Measures of Dispersion – SS2 Economics Lesson Note (First Term)

Measures of dispersion, also called measures of spread or variation, describe how far the data in a distribution are spread about the mean. They include the range, mean deviation, variance, standard deviation and coefficient of variation. This note shows how to calculate the range, mean deviation, variance and standard deviation for a grouped frequency distribution.

Behavioural objectives

By the end of the lesson, students should be able to:

• explain what measures of dispersion are;
• calculate the range of a set of data;
• calculate the mean deviation of a grouped frequency distribution;
• calculate the variance of a grouped frequency distribution;
• calculate the standard deviation and state how it is related to the variance.

What are measures of dispersion?

Measures of dispersion are also known as measures of spread or variation. They describe how the data in a distribution are spread about the mean, or the overall spread of the data. The measures are the range, mean deviation, standard deviation, variance, coefficient of variation, and so on.

The range

The range of data is the difference between the highest and the lowest value in the data. The formula is:

Range = Highest value – Lowest value

The mean deviation

The mean deviation measures the dispersion around the arithmetic mean. It tells us how far, on average, the individual observations are from the mean. For a grouped frequency distribution:

Mean deviation = ∑f|x – x̄| / ∑f

The variance

The variance is the average of the squares of the deviations of the measurements about their mean.

Variance = ∑f(x – x̄)² / ∑f

The standard deviation

The standard deviation is the square root of the average of the squares of the deviations of the measurements about their mean.

Standard deviation = √[∑f(x – x̄)² / ∑f]

NB: The standard deviation is the positive square root of the variance.

Worked example: weights of 50 students

The data below shows the weight of 50 students to the nearest kg.

65 58 51 36 23 40 53 59 70 51 46 59 50 67 46 39 61 62 73 60 71 51 47 32 48 40 40 51 58 67 60 69 43 52 37 26 38 50 59 40 44 54 42 68 74 45 39 48 55.

Prepare a grouped frequency table, then calculate:

1. the range
2. the mean deviation
3. the variance
4. the standard deviation

Grouped frequency table

Mean

Mean, x̄ = ∑fx / ∑f = 2535 / 50 = 50.7 kg

Solution

1. Range = Highest score – Lowest score = 74 – 23 = 51 kg
2. Mean deviation = ∑f|x – x̄| / ∑f = 529.60 / 50 = 10.59 kg
3. Variance = ∑f(x – x̄)² / ∑f = 7848 / 50 = 156.96 kg²
4. Standard deviation = √[∑f(x – x̄)² / ∑f] = √156.96 = 12.53 kg

Evaluation

1. What are measures of dispersion? Name four of them.
2. How is the range of a set of data calculated?
3. Write the formula for the mean deviation of a grouped frequency distribution.
4. How is the standard deviation related to the variance?

Assignment

1. How is the midpoint (x) ascertained in a grouped frequency table?
2. State two advantages of the mean over other measures of central tendency (that is, the median and the mode).
3. The table below shows the income of 40 workers in a factory in ₦.

   61 78 70 83 92 67 66 83
   76 68 79 84 82 86 81 60
   78 77 86 77 81 92 80 70
   70 40 75 60 74 82 77 87
   63 94 76 87 81 77 87 84

   Using class intervals of 40-49, 50-59 and so on:

   1. Construct a grouped frequency table of the distribution.
   2. Calculate the mean of the distribution.