Measures of Spread

Key Terms

Range: Maximum − minimum; simple but sensitive to outliers.
Quartiles: Q1 (25th percentile), Q2 = median (50th), Q3 (75th percentile); divide ordered data into four equal groups.
Interquartile range (IQR): IQR = Q3 − Q1; measures the spread of the middle 50% of data; resistant to outliers.
Standard deviation (s): Average distance of values from the mean; s = √[∑(x − &x̄)² / (n − 1)]. Sensitive to outliers.
Variance: s²; the square of the standard deviation. Measured in squared units.
Outlier rule: A value is an outlier if it is more than 1.5 × IQR below Q1 or above Q3.

Measuring the Spread of Data

Range = max − min    (simplest, most affected by outliers)

Interquartile Range (IQR) = Q3 − Q1    (spread of the middle 50%)

Outlier fences (IQR method):
• Lower fence = Q1 − 1.5 × IQR
• Upper fence = Q3 + 1.5 × IQR
Any value outside these fences is an outlier.

Standard deviation s = √[ Σ(x − x̅)² / (n − 1) ]    (QCAA Formula Sheet)
Measures the average distance of each value from the mean.

Finding Quartiles

Quartile	Meaning	How to find
Q1 (25th percentile)	Lower quartile — 25% of data below this	Median of the lower half of data
Q2 (50th percentile)	Median	Middle value of ordered data
Q3 (75th percentile)	Upper quartile — 75% of data below this	Median of the upper half of data

Worked Example 1 — IQR and Outlier Detection

Dataset: 12, 15, 17, 18, 20, 23, 24, 27, 60

Step 1: Order the data (already ordered). n = 9.

Step 2: Find Q1 and Q3
Lower half (below median 20): 12, 15, 17, 18 → Q1 = (15+17)/2 = 16
Upper half (above median 20): 23, 24, 27, 60 → Q3 = (24+27)/2 = 25.5

Step 3: IQR = 25.5 − 16 = 9.5

Step 4: Fences
Lower = 16 − 1.5(9.5) = 16 − 14.25 = 1.75
Upper = 25.5 + 1.5(9.5) = 25.5 + 14.25 = 39.75

Conclusion: 60 > 39.75, so 60 is an outlier. All other values are within fences.

Worked Example 2 — Standard Deviation

Dataset: 3, 5, 7, 7, 8 (n = 5)

Step 1: Mean = (3+5+7+7+8)/5 = 30/5 = 6

Step 2: Find each deviation squared:
(3−6)² = 9, (5−6)² = 1, (7−6)² = 1, (7−6)² = 1, (8−6)² = 4

Step 3: Sum of squared deviations = 9+1+1+1+4 = 16

Step 4: s = √(16/(5−1)) = √(16/4) = √4 = 2

On average, values in this dataset are 2 units from the mean of 6.

Hot Tip: The IQR is resistant to outliers — an extreme value cannot change Q1 or Q3 much. But the range and standard deviation are both heavily affected by outliers. When data has extreme values, prefer reporting IQR over range for spread, and median over mean for centre.

Full Lesson: Measures of Spread

Why Measure Spread?

Two datasets can have identical means but be completely different. Consider Test A scores: 68, 70, 70, 72 (mean = 70) vs Test B scores: 40, 55, 90, 95 (mean = 70). Both classes averaged 70%, but the performance in Test B was far more varied. Measures of spread quantify this difference.

The Five-Number Summary

The complete picture of a dataset's spread is captured by the five-number summary: minimum, Q1, median (Q2), Q3, maximum. These five values are used to draw a box plot (box-and-whisker plot).

Box plot structure:
• Box spans from Q1 to Q3 (contains middle 50% of data)
• Line inside box = median
• Whiskers extend to the smallest and largest non-outlier values
• Outliers are plotted as individual dots beyond the whiskers

Standard Deviation in Context

Standard deviation is the most informative measure of spread but the most complex to calculate. It measures the "average distance" each data value lies from the mean. A small SD means data is tightly packed; a large SD means data is widely spread.

Real-world examples:
• Manufacturing: small SD in bolt diameters means consistent quality
• Finance: large SD in share prices means high-risk investment
• Exams: SD of 5 means most students scored within 5 marks of the mean

Choosing the Right Measure

Use IQR when data has outliers or is skewed (pairs with median). Use standard deviation when data is roughly symmetric with no extreme outliers (pairs with mean). Always state which measure you are reporting and why.

Lesson Tip: When calculating quartiles by hand for small datasets, always order the data first, find the median to split the data into two halves, then find the median of each half. For even-sized datasets, the two halves do not include the median value.

Mastery Practice

See Answers ➔

Fluency
Find the range and IQR for the dataset: 3, 7, 8, 12, 14, 15, 18, 22, 25
Fluency
Find Q1, Q2 (median), Q3 and IQR for: 5, 9, 11, 13, 15, 17, 19, 23
Fluency
Calculate the standard deviation (to 2 d.p.) of: 2, 4, 4, 4, 5, 5, 7, 9
Hint: the mean is 5.
Fluency
A dataset has Q1 = 20 and Q3 = 35. (a) Find the IQR. (b) Calculate the lower and upper outlier fences.
Understanding
Heights (cm) of 9 basketball players: 172, 175, 178, 180, 182, 183, 185, 188, 220. Use the IQR outlier method to determine whether 220 cm is an outlier. Show all working.
Understanding
Two classes sat the same test. Class A has mean 70 and SD = 5. Class B has mean 70 and SD = 15. Interpret what each standard deviation tells you about the students in each class.
Understanding
Dataset: 10, 12, 13, 15, 16, 18, 45. Calculate the range and IQR. Which measure better represents the spread of the central data? Justify your answer.
Understanding
The five-number summary for a dataset is: Min = 8, Q1 = 15, Median = 22, Q3 = 30, Max = 45. (a) Find the IQR. (b) Use the fence test to check if there are any outliers. (c) Describe the shape of the distribution (use the median’s position relative to Q1 and Q3).
Problem Solving
Two investment portfolios both have a mean annual return of 8%. Portfolio A has SD = 2%, Portfolio B has SD = 9%. Explain which portfolio is more suitable for: (a) a retiree needing stable income, (b) a young investor comfortable with risk. Give reasons.
Problem Solving
A quality control inspector measures 8 bolts (lengths in mm): 50.1, 49.8, 50.2, 50.0, 49.9, 50.3, 49.7, 50.0.
1. Calculate the mean and standard deviation.
2. Bolts outside x̅ ± 2s are rejected. Identify any rejected bolts.
3. If the acceptable tolerance were tightened to x̅ ± 1s, how many bolts would be rejected?