Understanding Dispersion Measures in NumPy : Range, Variance, and Standard Deviation

What is Dispersion (Data Spread)?

In simple terms, dispersion (or spread) tells us how spread out the data values are.

• If data values are very close to each other → Low Dispersion
• If data values are far apart → High Dispersion

Example

Dataset A (Low Dispersion)

Marks: 85, 87, 86, 88, 84

All marks are close to each other. Average ≈ 86

This dataset has low dispersion.

Dataset B (High Dispersion)

Marks: 45, 92, 35, 95, 78

The marks vary a lot, from 35 to 95. Average ≈ 69

This dataset has high dispersion.

Common Ways to Measure Dispersion

1. Range (Simplest Method)

Range is the difference between the largest and smallest value.

Formula:

Range = Maximum Value − Minimum Value

Example:

Marks: 45, 67, 78, 89, 95

Range = 95 − 45 = 50

Limitation: It uses only two values and can be strongly affected by extreme values (outliers).

2. Variance

Variance measures how far data values are from the average (mean).

Example:

Data: 2, 4, 6, 8

Mean = 5

Differences from the mean:

2 − 5 = -3

4 − 5 = -1

6 − 5 = 1

8 − 5 = 3

Square the differences:

• 9, 1, 1, 9

Variance = (9 + 1 + 1 + 9) / 4 = 5

3. Standard Deviation

Standard deviation is the square root of variance.

Formula:

Standard Deviation = √Variance

Using the previous example:

Variance = 5

Standard Deviation = √5 ≈ 2.236

Meaning: On average, the data values are about 2.24 units away from the mean.

Real-Life Examples

Cricket Scores

Consistent Player (Low Dispersion)

Scores: 45, 52, 48, 50, 47

The scores are similar and predictable.

Inconsistent Player (High Dispersion)

Scores: 12, 87, 5, 102, 8

The scores vary greatly.

Stock Market

Low Risk Stock

Prices: 100, 102, 101, 99, 100

Prices change very little.

High Risk Stock

Prices: 80, 150, 60, 200, 50

Prices fluctuate significantly.

Key Idea

• Low Dispersion → Data is more consistent and predictable.
• High Dispersion → Data is more variable and less predictable.

Specific Example : Dispersion Measures

Dataset: Marks of 10 Students (Out of 100)

Data:

48, 52, 55, 60, 65, 72, 78, 80, 85, 90

Step 1: Range (Simplest Measure)

• Maximum value = 90
• Minimum value = 48

Range = 90 − 48 = 42

Meaning: The marks are spread across 42 points.

Step 2: Mean (Average)

We first calculate the mean because Variance and Standard Deviation depend on it.

Sum of all marks:

48 + 52 + 55 + 60 + 65 + 72 + 78 + 80 + 85 + 90 = 685

Mean = 685 ÷ 10 = 68.5

Step 3: Variance

Find the difference between each mark and the mean, square the difference, and then calculate the average.

| Marks | Difference from Mean | Squared Difference |
| ----- | -------------------- | ------------------ |
| 48    | 48 − 68.5 = -20.5    | 420.25             |
| 52    | -16.5                | 272.25             |
| 55    | -13.5                | 182.25             |
| 60    | -8.5                 | 72.25              |
| 65    | -3.5                 | 12.25              |
| 72    | +3.5                 | 12.25              |
| 78    | +9.5                 | 90.25              |
| 80    | +11.5                | 132.25             |
| 85    | +16.5                | 272.25             |
| 90    | +21.5                | 462.25             |

Total of squared differences = 1928.5

Variance = 1928.5 ÷ 10 = 192.85

Step 4: Standard Deviation (Most Important)

Standard Deviation = √Variance

SD = √192.85 ≈ 13.89

Meaning: On average, each student’s score is about 13.89 points away from the mean (68.5).

NumPy Code for Dispersion Measures

Here is a clean Python program that calculates the main dispersion measures using NumPy.

import numpy as np

# Dataset
marks = np.array([48, 52, 55, 60, 65, 72, 78, 80, 85, 90])

print("Dataset:", marks)
print("=" * 40)

# 1. Range
data_range = np.max(marks) - np.min(marks)
print(f"1. Range              : {data_range}")

# 2. Mean
mean = np.mean(marks)
print(f"2. Mean               : {mean}")

# 3. Variance
variance = np.var(marks)  # Population variance (default)
print(f"3. Variance           : {variance:.4f}")

# 4. Standard Deviation
std_dev = np.std(marks)   # Population standard deviation (default)
print(f"4. Standard Deviation : {std_dev:.4f}")

# 5. Sample Variance
variance_sample = np.var(marks, ddof=1)
print(f"5. Sample Variance    : {variance_sample:.4f}")

# 6. Sample Standard Deviation
std_dev_sample = np.std(marks, ddof=1)
print(f"6. Sample Std Dev     : {std_dev_sample:.4f}")

Expected Output

Dataset: [48 52 55 60 65 72 78 80 85 90]
========================================
1. Range              : 42
2. Mean               : 68.5
3. Variance           : 192.8500
4. Standard Deviation : 13.8863
5. Sample Variance    : 214.2778
6. Sample Std Dev     : 14.6375

Understanding the Results

Range

Shows the difference between the highest and lowest values.

Range = Maximum Value - Minimum Value

In this dataset

90 - 48 = 42

Mean

The average value of all marks.

Mean = Sum of Values / Number of Values

Result

68.5

Variance

Measures how far the data values are spread from the mean.

np.var(marks)

Result

192.85

A larger variance means the data is more spread out.

Standard Deviation

Shows the spread of the data in the same unit as the original values.

np.std(marks)

Result

13.8863

On average, the marks are about 13.89 points away from the mean.

Summary

These measures help you understand how closely grouped or widely spread your data is around the average value.