Range, variance, and standard deviation.

Grade 12 Statistics Lesson: Range, Variance, and Standard Deviation

Lesson Objectives

Understand the concepts of range, variance, and standard deviation.
Calculate range, variance, and standard deviation from data sets.
Interpret measures of dispersion to analyze data variability.

Lesson Introduction

In statistics, measures of dispersion describe the spread or variability of data.

Range is the difference between the highest and lowest values.
Variance measures the average squared deviation from the mean.
Standard Deviation is the square root of the variance, showing spread in the original units.

Range

Example 1: Find the range of the data set: 12, 15, 20, 22, 18.

Step 1: Identify the maximum and minimum values.
Max = 22, Min = 12
Step 2: Calculate the range:

\text{Range} = \text{Max} - \text{Min} = 22 - 12 = 10

Example 2: Find the range of the data: 5, 8, 6, 9, 7, 4.

Max = 9, Min = 4

\text{Range} = 9 - 4 = 5

Example 3: Calculate the range for: 100, 95, 102, 98, 99.

Max = 102, Min = 95

\text{Range} = 102 - 95 = 7

Example 4: Data: 3.5, 4.2, 4.8, 3.9, 4.0

Max = 4.8, Min = 3.5

\text{Range} = 4.8 - 3.5 = 1.3

Example 5: Find the range: 25, 28, 24, 30, 27.

Max = 30, Min = 24

\text{Range} = 30 - 24 = 6

Variance

Example 1: Find the variance for data set: 3, 5, 7, 9.

Step 1: Calculate the mean

\bar{x} = \frac{3 + 5 + 7 + 9}{4} = \frac{24}{4} = 6

Step 2: Calculate squared deviations

(3 - 6)^2 = 9, \quad (5 - 6)^2 = 1, \quad (7 - 6)^2 = 1, \quad (9 - 6)^2 = 9

Step 3: Find the variance (population)

\sigma^2 = \frac{9 + 1 + 1 + 9}{4} = \frac{20}{4} = 5

Example 2: Find variance for 4, 8, 6.

\bar{x} = \frac{4+8+6}{3} = 6

Squared deviations:

(4 - 6)^2 = 4, (8 - 6)^2 = 4, (6 - 6)^2 = 0

Variance:

\sigma^2 = \frac{4 + 4 + 0}{3} = \frac{8}{3} \approx 2.67

Example 3: Data: 10, 15, 20

\bar{x} = \frac{10+15+20}{3} = 15

Squared deviations:

(10 - 15)^2=25, (15-15)^2=0, (20-15)^2=25

Variance:

\sigma^2 = \frac{25+0+25}{3} = \frac{50}{3} \approx 16.67

Example 4: Calculate variance for: 2, 5, 9, 4.

\bar{x} = \frac{2+5+9+4}{4} = 5

Squared deviations:

(2-5)^2=9, (5-5)^2=0, (9-5)^2=16, (4-5)^2=1

Variance:

\sigma^2 = \frac{9 + 0 + 16 + 1}{4} = \frac{26}{4} = 6.5

Example 5: Find variance of: 7, 7, 7, 7.

Mean:

\bar{x} = 7

Squared deviations:

(7-7)^2=0

for all values.
Variance:

\sigma^2 = 0

Standard Deviation

Example 1: Find the standard deviation for the data: 3, 5, 7, 9.

Variance from previous example = 5
Standard deviation:

\sigma = \sqrt{5} \approx 2.236

Example 2: Calculate standard deviation for: 4, 8, 6.

Variance = 2.67
Standard deviation:

\sigma = \sqrt{2.67} \approx 1.633

Example 3: Data: 10, 15, 20.

Variance = 16.67
Standard deviation:

\sigma = \sqrt{16.67} \approx 4.082

Example 4: Data: 2, 5, 9, 4.

Variance = 6.5
Standard deviation:

\sigma = \sqrt{6.5} \approx 2.55

Example 5: Data: 7, 7, 7, 7.

Variance = 0
Standard deviation:

\sigma = 0

Exercise

Calculate the variance of the data: 4, 7, 10, 6.
Find the standard deviation of: 12, 15, 20, 22, 18.
Determine the range for the data: 35, 28, 32, 40, 30.
Calculate the variance of the data: 3, 5, 8, 6, 7.
Find the standard deviation for the data: 9, 11, 15, 14, 10.
Calculate the range of: 100, 105, 110, 95, 102.
Find the variance of: 8, 12, 15, 10, 9.
Calculate the standard deviation of: 20, 18, 22, 24, 19.
[NECO] Find the range, variance, and standard deviation of: 6, 6, 6, 6, 6. (Past Question)
The marks scored by 6 students in a test are: 65, 70, 75, 80, 85, 90. Calculate the variance and standard deviation of the marks.

Conclusion/Recap

Range, variance, and standard deviation measure how data is spread out. Range is the difference between the highest and lowest values. Variance calculates the average squared differences from the mean. Standard deviation is the square root of the variance and shows the typical distance of data points from the mean. Understanding these helps analyze data variability effectively..