Measures of Dispersion (Spread)

Master range, variance, standard deviation, and interquartile range

CAPS Grade 10 Mathematics

Dispersion, or spread, describes the extent to which data points differ from each other and from the central tendency. Understanding these measures helps in assessing the variability and reliability of data.

Understanding Range

The range is the difference between the maximum and minimum values.

Understanding Variance & Standard Deviation

Variance measures average squared distance from mean. Standard deviation is the square root, giving spread in original units.

Understanding IQR (Interquartile Range)

The IQR measures the spread of the middle 50% of the data, from Q1 to Q3.

Learning Outcomes

Calculate and interpret the range of a dataset
Understand variance and standard deviation
Compute standard deviation for ungrouped data
Find quartiles and interquartile range (IQR)
Calculate coefficient of variation (CV)
Interpret what each measure tells us about data spread

1. Range

The range is the simplest measure of dispersion.

Range = Maximum Value - Minimum Value

Range Example

Data

3, 7, 8, 12, 15

Solution

Max = 15, Min = 3

Range = 15 - 3 = 12

Quiz 1 - Range

Find the range: 4, 9, 12, 18, 21, 25

A) 17

B) 21

C) 25

D) 29

2. Variance

Variance measures the average squared deviation of each data point from the mean.

Population Variance

σ² = Σ(xᵢ - μ)² / N

Sample Variance

s² = Σ(xᵢ - x̄)² / (n-1)

Variance Calculation

Data

4, 8, 6, 5, 3

Solution

Mean = (4+8+6+5+3)/5 = 5.2

Deviations: -1.2, 2.8, 0.8, -0.2, -2.2

Squared: 1.44, 7.84, 0.64, 0.04, 4.84

Sum of squares = 14.8

Sample variance s² = 14.8/4 = 3.7

3. Standard Deviation

Standard deviation is the square root of variance and provides a measure of dispersion in the same units as the original data.

s = √s²

Standard Deviation Example

From previous example

Variance s² = 3.7

Solution

s = √3.7 ≈ 1.92

Quiz 2 - Standard Deviation

If variance = 9, what is standard deviation?

A) 3

B) 4.5

C) 9

D) 81

4. Interquartile Range (IQR)

The interquartile range measures the spread of the middle 50% of a dataset.

IQR = Q3 - Q1

IQR Example

Data

1, 3, 5, 7, 9, 11, 13, 15

Solution

Q1 (25th percentile) = 4

Q3 (75th percentile) = 12

IQR = 12 - 4 = 8

Quiz 3 - IQR

Q1 = 10, Q3 = 30. What is IQR?

A) 10

B) 20

C) 30

D) 40

5. Coefficient of Variation (CV)

The coefficient of variation expresses standard deviation as a percentage of the mean.

CV = (s / x̄) × 100%

CV Example

Data

Mean = 50, Standard deviation = 10

Solution

CV = (10/50) × 100% = 20%

Comparison of Dispersion Measures

Measure	What it tells us	Formula	Affected by outliers?
Range	Total spread	Max - Min	Yes
Variance	Average squared deviation	Σ(x-x̄)²/(n-1)	Yes
Standard Deviation	Typical deviation	√variance	Yes
IQR	Middle 50% spread	Q3 - Q1	No
CV	Relative variability	(s/x̄)×100%	Yes

Practice & Assess

Test your knowledge with these interactive games.

Match - Measure to Formula

Range

Max - Min

Variance

Average squared deviation

Standard deviation

√variance

IQR

Q3 - Q1

Fill - Coefficient of Variation

CV = (s / x̄) × ___%

Practice Questions

Find range: 10, 15, 20, 25, 30, 35

Answer: 25

Variance = 25. Find standard deviation.

Answer: 5

Q1 = 15, Q3 = 45. Find IQR.

Answer: 30

Summary of Key Concepts

Range: Max - Min (simplest, but sensitive to outliers)

Variance: Average squared deviation from mean

Standard Deviation: √variance (same units as data)

IQR: Q3 - Q1 (middle 50% spread)

CV: (s/mean) × 100% (relative variability)

Key Terms

Range Variance Standard deviation Quartile IQR Coefficient of variation Deviation Spread Dispersion Outlier Percentile

Central Tendency Statistics