Location: Measurement of Central Tendency – SS1 Further Mathematics Lesson Note (Third Term)

Measures of central tendency show how data are centrally placed. The three commonest are the arithmetic mean, median and mode. The mean is the sum of all items divided by the number of items. The median is the middle value of ordered data. The mode is the value that occurs most often.

Behavioural objectives

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

• State what is meant by the mean, median and mode.
• Calculate the mean, median and mode of ungrouped data.
• Calculate the mean of grouped data using the class mark method and the assumed mean method.
• State the formulae for the mode and median of grouped data by interpolation, and explain each symbol.

What are measures of central tendency?

A measure of central tendency shows how the data are centrally placed. The three commonest measures of location, depending on the information required, are the arithmetic mean, median and mode.

How do you find the mean, mode and median of ungrouped data?

Mean

The mean is the most widely used measure and is sometimes called the arithmetical average. The mean of the numbers x₁, x₂, x₃, x₄, …, x_n is given by:

Mean (x̄) = ∑x/n

where ∑x is the sum of all items and n is the number of items.

When the data involve frequency, mean = ∑fx/∑f.

Example 1

Calculate the mean of the numbers 15, 17, 19, 21, 23, 25, 27, 29.

Solution:

Mean (x̄) = (15 + 17 + 19 + 21 + 23 + 25 + 27 + 29) ÷ 8 = 176/8 = 22

Example 2

The table shows the number of suitcases possessed by a group of travellers.

Calculate the mean to the nearest whole number.

Solution:

Mean (x̄) = ∑fx/∑f = 84/30 = 2.8, which is 3 to the nearest whole number.

Mode

The mode of a distribution is the value of the variable which occurs most often in the distribution. A distribution can have more than one mode if more than one item has the highest frequency.

Example

Find the mode of the data 5, 4, 8, 9, 6, 8, 9, 3, 8.

The mode is 8, because it appears 3 times, more often than any other value.

Median

The median is the middle value of a set of data when the data are arranged in ascending or descending order. When there is an even number of items, the median is the mean of the two middle values.

Example

Find the median of these numbers:

1. 35, 28, 42, 28, 56, 70, 35
2. 18, 20, 25, 30, 22, 25, 28, 15

Solution:

(1) Rearranging the numbers: 70, 56, 42, 35, 35, 28, 28. The median is 35.

(2) Rearranging the numbers: 15, 18, 20, 22, 25, 25, 28, 30.

Median = (22 + 25)/2 = 47/2 = 23.5

How do you find the mean, mode and median of grouped data?

Mean of grouped data

The arithmetic mean of a grouped frequency distribution can be obtained using either of two methods.

1. Class mark method: x̄ = ∑fx/∑f, where x is the midpoint (class mark) of the class interval.
2. Assumed mean method (also called the working mean method): x̄ = A + (∑fd/∑f), where d = x – A, x = class mark and A = assumed mean.

Example

The numbers of matches in 100 boxes are counted and the results are shown in the table below.

Calculate the mean (i) using the class mark method and (ii) using the assumed mean method, given that the assumed mean is 30.5.

Solution:

1. Class mark method: x̄ = ∑fx/∑f = 3214/100 = 32.14, which is 32 matches per box to the nearest whole number.
2. Assumed mean method: x̄ = A + (∑fd/∑f) = 30.5 + (164/100) = 30.5 + 1.64 = 32.14, which is 32 matches per box to the nearest whole number.

Mode of grouped data

The mode of a grouped frequency distribution can be found geometrically (from a histogram) and by the interpolation method.

1. Mode from a histogram: The highest bar is the modal class. Draw a straight line from the top right corner of the modal bar to the top right corner of the adjacent bar on the left. Draw another line from the top left corner of the modal bar to the top left corner of the adjacent bar on the right.
2. Mode from interpolation: The mode can be obtained using the formula:

Mode = L_m + [Δ₁/(Δ₁ + Δ₂)] × C

• L_m = lower class boundary of the modal class.
• Δ₁ = difference between the frequency of the modal class and the class before it.
• Δ₂ = difference between the frequency of the modal class and the class after it.
• C = class width of the modal class.

Median of grouped data from the interpolation formula

Median = L₁ + [(N/2 – Cf_m)/f_m] × C

• L₁ = lower class boundary of the median class.
• Cf_m = cumulative frequency of the class before the median class.
• f_m = frequency of the median class.
• C = class width of the median class.
• N = total frequency.

Evaluation

1. Define the mean, the median and the mode.
2. From the suitcase table, which number of suitcases is the mode?
3. When does a distribution have more than one mode?
4. State the formula for the mean of grouped data by the assumed mean method and explain each symbol.
5. State the interpolation formula for the mode and explain Δ₁, Δ₂ and C.

Assignment

1. Write down the interpolation formula for the median of grouped data and explain each symbol.
2. From the table of matches in 100 boxes, state the modal class and its frequency.
3. Explain how to find the median of a set of data with an even number of items.