Measures of dispersion: range, interquartile range, standard deviation

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Statistics – Measures of Dispersion (IGCSE Mathematics 0580)

Quick note before we dive in
This checklist helps you verify that your lecture‑notes cover everything required by the Cambridge IGCSE 0580 syllabus. Use it to audit each topic and spot any gaps in depth or structure.
Syllabus unit Core (C‑G) ✔︎ Extended (A*‑C) ✔︎ Key points to check
9 Measures of Dispersion ✓ ✓
  • Range – definition, calculation, interpretation.
  • IQR – quartile method, box‑plot construction, outlier rules.
  • Standard deviation – population & sample formulas, step‑by‑step example (optional).
Scope for the 2025‑27 syllabus
  • Core – Range only (C9.3).
  • Extended – Range + Inter‑quartile range (IQR) and the ability to draw & interpret a box‑plot (C9.3, C9.5).
  • Optional (teacher‑resource) – Standard deviation (useful for deeper understanding but not required for the exam).

1. Range (Core)

  • Definition: Range = Maximum value – Minimum value
  • Gives a quick impression of the total spread of the data set.
  • Very sensitive to extreme values (outliers).

How to calculate the range

  1. Identify the largest (maximum) and smallest (minimum) observations.
  2. Subtract the minimum from the maximum.

Worked example – Range

Data (test scores): 62, 70, 75, 80, 85, 90, 95

  • Maximum = 95
  • Minimum = 62
  • Range = 95 − 62 = 33

2. Inter‑quartile Range (IQR) (Extended)

The IQR measures the spread of the middle 50 % of the data. It is the difference between the third quartile (Q₃) and the first quartile (Q₁).

IQR = Q₃ – Q₁

Steps to find the quartiles

  1. Arrange the data in ascending order.
  2. Find the median – this is the second quartile (Q₂).
  3. Split the ordered list into a lower half and an upper half:
    • If the number of observations is odd, exclude the median when forming the halves.
    • If the number is even, split directly down the middle.
  4. Q₁ = median of the lower half.
  5. Q₃ = median of the upper half.

Worked example – Quartiles & IQR

Data (test scores): 62, 70, 75, 80, 85, 90, 95

  1. Ordered list: 62, 70, 75, 80, 85, 90, 95
  2. Median (Q₂) = 80
  3. Lower half (exclude Q₂) = 62, 70, 75 → Q₁ = median = 70
  4. Upper half = 85, 90, 95 → Q₃ = median = 90
  5. IQR = 90 − 70 = 20

Box‑plot (C9.5)

  • The box‑plot displays Q₁, Q₂ (median), Q₃ and “whiskers” that reach the minimum and maximum (or the fences if outliers are shown).
  • Interpretation cues:
    • Length of the box = IQR (spread of the central 50 %).
    • Short whiskers relative to the box → low overall spread; long whiskers → high spread.
    • Points beyond the whiskers are plotted individually as outliers.

Box‑plot sketch (hand‑drawn in the exam)

Simple box‑plot sketch
Box‑plot showing Q₁ = 70, Q₂ = 80, Q₃ = 90, whiskers to the minimum (62) and maximum (95).

3. Standard Deviation (Optional – teacher resource)

The standard deviation measures the average distance of each observation from the mean. It provides a more detailed picture of spread than the range or IQR.

Formulas

  • Population (size = \(n\), mean = \(\mu\)): $$\displaystyle \sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\mu)^2}{n}}$$
  • Sample (size = \(n\), mean = \(\bar{x}\)) – use Bessel’s correction: $$\displaystyle s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}$$

Step‑by‑step calculation (sample)

Step Action Result
1 Calculate the sample mean \(\bar{x}\). \(\displaystyle\bar{x}= \frac{\sum x_i}{n}\)
2 Find each deviation \((x_i-\bar{x})\). List of deviations
3 Square each deviation. \((x_i-\bar{x})^2\)
4 Add all squared deviations. \(\displaystyle\sum (x_i-\bar{x})^2\)
5 Divide by \(n-1\) → sample variance \(s^2\). \(\displaystyle s^2 = \frac{\sum (x_i-\bar{x})^2}{n-1}\)
6 Take the square root → \(s\). \(\displaystyle s = \sqrt{s^2}\)

Worked example – Sample standard deviation

Data (test scores): 62, 70, 75, 80, 85, 90, 95

Mean \(\displaystyle\bar{x}= \frac{62+70+75+80+85+90+95}{7}=78.14\) (2 d.p.)

\(x_i\) \(x_i-\bar{x}\) \((x_i-\bar{x})^2\)
62‑16.14260.5
70‑8.1466.3
75‑3.149.9
801.863.5
856.8647.0
9011.86140.7
9516.86284.4
Sum of squares 812.3

Sample variance \(s^2 = \dfrac{812.3}{7-1}=135.38\)

Sample standard deviation \(s = \sqrt{135.38}=11.63\) (2 d.p.)

When to use standard deviation (optional)

  • When you need a measure that reflects how tightly the data cluster around the mean.
  • Useful for comparing the spread of two different data sets with similar means.
  • Not required for the IGCSE exam, but valuable for deeper statistical insight.

Summary of Key Points

Measure Formula / Procedure When to use
Range (Core) Maximum – Minimum Quick estimate of total spread; avoid if data contain outliers.
IQR (Extended) Q₃ – Q₁ (see quartile steps) Robust measure of central spread; essential for box‑plots and when outliers are present.
Standard deviation (Optional) Population: \(\sigma =\sqrt{\frac{\sum (x_i-\mu)^2}{n}}\)
Sample: \(s =\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}\) 		Provides an average distance from the mean; useful for deeper analysis or comparing different data sets.

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