Mathematics · Lesson 25

Mean, median, mode, range

JAMBWAECNECO

What the syllabus asks

JAMB: Statistics: representation of data, measures of location and dispersion

WAEC: Statistics (frequency tables, grouped data, histograms, frequency polygons, pie charts, mean, median, mode, cumulative frequency and ogive, quartiles, range, mean deviation, variance, standard deviation)

NECO: Statistics (frequency tables, grouped data, histograms, frequency polygons, pie charts, mean, median, mode, cumulative frequency and ogive, quartiles, range, mean deviation, variance, standard deviation)

What you will learn

Build frequency tables and draw histograms, pie charts and bar charts.
Calculate the mean, median and mode of raw, tabulated and grouped data.
Use an assumed mean to simplify calculations.
Draw an ogive and read off the median, quartiles and percentiles.
Calculate the range, mean deviation, variance and standard deviation.
Interpret statistics in real Nigerian contexts.

Key points

Mean: Σfx/Σf; for grouped data use class marks; the assumed mean A gives x̄ = A + Σfd/Σf.
Median: middle value of ordered data; grouped: L + [(N/2 − F)/f]c.
Mode: most frequent value; modal class for grouped data.
Pie chart angle = f/total × 360°.
Ogive: c.f. against upper class boundary; median at N/2, Q₁ at N/4, Q₃ at 3N/4.
Range, mean deviation Σ|x − x̄|/n, variance and σ = √[Σf(x − x̄)²/Σf].
Adding a constant shifts the mean only; multiplying by c scales both the mean and σ.

Start here: Collecting and organising data

A class teacher in Ibadan records the number of siblings each of her 20 students has: 2, 3, 1, 4, 2, 2, 5, 3, 0, 2, 1, 3, 2, 4, 6, 2, 3, 1, 2, 3. As a list, these numbers say very little. Organised into a frequency table, they tell a story at a glance.

Go through the list once, making a tally mark for each value, then count the tallies.

Siblings (x)	0	1	2	3	4	5	6
Frequency (f)	1	3	7	5	2	1	1

Check: 1 + 3 + 7 + 5 + 2 + 1 + 1 = 20 ✓. Always check that the frequencies add up to the total. It takes five seconds and catches a missed tally.

Grouped data

When the data spread widely, such as exam scores from 0 to 100, we group them into classes like 40–49 and 50–59. Each class has a class mark (mid-point), such as (40 + 49)/2 = 44.5, and class boundaries 39.5 and 49.5, which close the gaps between classes.

Worked example 25.1

Find the class mark, boundaries and class width of the class 20–24.

Class mark: (20 + 24)/2 = 22.
Boundaries: 19.5 and 24.5.
Width: 24.5 − 19.5 = 5. Use the boundaries, not 24 − 20 = 4.

22; 19.5–24.5; 5

⚠️ Common mistake: class width = 24 − 20 = 4. The class 20–24 contains five whole numbers (20, 21, 22, 23 and 24), so the width is 5.

Try it yourself. From the siblings table, what fraction of the students have more than 3 siblings?

Show solution

2 + 1 + 1 = 4 out of 20 = 1/5.

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Practice questions

1. Find the mean of 12, 15, 11, 18, 14.

Show worked answer

70/5 = 14.

2. Find the median of 8, 3, 10, 5, 7, 12.

Show worked answer

3, 5, 7, 8, 10, 12, so (7 + 8)/2 = 7.5.

3. A sector of 54° represents ₦27,000 of a budget. Find the total budget.

Show worked answer

27,000 × 360/54 = ₦180,000.

4. Find the standard deviation of 4, 6, 8, 10, 12.

Show worked answer

x̄ = 8; Σd² = 16 + 4 + 0 + 4 + 16 = 40; σ = √8 ≈ 2.83.

5. The mean of 8 numbers is 12. When one number is removed, the mean becomes 11. Find the removed number.

Show worked answer

96 − 77 = 19.

Key words

Data: Facts or numbers collected for study.
Frequency (f): How many times a value occurs.
Class interval: A group of values such as 10–19.
Class mark (mid-point): The middle of a class, used as x for grouped data.
Class boundaries: The exact limits between classes, such as 9.5 and 19.5.
Mean (x̄): The sum of the values divided by the number of values.
Median: The middle value of ordered data.
Mode: The most frequent value.
Cumulative frequency: A running total of frequencies.
Ogive: The graph of cumulative frequency against upper class boundary.
Quartiles: Values that cut ordered data into four equal parts: Q₁, Q₂ (the median) and Q₃.
Interquartile range: Q₃ − Q₁.

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