Mathematics · Lesson 26

Statistics II: dispersion, grouped data, charts, ogive

JAMBWAECNECO

What the syllabus asks

JAMB: representation of data, measures of location and dispersion (range, mean deviation, variance, standard deviation), cumulative frequency

WAEC: statistics: frequency tables, grouped data, class boundaries, mean (including assumed mean), median and mode of grouped data, histograms, frequency polygons, pie charts, cumulative frequency curve (ogive), quartiles and percentiles, measures of dispersion: range, interquartile range, mean deviation, variance and standard deviation

NECO: statistics: frequency tables, grouped data, class boundaries, mean (including assumed mean), median and mode of grouped data, histograms, frequency polygons, pie charts, cumulative frequency curve (ogive), quartiles and percentiles, measures of dispersion: range, interquartile range, mean deviation, variance and standard deviation

What you will learn

Construct frequency tables for grouped data with class boundaries and mid-points.
Calculate the mean of grouped data directly and by an assumed mean.
Estimate the median and mode of grouped data by formula and from graphs.
Draw and interpret histograms, frequency polygons and pie charts.
Draw an ogive and read the median, quartiles, percentiles and interquartile range.
Calculate range, mean deviation, variance and standard deviation for ungrouped and grouped data.
Compare data sets using measures of location and spread.

Key points

Location (mean, median, mode) and dispersion (range, IQR, MD, σ) together describe data.
Class boundaries close gaps (10.5–20.5); width = boundary difference; mid-points used for calculation.
x̄ = Σfx/Σf, or A + Σfd/Σf with an assumed mean.
Median = L + ((n/2 − F)/f)c; mode = L + (d₁/(d₁ + d₂))c.
Histogram: no gaps, area ∝ frequency; use frequency density for unequal widths.
Pie chart angle = (f/total) × 360°; check the angles sum to 360°.
Ogive: cumulative frequency vs upper boundary; Q₁ at n/4, median n/2, Q₃ 3n/4.
IQR = Q₃ − Q₁; mean deviation = Σf|x − x̄|/Σf.
σ² = Σf(x − x̄)²/Σf = Σfx²/Σf − x̄²; σ = √σ².
Adding a constant leaves σ unchanged; multiplying by k multiplies σ by k.

Start here: Why we need more than an average

Two SS3 classes both score an average of 60% in a Mathematics test. Are they equally good? Not necessarily. In one class every student scored between 55 and 65; in the other, half scored above 90 and half below 30. An average alone hides this. We also need to know how spread out (dispersed) the data are.

Statistics therefore asks two questions about any data set:

Where is the centre? Measures of location (central tendency): mean, median, mode.
How spread out is it? Measures of dispersion: range, interquartile range, mean deviation, variance, standard deviation.

For large data sets, such as the ages of all patients at a clinic in Kaduna, or the marks of 400 candidates, we first organise the data into a grouped frequency table, then calculate from the table.

Worked example 26.1 (seeing spread)

Class A scores: 7, 8, 8, 9, 9 (mean 8.2). Class B scores: 2, 4, 9, 13, 14 (mean 8.4). Compare their ranges.

Range = highest − lowest.
A: 9 − 7 = 2. B: 14 − 2 = 12.
The averages are almost the same, but B is six times as spread out. Class A is consistent; class B has very strong and very weak students.

Ranges 2 and 12: B is far more dispersed

Statistics is used every day in Nigeria. The National Bureau of Statistics reports inflation and unemployment; hospitals track how long patients wait; WAEC analyses candidates' scores to set grade boundaries; a poultry farmer records how many eggs each pen lays per week. In each case, a single "average" is not enough: the farmer wants to know whether production is steady or wildly up and down, and the hospital wants to know not only the average wait but how many people wait far longer than average.

In this lesson we build a full toolkit, step by step, using one data set (the marks of 40 students) all the way through, so you can see how each measure relates to the others.

⚠️ Common mistake: concluding that two groups are "the same" because their means are equal. Always look at spread too.

Try it yourself. Find the range of 15, 22, 9, 31, 18.

Show solution

31 − 9 = 22.

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Practice questions

1. Find the mean of x = 2, 4, 6, 8 with f = 3, 7, 6, 4.

Show worked answer

Σfx = 6 + 28 + 36 + 32 = 102; Σf = 20; 5.1.

2. Find the variance and standard deviation of 3, 5, 7, 9, 11.

Show worked answer

x̄ = 7; Σd² = 16 + 4 + 0 + 4 + 16 = 40; σ² = 8; σ ≈ 2.83.

3. Find the mean deviation of 10, 12, 15, 19.

Show worked answer

x̄ = 14; |d| = 4, 2, 1, 5; MD = 3.

4. In a pie chart of 60 items, a sector is 72°. How many items does it show?

Show worked answer

72/360 × 60 = 12.

5. Classes 0–9, 10–19, 20–29 have f = 6, 10, 4. Find the median.

Show worked answer

n/2 = 10; class 10–19, L = 9.5, F = 6, f = 10, c = 10; 9.5 + 4 = 13.5.

Key words

Frequency: The number of times a value or class occurs.
Class interval: A group of values, e.g. 11–20.
Class boundaries: The true limits of a class, e.g. 10.5 and 20.5 for 11–20.
Class width: Upper boundary minus lower boundary.
Class mark (mid-point): The middle of a class, (lower + upper limit)/2.
Cumulative frequency: A running total of frequencies.
Ogive: The graph of cumulative frequency against upper class boundary.
Quartiles: Values dividing ordered data into four equal parts: Q₁, Q₂ (median), Q₃.
Percentile: A value below which a given percentage of the data lies.
Interquartile range: Q₃ − Q₁, the spread of the middle half of the data.
Mean deviation: The average distance of the values from the mean.
Variance: The mean of the squared deviations from the mean.

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