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

A measure of central tendency is a single value that shows how a set of data is centrally placed.

Behavioural objectives

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

1. State what is meant by the mean, median and mode.
2. Calculate the mean, median and mode of ungrouped data.
3. Calculate the mean of grouped data using the class mark method and the assumed mean method.
4. State the formulae for the mode and median of grouped data by interpolation, and explain each symbol.
5. Use the interpolation formulae to find the mode and median of a grouped frequency distribution.

What are measures of central tendency?

Measures of central tendency are values that show where the centre of a set of data lies.

The three commonest measures of location are the arithmetic mean, the median and the mode. The one used depends on the information required.

• Mean: the sum of all items divided by the number of items.
• Median: the middle value of data arranged in order.
• Mode: the value that occurs most often.

How do the three measures compare?

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

Ungrouped data are raw values listed one by one, or listed with their frequencies.

What is the mean?

The mean is the sum of all the items divided by the number of items. It is sometimes called the arithmetical average.

• Raw data: mean (x̄) = ∑x/n, for the numbers x₁, x₂, x₃, …, x_n.
• ∑x: the sum of all items.
• n: the number of items.
• Data with frequency: mean (x̄) = ∑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.

What is the mode?

The mode is the value of the variable that occurs most often in a distribution.

• One mode: one value has the highest frequency.
• More than one mode: two or more values share the highest frequency.

Example

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

Solution:

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

What is the median?

The median is the middle value of a set of data arranged in ascending or descending order.

1. Arrange the data in order of size.
2. Count the number of items, n.
3. If n is odd, the median is the middle item, which is the ((n + 1)/2)th value.
4. If n is even, the median is the mean of the two middle items.

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?

Grouped data are values arranged in class intervals with a frequency for each class.

How do you find the mean of grouped data?

The mean of a grouped frequency distribution is found by either the class mark method or the assumed mean method.

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

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.

How do you find the mode of grouped data?

The mode of grouped data is found from a histogram or by the interpolation formula.

Mode from a histogram

1. Draw the histogram. The highest bar is the modal class.
2. 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.
3. Draw another line from the top left corner of the modal bar to the top left corner of the adjacent bar on the right.
4. Drop a vertical line from the point where the two lines cross to the horizontal axis.
5. Read the mode at the foot of that vertical line.

Mode from the interpolation 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.

Example

Find the mode of the matches data above.

Solution:

1. The highest frequency is 37, so the modal class is 33 – 36.
2. The class boundaries are 32.5 and 36.5, so L_m = 32.5.
3. Δ₁ = 37 – 34 = 3 and Δ₂ = 37 – 11 = 26.
4. C = 36.5 – 32.5 = 4.
5. Mode = 32.5 + [3/(3 + 26)] × 4 = 32.5 + 12/29 = 32.5 + 0.41 = 32.91 (2 d.p.).

How do you find the median of grouped data?

The median of grouped data is found from the interpolation formula, using the cumulative frequency.

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.

Example

Find the median of the matches data above.

Solution:

1. N = 100, so N/2 = 50.
2. The 50th item falls in the class 29 – 32, because the cumulative frequency reaches 52 there. This is the median class.
3. L₁ = 28.5, Cf_m = 18, f_m = 34 and C = 4.
4. Median = 28.5 + [(50 – 18)/34] × 4 = 28.5 + 128/34 = 28.5 + 3.76 = 32.26 (2 d.p.).

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.