MEASURES OF DISPERSION Lesson Note
This SS2 Economics lesson note covers MEASURES OF DISPERSION for First Term, Week 2.
Learning Objectives
- By the end of the lesson, students should be able to:
- Define dispersion and explain its importance in economic data analysis.
- Calculate the range, variance, mean deviation and standard deviation for simple ungrouped data.
- Interpret the size of dispersion in a data set.
- Explain a simple linear equation and identify its components.
- Solve pairs of simultaneous linear equations using the substitution
Lesson Note
Measures of Dispersion
The measures of dispersion also known as the measures of variability is the degree of spread of the numerical value in a distribution. It measures the variation that occurs in a given set of data.
Measures of dispersion includes range, the quartile mean deviation, variance and standard deviation
A. Range
Range is the difference between the highest value and the lowest value in a data set.
Interpretation: A larger range generally indicates greater spread, while a smaller range indicates less spread.
B. Mean Deviation
Mean deviation is the arithmetic mean of the absolute deviations of the observations from a chosen central value, usually the mean.
Where x = each observation, x̄ = mean, and n = number of observations.
Calculate the mean deviation of the following age of some pupils in Pen-Ark schools: 4, 5, 6 , 8,1 0, 3
Find the Arithmetic mean
x = 4 + 5 + 6 + 8 +10 + 3
6
x = 36
6
X = 6
Mean deviation using the formula above
M.D = [4 – 6] + [5 - 6] + [6 - 6] + [8 - 6] + [10 - 6] + [3 - 6]
6
M.D = 2 + 1 + 0 + 2 + 4 + 3
6
M.D = 12
6
M.D = 2
C. Variance
Variance refers to the Arithmetic mean of the squares of the deviation of the observation from the true mean. It is also referred to as the mean square deviation
Example: calculate the variance of the following set of data: 3, 5, 8, 5, 6, 9
Step 1 calculate the Arithmetic mean
x = 3 + 5 + 8 + 5 + 6 + 9
6
Step 2 calculate the deviation (x - x̄) and find the square of each and find the sum
(x - x̄) = [3 - 6], [5-6], [8-6], [5-6], [6-6], [9-6]
(x - x̄)2 = (-3)2 + (-1)2 + (2)2 + (-1)2 + (0)2 + (3)2
Step 3: 9 + 1 + 4 + 1 + 0 + 9= 24
Variance = = 24
n 6
Variance = 4
D. Standard Deviation
Standard deviation is the positive square root of the variance. It is one of the most widely used measures of dispersion.
Calculate the standard deviation of the following set of data 3, 5, 8, 5, 6, 9
Based on the calculation above:
S.D = Variance
S.D =
E. Summary Table
| Measure | Meaning | Formula |
|---|---|---|
| Range | Difference between highest and lowest values | H − L |
| Mean Deviation | Average absolute distance from the mean | Σ|x − x̄| / n |
| Variance | Average squared distance from the mean | Σ(x − x̄)² / n |
| Standard Deviation | Square root of variance | √Variance |
F. Economic Importance of Measures of Dispersion
They help economists compare the stability or variability of prices,
They help businesses assess fluctuations in sales, costs and profits.
They help government and researchers study inequality and variations in economic data.
They help in decision-making where risk or uncertainty is involved.
G. Conclusion
Measures of dispersion provide information about how widely observations are spread around their central value. Range is the simplest measure, while variance and standard deviation provide more comprehensive measures of spread. Simple linear and simultaneous equations provide useful mathematical tools for solving problems involving unknown economic quantities
Evaluation
Choose the correct answer from options A–D.
1. Measures of dispersion refer to the degree of ______ in a set of data.
A. average
B. spread
C. frequency
D. centrality
2. Which of the following is a measure of dispersion?
A. Mean
B. Median
C. Range
D. Mode
3. The range of a data set is calculated by:
A. Highest value + Lowest value
C. Lowest value × Highest value
D. Highest value ÷ Lowest value
4. Find the range of 4, 7, 10, 15, 20.
A. 14
B. 16
C. 18
D. 24
5. Mean deviation is the average of the ______ deviations from the mean.
A. squared
B. absolute
C. negative
D. total
6. Variance is also known as the:
A. mean square deviation
B. mean absolute deviation
C. average range
D. standard deviation
7. Standard deviation is the positive square root of:
A. range
B. mean
C. variance
D. mean deviation
8. If the variance of a distribution is 36, what is its standard deviation?
A. 3
B. 6
C. 12
D. 18
9. Find the range of the following data: 8, 12, 15, 20, 25, 30.
A. 20
B. 22
C. 24
D. 28
10. If the highest value in a data set is 45 and the lowest value is 18, what is the range?
A. 17
B. 27
C. 37
D. 63
Theory Questions
1. Calculate the mean, range, mean deviation, variance and standard deviation of: 3, 5, 7, 9, 11, 6, 8, 12, 4