9.2Measures of Spread
Calculate and interpret range, IQR, MAD, and standard deviation. Compare data sets using measures of variability and understand how spread describes a distribution.
Why This Matters
Measures of spread tell you how consistent or variable a data set is — critical in quality control, medical research, and finance. Standard deviation and IQR are foundational concepts in AP Statistics and data science.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do measures of spread describe the variability of a data set, and why does it matter which measure we choose?
Lesson Overview
While measures of center (mean, median, mode) tell us where data is located, measures of spread tell us how much the data varies. Two data sets can have the same mean but look completely different — one tightly clustered, the other widely scattered. The four main measures of spread are range (the total span), IQR (the spread of the middle 50%), MAD (average distance from the mean), and standard deviation (the most precise measure of typical distance from the mean).
Measures of Spread — Formulas & Steps
Range
Range = Maximum − Minimum
Identify max → identify min → subtract.
IQR
IQR = Q3 − Q1
Order data → find median (Q2) → Q1 = median of lower half → Q3 = median of upper half → subtract.
Outlier rule: below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.
MAD
MAD = (Σ |xᵢ − x̄|) ÷ n
Find mean → subtract mean from each value → take absolute value → sum → divide by n.
Standard Deviation (conceptual)
Similar to MAD but uses squared differences. Larger value = more spread. At Algebra 1 level, focus on interpreting and comparing, not computing by hand.
Range — Total Span of Data
Box Plot — IQR (Middle 50%)
2Q1
6Med
10Q3
15Max
20
Effect of an Outlier on Measures of Spread
| Data Set | Range | IQR | MAD |
|---|---|---|---|
| 5, 7, 8, 9, 10, 11 | 6 | 3 | 1.5 |
| 5, 7, 8, 9, 10, 50 | 45 ↑↑ | 3 (same) | 10.3 ↑ |
Range is severely affected. IQR is resistant (unchanged). MAD increases but less dramatically.
Worked Examples
Find the range of: 14, 8, 22, 5, 17, 11, 30
Identify the maximum value: 30.
Identify the minimum value: 5.
Range = Maximum − Minimum = 30 − 5 = 25.
Find the five-number summary and IQR for: 3, 7, 8, 12, 15, 18, 21, 24
Data is already ordered: 3, 7, 8, 12, 15, 18, 21, 24.
Minimum = 3. Maximum = 24.
Median (Q2): 8 values — average of 4th and 5th: (12 + 15) ÷ 2 = 13.5.
Lower half: 3, 7, 8, 12. Q1 = (7 + 8) ÷ 2 = 7.5.
Upper half: 15, 18, 21, 24. Q3 = (18 + 21) ÷ 2 = 19.5.
IQR = Q3 − Q1 = 19.5 − 7.5 = 12.
Find the IQR for: 10, 14, 18, 22, 26, 30, 34
7 values — median is the 4th value: 22.
Lower half (exclude median): 10, 14, 18. Q1 = 14.
Upper half (exclude median): 26, 30, 34. Q3 = 30.
IQR = Q3 − Q1 = 30 − 14 = 16.
Calculate the MAD for: 4, 6, 8, 10, 12
Step 1 — Mean: (4 + 6 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8.
Step 2 — Absolute deviations: |4−8| = 4, |6−8| = 2, |8−8| = 0, |10−8| = 2, |12−8| = 4.
Step 3 — Sum: 4 + 2 + 0 + 2 + 4 = 12.
Step 4 — MAD = 12 ÷ 5 = 2.4.
Compare the spread of Set A: {2, 4, 6, 8, 10} and Set B: {1, 3, 6, 9, 11}. Use range and MAD.
Set A — Range: 10 − 2 = 8. Mean = 6. Deviations: 4, 2, 0, 2, 4. Sum = 12. MAD = 12 ÷ 5 = 2.4.
Set B — Range: 11 − 1 = 10. Mean = 6. Deviations: 5, 3, 0, 3, 5. Sum = 16. MAD = 16 ÷ 5 = 3.2.
Both sets have mean = 6, but Set B has a larger range (10 vs. 8) and larger MAD (3.2 vs. 2.4).
Guided Practice
Guided Practice Video: Measures of Spread
Review range, interquartile range, mean absolute deviation, and standard deviation — and how spread describes data variability — before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Find the range of: 9, 3, 17, 5, 12, 8, 21
Hint: Identify the maximum and minimum values, then subtract.
Find the five-number summary for: 6, 10, 14, 18, 22, 26, 30
Hint: Order the data. Find the median first, then split into lower and upper halves to find Q1 and Q3.
Find the IQR for: 5, 8, 11, 14, 17, 20, 23, 26
Hint: Find Q1 (median of lower half) and Q3 (median of upper half). IQR = Q3 − Q1.
Calculate the MAD for: 2, 4, 6, 8, 10
Hint: Step 1: Find the mean. Step 2: Find |each value − mean|. Step 3: Average those absolute deviations.
Data: 5, 7, 8, 9, 10, 11, 50. Find the range and IQR. Which is more affected by the outlier 50?
Hint: Calculate both. Range uses max and min — which includes the outlier. IQR uses Q1 and Q3 — does the outlier affect those?
Key Vocabulary
Range
Max − Min. The total span of the data. Sensitive to outliers.
Interquartile Range (IQR)
Q3 − Q1. The spread of the middle 50% of data. Resistant to outliers.
Mean Absolute Deviation (MAD)
Average of |each value − mean|. Measures how far, on average, each value is from the mean.
Five-Number Summary
Min, Q1, Median, Q3, Max — five values that fully describe a data set's spread.
Outlier
A value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR. Strongly affects range but not IQR.
Standard Deviation
Measures typical distance from the mean (similar to MAD). Larger value = more spread.
Interactive Practice — 5 Questions
What is the range of: 4, 11, 7, 19, 3, 15?
For the data set 2, 5, 8, 11, 14, 17, 20, what is Q1?
Which measure of spread is MOST resistant to outliers?
A student calculates MAD for {3, 5, 7} as: deviations = −2 + 0 + 2 = 0, MAD = 0. What is the error?
Set A has MAD = 1.4. Set B has MAD = 8.2. Both have the same mean. Which has greater variability?
Independent Practice
Independent Practice
Find the five-number summary and IQR for: 4, 8, 12, 16, 20, 24, 28. Show all steps — order the data, find the median, then split into halves.
Outlier effect: Data set A is 5, 7, 8, 9, 10, 11. Data set B replaces 11 with 50: 5, 7, 8, 9, 10, 50. Find the range and IQR for both. Describe how the outlier changed each measure.
Calculate the MAD for: 10, 20, 30, 40, 50. Show every step including each absolute deviation. Interpret your answer in a sentence.
Outlier test: Data set is 20, 22, 25, 28, 30, 55. Use the 1.5 × IQR rule to determine whether 55 is an outlier. Show all work (find Q1, Q3, IQR, fences).
Compare variability: Set A: {5, 6, 7, 8, 9}. Set B: {1, 4, 7, 10, 13}. Both have mean = 7. Find the MAD for each. Which has greater variability? Explain what this means in context.
Common Mistakes
Not ordering data before finding quartiles — Q1 and Q3 come out wrong if the data isn't sorted.
Always sort the data from least to greatest before finding the median, Q1, or Q3.
Including the median in both halves when finding Q1 and Q3 for an odd-count data set.
For an odd number of values, exclude the median from both the lower and upper halves.
Forgetting absolute values in MAD — deviations cancel out and MAD comes out as 0.
Each deviation is |xᵢ − mean|. Always take the absolute value before summing.
Confusing range and IQR — using range when asked for the middle spread, or vice versa.
Range = Max − Min (entire data, sensitive to outliers). IQR = Q3 − Q1 (middle 50%, resistant to outliers).
Math Tips
Always order the data first before finding quartiles or IQR — skipping this step causes errors.
IQR is resistant to outliers because it only uses the middle 50% of data (Q1 to Q3).
Range is sensitive to outliers — one extreme value can make the range very large even if most data is clustered.
MAD uses absolute values — deviations can be positive or negative, but always take |xᵢ − mean| before averaging.
When comparing two data sets, the one with the larger IQR or MAD has greater variability.
Standard deviation and MAD tell the same story — both measure average distance from the mean. Larger = more spread.