

3. Platykurtic (Low kurtosis)

• Very few extreme values

Numerical example: Data: 48, 50, 52, 54, 56

All values are close. No extreme values. This data is platykurtic.

4. Measures of Relationship

(How Two Variables Are Related)

Looking at one variable gives limited insight.

Understanding how two variables move together gives better understanding.

1. Covariance

Covariance values are hard to interpret directly

Formula:

Where:

• x , y = values of two different variables
• xˉ, yˉ​ = mean of each variable
• n = number of observations

Meaning:

• Positive → both increase together
• Negative → one increases, other decreases

2. Correlation

Formula:

Where:

• σx = standard deviation of variable x
• σy​ = standard deviation of variable y

Range: -1 to +1

• +1 → strong positive relationship
• -1 → strong negative relationship
• 0 → no relationship

Correlation does NOT mean causation.

3. Spearman Rank Correlation

(Understanding Relationship Using Ranks)

Spearman Rank correlation is used to measure the relationship between two variables using ranks instead of actual values.

It tells us:

• Whether two variables move in the same direction
• Or move in opposite directions

This method is very useful when:

• Data is ranked
• Relationships are not linear
• Data is ordinal (positions, order, ratings)

When Do We Use Spearman Rank Correlation?

We use Spearman rank correlation when:

• Data is already given in rank form
• Data has order, but exact difference does not matter
• A relationship is not a straight line (non-linear)

Common examples:

• Student rankings
• Survey ratings (1 to 5)
• Employee performance ranking

Formula

Where:

• rs = Spearman rank correlation coefficient
• d = difference between ranks of each item
• d² = square of rank difference
• ∑d² = sum of squared rank differences
• n = total number of observations

Step-by-Step Solved Example

Suppose we want to check the relationship between math marks ranking and science marks ranking of students.

Apply the Formula:

This means:

• Strong positive relationship
• Students who rank high in math also tend to rank high in science.

Understanding the Value of Spearman Correlation

• rs=+1 → Perfect positive relationship
• rs=−1 → Perfect negative relationship
• rs=0 → No relationship

Conclusion

Descriptive statistics is the foundation of data analysis. It helps us understand data before making any decisions.

Measures like mean, median, and mode show typical values; dispersion measures explain variation; shape measures describe data behavior; and relationship measures show how variables are connected.

For a data analyst, these concepts help summarize data, find patterns and outliers, understand risk, and explain insights clearly. Without descriptive statistics, data analysis becomes guesswork.