Measures of Spread
Two classes both have a mean test score of 75. But in Class A every student scored between 70 and 80. In Class B scores ranged from 40 to 100. The mean alone hides this crucial difference. Measures of spread tell us how varied or consistent data is — whether values cluster tightly around the centre or are widely scattered.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
Class X scores: 70, 72, 75, 78, 80. Class Y scores: 50, 65, 75, 85, 100. Both have mean = 75. Which class performed more consistently? How would you measure this?
Before reading on — write your gut feeling. We will revisit this at the end of the lesson.
Three measures of spread are tested in HSC Maths Standard. Each captures something different about how data is distributed.
Range: $R = \text{max} - \text{min}$ — simple, but heavily affected by outliers.
IQR: $IQR = Q_3 - Q_1$ — spread of the middle 50%; robust to outliers.
Standard deviation: $s = \sqrt{\dfrac{\sum(x - \bar{x})^2}{n-1}}$ (sample) — average distance from the mean; uses every value.
Key facts
- Range, IQR, standard deviation
- Quartiles and percentiles
- Sample vs population SD formulas
Concepts
- Why spread matters alongside centre
- How outliers affect different measures
- When to use each measure of spread
Skills
- Calculate range, IQR, standard deviation
- Find quartiles from ordered data
- Compare consistency of data sets
The range is the difference between the maximum and minimum values:
$$R = \text{max} - \text{min}$$Example: 12, 15, 18, 22, 30 → Range = 30 − 12 = 18
What to write in your book
- Range = max − min. Quick but unreliable when outliers are present.
- Only uses two data values — completely ignores all values in between.
- A large range does not always mean data is widely spread throughout.
Quick check: For the data set 3, 7, 7, 8, 10, 25, what is the range?
Quartiles divide ordered data into four equal parts:
- Q1 (Lower quartile): Median of the lower half = 25th percentile
- Q2 (Median): 50th percentile
- Q3 (Upper quartile): Median of the upper half = 75th percentile
Example: 8, 12, 15, 18, 20, 22, 25, 30
- Median = (18 + 20) ÷ 2 = 19
- Lower half: 8, 12, 15, 18 → Q1 = (12 + 15) ÷ 2 = 13.5
- Upper half: 20, 22, 25, 30 → Q3 = (22 + 25) ÷ 2 = 23.5
- IQR = 23.5 − 13.5 = 10
What to write in your book
- Order the data first. Find the median (Q2), then find the medians of the lower and upper halves.
- $IQR = Q_3 - Q_1$. Shows spread of the middle 50%.
- IQR is robust — outliers do not change it.
True or false: The IQR is affected by outliers in the data set.
Standard deviation measures how far data values typically are from the mean. There are two versions:
$$\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}} \quad \text{(population)}$$ $$s = \sqrt{\frac{\sum(x - \bar{x})^2}{n-1}} \quad \text{(sample)}$$Steps to calculate:
- Find the mean $\bar{x}$
- Find each deviation: $(x - \bar{x})$
- Square each deviation: $(x - \bar{x})^2$
- Sum all squared deviations
- Divide by $n$ (population) or $n-1$ (sample) to get the variance
- Take the square root
Example: 5, 7, 9
Mean = 7. Deviations: −2, 0, 2. Squared: 4, 0, 4. Sum = 8.
Population SD: $\sigma = \sqrt{8/3} \approx 1.63$. Sample SD: $s = \sqrt{8/2} = 2.00$.
What to write in your book
- SD = average distance from the mean. Never negative.
- Population formula divides by $n$; sample formula divides by $n - 1$.
- Larger SD = more variable data. Smaller SD = more consistent data.
Fill the gap: For data 4, 6, 8, 10, 12, the mean is 8. The sum of squared deviations is 40. The population standard deviation is (to 2 d.p.).
Worked examples · reveal each step
Weekly sales ($): 1200, 1350, 1400, 1450, 1500, 1550, 1600, 2500. Find the range, Q1, Q3 and IQR.
For the data 10, 12, 15, 18, 20, find the population standard deviation.
Common errors · the 3 traps that cost marks
What to write in your book
- Always order data before finding quartiles.
- Range is dramatically affected by outliers; IQR is not.
- When comparing two data sets with same mean, the one with smaller SD is more consistent.
Match each measure to its description:
Quick-fire practice · 2 activities
For the data set 5, 8, 10, 12, 15, 18, 20, 25: find the range, Q1, Q3 and IQR. Then calculate the population SD for 4, 6, 8, 10, 12.
Two data sets both have mean = 50. Set A: range = 10, SD = 3. Set B: range = 40, SD = 15. Which is more consistent? Why is IQR often preferred over range when comparing salaries across companies?
Top 3 list: Name THREE real-world situations where knowing the spread of data is just as important as knowing the average. For each, state which measure of spread you would use and why.
Class X is more consistent. The scores range from 70 to 80 (range = 10), while Class Y ranges from 50 to 100 (range = 50). Measuring with standard deviation confirms this: Class X SD ≈ 3.16, Class Y SD ≈ 18.71. The much smaller SD for Class X shows that its scores cluster tightly around the mean, while Class Y has wide variation. Both classes have the same average ability, but Class X is far more uniform in performance.
What has changed in your understanding? What did you get right? What surprised you?
Pick your answer, then rate your confidence — that tells the system what to drill next.
Q1. For the data set 8, 12, 15, 18, 20, 22, 25, 30, what is the IQR?
Q2. Which measure of spread is most resistant to the effect of outliers?
Q3. For data set 5, 7, 9 the population standard deviation is approximately:
Q4. Two classes both have mean = 70. Class A has SD = 4; Class B has SD = 12. Which statement is correct?
Q5. The sample SD formula uses $n-1$ in the denominator (rather than $n$) because:
SA 1. For the data set 8, 12, 15, 18, 20, 22, 25, 30, 35, 50: (a) Find range, Q1, Q3 and IQR. (b) Find the mean and population standard deviation. (c) Remove the outlier (50) and recalculate range and IQR. Which measure changes more? (2 marks)
SA 2. Two basketball players both average 15 points per game. Player A scores: 12, 14, 15, 16, 18. Player B scores: 5, 10, 15, 20, 25. (a) Calculate the range and population standard deviation for each player. (b) Which player is more consistent? (c) Which player would you want in a close final, and why? (2 marks)
SA 3. A company reports employee salaries with mean = $80K and median = $55K. (a) What does this tell you about the shape of the salary distribution? (b) SD = $45K and IQR = $20K — which measure of spread is more useful for a prospective employee, and why? (c) Discuss how reporting different combinations of centre and spread can create very different impressions of the same data. (3 marks)
Comprehensive answers (click to reveal)
MC 1 — B: Q1 = (12+15)/2 = 13.5; Q3 = (22+25)/2 = 23.5; IQR = 10.
MC 2 — C: IQR only uses the middle 50% of data, so extreme values do not affect it.
MC 3 — A: Mean = 7; deviations: −2, 0, 2; squared: 4, 0, 4; sum = 8; variance = 8/3; SD = √2.67 ≈ 1.63.
MC 4 — D: Same mean but Class A has a smaller SD, so its scores are clustered more tightly — more consistent.
MC 5 — B: Using n−1 (Bessel's correction) produces an unbiased estimate of the population variance from a sample.
SA 1 (2 marks): (a) Range = 42; Median = 21; Q1 = 15; Q3 = 30; IQR = 15 [0.5]. (b) Mean = 23.5; SD ≈ 11.69 [0.5]. (c) Without 50: range = 27 (drops by 15); IQR = 15 (unchanged). Range changes far more — IQR is robust to outliers [1].
SA 2 (2 marks): (a) A: range = 6, SD = 2; B: range = 20, SD = 7.07 [0.5]. (b) Player A — smaller range and SD [0.5]. (c) Player A for reliability; Player B if you need a potential 25-point game but accept the risk [1].
SA 3 (3 marks): (a) Mean > median indicates right skew — few high earners pull mean above the typical salary [1]. (b) IQR ($20K) is more useful — SD is inflated by executive outliers [0.5]. (c) Mean + SD makes variation and centre appear large; median + IQR gives a more modest picture. Different choices serve different narratives [1.5].
Five timed questions on range, IQR, standard deviation and choosing the right measure. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms using range, IQR and standard deviation. Pool: lesson 3.
Mark lesson as complete
Tick when you've finished the practice and review.