Basic Tools for Economic Analysis – SS2 Economics Lesson Note (First Term)

Measures of central tendency are statistical values that give the middle, centre or average of a set of data. They show where the values of a set gather around a central point. The three measures are the arithmetic mean, the median and the mode. This note gives the formula for each and works one example with grouped data.

Behavioural objectives

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

• state what measures of central tendency are;
• name the three measures of central tendency;
• state the formulas for the mean, median and mode of grouped data;
• calculate the mean, median and mode from a grouped frequency table.

What are measures of central tendency?

Measures of central tendency are the 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 middle, centre or average of a set of data. They include 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 (for grouped data, the class mid-point)

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 = summation 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 said to be bi-modal for two modes and tri-modal for three. 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: how do you find the mean, median and mode of grouped data?

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

Use the information above to calculate:

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

Solution 1: the mean

Mean = ∑fx ÷ ∑f = 506 ÷ 27 = 18.7 (to 1 decimal place)

Solution 2: the median

• L = 15.5
• N/2 = 27/2 = 13.5
• Fb = 9
• f = 5
• c = 5

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

Solution 3: the mode

• 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

Evaluation

1. What are measures of central tendency?
2. Name the three measures of central tendency.
3. State the formula for the mean of grouped data and say what each symbol stands for.
4. What is meant by a bi-modal series?
5. In the worked example, which class is the median class, and why?

Assignment

1. State the formulas for the median and the mode of grouped data and explain each symbol.
2. Using the frequency table in the worked example, find the number of students who scored 16 marks or more.