Statistics: Distributions, Measures, and Data Interpretation — ACT Math

Introduction

Statistics questions on the ACT look like 'easy reading comprehension' but hide calculation traps — especially in questions about mean vs median and standard deviation.

Statistics and data questions account for 6-9 questions per ACT and range from easy (reading a bar chart) to hard (computing weighted mean or comparing standard deviations).

By the end of this lesson you will be able to:

Calculate and interpret mean, median, mode, and range from data sets
Interpret histograms, box plots, scatter plots, and frequency tables
Understand standard deviation conceptually and compare spreads of distributions

You'll find a missing value in a data set given that the mean must equal a specific target — using the 'sum method' for mean problems.

The Concept

The Core Rule

Mean = sum of all values / number of values. Median = middle value when sorted (average of two middle values for even n). Mode = most frequent value. Range = max − min. Standard deviation measures spread — larger SD means more spread out. Mean is sensitive to outliers; median is not.

How the ACT tests this

Asking you to find a missing value given a target mean (use sum = mean × n)
Presenting a data display and asking which measure of center best represents skewed data
Comparing the standard deviation or range of two data sets shown in graphs

Measures of Center

Mean: add all values and divide by n. For grouped data, use sum = mean × n to find a missing value. Median: sort the data, find the middle. If n is even, average the two middle values. Mode: the value that appears most often.

Adding a constant c to every value increases mean and median by c; range unchanged
Multiplying every value by k multiplies mean, median, and range by k
Outliers pull the mean toward them; median is resistant to outliers

Data Displays

Histogram: bars show frequency by interval. Box plot: shows minimum, Q1, median, Q3, maximum. Scatter plot: shows correlation (positive, negative, or none). Line of best fit: use the trend to predict values.

IQR = Q3 − Q1 (measures middle 50% spread)
Outlier rule: more than 1.5 × IQR above Q3 or below Q1
Correlation ≠ causation (frequently tested as a conceptual trap)

Standard Deviation and Normal Distribution

Standard deviation (SD) = typical distance from the mean. You do NOT need to calculate SD on the ACT — only compare or interpret it. In a normal distribution: 68% of data within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

Two data sets with same mean but different SD: larger SD is more spread out
Shifting all data by a constant does NOT change SD
Multiplying all data by k multiplies SD by |k|

Your strategy

For missing-value mean problems: missing value = (target mean × n) − (sum of known values)
Always sort data before finding median — the ACT sometimes gives unsorted lists
For data display questions, read axis labels and scales carefully before answering
When comparing spreads, look at which data set is more clustered or more dispersed

Worked Examples

Easy Example 1 Finding The Mean Instead Of The Median, Or Failing To Sort Before Identifying The Middle Value

The test scores of 5 students are 72, 85, 90, 68, and 95. What is the median score?

A. 82
B. 85 (Correct answer)
C. 87
D. 90
E. 68

Step 1

Sort the scores: 68, 72, 85, 90, 95

Step 2

With 5 values, the median is the 3rd value

Step 3

Median = 85

Correct answer: B

Why B is correct

Correct: sorted 3rd value = 85

Why other options are wrong

A: Mean of all five scores: (72+85+90+68+95)/5 = 410/5 = 82 — that's the mean, not median

C: Average of 85 and 90; only correct for n=2 scenario

D: 4th value after sorting, not the middle

E: The minimum value, not the median

⚠ Trap: Finding the mean instead of the median, or failing to sort before identifying the middle value

Medium Example 2 Using N = 5 Instead Of N = 6 When Computing The Target Sum

The mean of 6 numbers is 14. Five of the numbers are 8, 12, 15, 18, and 20. What is the sixth number?

A. 9
B. 11 (Correct answer)
C. 13
D. 15
E. 17

Step 1

Target sum: mean × n = 14 × 6 = 84

Step 2

Sum of known five: 8 + 12 + 15 + 18 + 20 = 73

Step 3

Sixth number: 84 − 73 = 11

Correct answer: B

Why B is correct

Correct: 84 − 73 = 11

Why other options are wrong

A: Used mean × 5 = 70 as target sum; wrong n

C: Arithmetic error in summing the five known values

D: Same as one of the given values; arithmetic error

E: Used incorrect target sum

⚠ Trap: Using n = 5 instead of n = 6 when computing the target sum

Hard Example 3 Assuming Equal Means Implies Equal Standard Deviations

Dataset A: {2, 4, 6, 8, 10}. Dataset B: {4, 5, 6, 7, 8}. Which statement is true?

A. Dataset A and B have the same mean and the same standard deviation
B. Dataset A and B have the same mean but Dataset A has a larger standard deviation (Correct answer)
C. Dataset A has a larger mean and a larger standard deviation
D. Dataset B has a larger mean and Dataset A has a smaller standard deviation
E. Dataset A and B have the same median but Dataset B has a larger standard deviation

Step 1

Mean of A: (2+4+6+8+10)/5 = 30/5 = 6. Mean of B: (4+5+6+7+8)/5 = 30/5 = 6. Same mean.

Step 2

Range of A: 10−2 = 8. Range of B: 8−4 = 4. Dataset A is more spread out.

Step 3

Therefore A has the larger standard deviation. Answer: B.

Correct answer: B

Why B is correct

Correct: same mean (6), A more spread → larger SD

Why other options are wrong

A: Means are equal (both 6) but standard deviations differ — A is more spread out

C: Means are actually equal, not different

D: Means are equal; and A has larger (not smaller) SD

E: Both medians are 6 (same) but B has SMALLER SD, not larger

⚠ Trap: Assuming equal means implies equal standard deviations

Strategy Tips

For missing-value mean: total needed = mean × n; subtract known values to find the missing one
Always sort before finding median — even if the list looks sorted, verify
Range and IQR measure spread; mean and median measure center — keep them separate
Standard deviation: more spread = larger SD; you never need to calculate it on ACT, only compare
For scatter plots, the line of best fit's slope tells you direction of association

Common pitfalls

Finding the mean when the question asks for the median

Not sorting the data before finding the median

Confusing range (max−min) with IQR (Q3−Q1)

Reading a data display: 30 seconds. Mean/median/mode calculation: 45 seconds. Missing-value mean: 60 seconds. Identify what each part is asking (center? spread? specific value?) before computing.

Summary

Mean = sum / n; to find a missing value, use sum = mean × n then subtract
Median requires sorted data; it is resistant to outliers unlike the mean
Standard deviation measures spread — you compare it, you don't calculate it on the ACT

You have 7 quiz scores with a current mean of 78. What score do you need on the 8th quiz to raise your mean to 80? Set up the equation using sum = mean × n.

Next: Probability and Counting All ACT Math lessons