Statistics – SS2 Mathematics Lesson Note (Third Term)

Measures of central tendency are values that show the middle or average of a set of data. They include the mean, median and mode. Measures of dispersion, also called measures of spread or variation, describe how the data are spread about the mean. They include the range, mean deviation, variance and standard deviation.

Behavioural objectives

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

• state what measures of central tendency and measures of dispersion are;
• calculate the mean, median and mode of grouped data;
• calculate the range, mean deviation, variance and standard deviation of a set of data;
• state the relationship between variance and standard deviation.

What are measures of central tendency?

These are values which show the degree to which a given set of data converges toward the central point of the data. They are also called measures of location. They give the centre or average of a set of data. The measures of central tendency are the arithmetic mean, the median and the mode.

Mean

The mean is the average of the variables obtained in a study. It is the most common kind of average. For grouped data, the formula for calculating the mean is:

Mean = ∑fx ÷ ∑f

Where:

• ∑ = summation
• f = frequency
• x = observation (the class mid-point for grouped data)

Median

The median is the middle number in any given distribution. For grouped data, the formula is:

Median = L + [(N/2 – Fb) ÷ f] × c

Where:

• L = lower class boundary of the median class
• N = sum of the frequencies
• Fb = cumulative frequency before the median class
• f = frequency of the median class
• c = class size

Mode

The mode is the number that appears most in any given distribution, that is, the number with the greatest frequency. When a series has more than one mode, it is called bi-modal (two modes) or tri-modal (three modes). For grouped data, the formula is:

Mode = L + [D1 ÷ (D1 + D2)] × c

Where:

• L = lower class boundary of the modal class
• D1 = frequency of the modal class minus the frequency of the class before it
• D2 = frequency of the modal class minus the frequency of the class after it
• c = width of the modal class

Worked example: mean, median and mode

The table below shows the marks of JSS 3 mathematics students.

Use the information above to calculate:

1. the mean
2. the median
3. the mode

Solution 1: the mean

Total: ∑f = 27 and ∑fx = 506

Mean = ∑fx ÷ ∑f = 506 ÷ 27 = 18.7

Solution 2: the median

N/2 = 27/2 = 13.5, so the median class is 16-20.

• L = 15.5
• Fb = 9
• f = 5
• c = 5

Median = 15.5 + [(13.5 – 9) ÷ 5] × 5 = 15.5 + 4.5 = 20

Solution 3: the mode

The modal class is 26-30, because it has the greatest frequency.

• L = 25.5
• D1 = 7 – 6 = 1
• D2 = 7 – 0 = 7
• c = 5

Mode = 25.5 + [1 ÷ (1 + 7)] × 5 = 25.5 + 0.625 = 26.125

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. They include the range, mean deviation, variance, standard deviation and coefficient of variation.

Range

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

Range = Highest value – Lowest value

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

Variance

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

Variance = ∑f(x – x̄)² ÷ ∑f

Standard deviation

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

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

Note that the standard deviation is the positive square root of the variance.

Worked example: range, mean deviation, variance and standard deviation

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

Solution

Mean, x̄ = ∑fx ÷ ∑f = 2535 ÷ 50 = 50.7

1. Range = Highest value – Lowest value = 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. State the formula for the mean of grouped data and say what each symbol means.
2. What is a bi-modal distribution?
3. What is the difference between measures of central tendency and measures of dispersion?
4. State the relationship between the variance and the standard deviation.

Assignment

1. Write down the formulae for the median and the mode of grouped data, and explain each symbol.
2. Using the marks table in the first worked example, calculate the mean deviation.