Further Mathematics · Lesson 16

Statistics: dispersion, correlation & regression

WAECNECO

What the syllabus asks

WAEC: Statistics: measures of location and dispersion for ungrouped and grouped data (range, quartiles, interquartile and semi-interquartile range, mean deviation, variance, standard deviation, coefficient of variation), use of assumed mean and coding; correlation (scatter diagrams, Spearman’s rank correlation coefficient, product-moment correlation), regression line of y on x by least squares and its…

NECO: Statistics: measures of location and dispersion for ungrouped and grouped data (range, quartiles, interquartile and semi-interquartile range, mean deviation, variance, standard deviation, coefficient of variation), use of assumed mean and coding; correlation (scatter diagrams, Spearman’s rank correlation coefficient, product-moment correlation), regression line of y on x by least squares and its…

What you will learn

Calculate range, quartiles, interquartile and semi-interquartile range for ungrouped and grouped data.
Calculate mean deviation, variance and standard deviation for ungrouped and grouped data.
Use an assumed mean and coding to simplify calculations.
State the effect of adding or multiplying by a constant on the mean and standard deviation, and compare data with the coefficient of variation.
Draw and interpret scatter diagrams.
Calculate and interpret Spearman’s rank correlation coefficient and the product-moment correlation coefficient.
Find and use the least-squares regression line of y on x.

Key points

Dispersion measures spread: range, IQR, mean deviation, variance, standard deviation.
IQR = Q₃ − Q₁; semi-IQR = ½(Q₃ − Q₁); grouped quartiles: L + [(pos − F)/f]c.
Mean deviation = Σ|x − x̄|/n.
σ² = Σ(x − x̄)²/n = Σx²/n − x̄²; σ = √σ².
Grouped: use mid-points; coding d = (x − A)/c, σ = c√[Σfd²/Σf − (Σfd/Σf)²].
Adding k: σ unchanged; multiplying by k: σ × |k|, variance × k².
CV = σ/x̄ × 100% compares relative spread.
Scatter diagrams show positive, negative or no correlation; correlation is not causation.
Spearman: rₛ = 1 − 6Σd²/[n(n² − 1)]; Pearson r from deviations; −1 ≤ r ≤ 1.
Regression y on x: b = Sxy/Sxx, a = ȳ − bx̄; passes through (x̄, ȳ); avoid extrapolation.

Start here: Why dispersion matters

Two SS3 classes both have a mean mark of 60% in a mock exam. In class A, every student scored between 55 and 65. In class B, marks ranged from 20 to 98. The mean alone hides a huge difference: class B has students in serious danger and others ready for distinctions. A school principal planning extra lessons needs to know the spread as well as the average.

Measures of dispersion describe how spread out data are around a central value. The main ones are the range, interquartile range, mean deviation, variance and standard deviation.

The range (highest − lowest) is quick but depends only on two extreme values, so one unusual result distorts it. The interquartile range ignores the extreme quarters. The standard deviation uses every value and is the most important measure in advanced work.

Worked example 16.1

Find the range of 23, 41, 17, 35, 29, 52.

Highest = 52, lowest = 17.
Range = 52 − 17 = 35.

Try it yourself. Give one disadvantage of the range.

Show solution

It uses only the two extreme values, so it is strongly affected by outliers and ignores the rest of the data.

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Practice questions

1. Find the mean deviation of 3, 5, 7, 9, 11.

Show worked answer

Mean 7; |dev| 4, 2, 0, 2, 4 = 12; 2.4.

2. Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

Show worked answer

Mean 5; Σ(x − 5)² = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32; variance 4; σ = 2.

3. Data have mean 20 and σ = 3. Each value is multiplied by 2 and 1 is subtracted. Find the new mean and σ.

Show worked answer

Mean 39; σ 6.

4. With A = 25, c = 10, Σf = 50, Σfd = 10, Σfd² = 70, find x̄ and σ.

Show worked answer

x̄ = 25 + 10(0.2) = 27; σ = 10√(1.4 − 0.04) = 10 × 1.166 = 11.7.

5. For 8 pairs, Σd² = 21. Find Spearman’s rₛ.

Show worked answer

1 − 126/504 = 0.75.

Key words

Dispersion: The spread or scatter of data about a central value.
Range: Highest value minus lowest value.
Quartiles: Values dividing ordered data into four equal parts: Q₁, Q₂ (median), Q₃.
Interquartile range: Q₃ − Q₁; the spread of the middle half of the data.
Semi-interquartile range: ½(Q₃ − Q₁), also called the quartile deviation.
Mean deviation: The mean of the absolute deviations from the mean, Σ|x − x̄|/n.
Variance: The mean of the squared deviations from the mean.
Standard deviation: The square root of the variance.
Coefficient of variation: σ/x̄ × 100%; compares spread relative to the mean.
Coding: Transforming data by d = (x − A)/c to simplify calculation.
Scatter diagram: A graph of paired data points used to see correlation.
Correlation: A measure of how closely two variables are linearly related.

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