AP Statistics Five-Number Summary, Boxplots and Comparing Distributions Worksheet Solutions (PDF)

AP Statistics math worksheet solutions: Five-Number Summary, Boxplots and Comparing Distributions – Maths PDF

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This boxplots compare answers for AP Statistics covers five-number summary, boxplots and comparing distributions (Common Core AP STAT 1.8, AP STAT 1.9, HSS-ID.A.1, HSS-ID.A.2), with a free PDF to print.

Each solution lists the sorted data, the quartiles, and any fences before it draws a conclusion. Use full comparison sentences, because the AP exam rewards context and comparison words.

AP Statistics · Common Core State Standards: AP STAT 1.8, AP STAT 1.9, HSS-ID.A.1, HSS-ID.A.2

Boxplots compare answers: exercises 1 to 10

Exercise 1 – Five-number summary

The data are already ordered. The median is the 6th value, 34.

Lower half (22, 25, 27, 30, 31): \(Q_1=27\). Upper half (36, 38, 41, 45, 60): \(Q_3=41\).

22, 27, 34, 41, 60.

Exercise 2 – Read a boxplot

Quiz scores.

(a) 22. (b) \(40-12=28\). (c) \(30-18=12\). (d) The box holds the middle half of the data.

22; 28; 12; about 50%.

Exercise 3 – Draw a boxplot

Boxplot of the data.

\(IQR=20-9=11\). Fences: \(9-16.5=-7.5\) and \(20+16.5=36.5\). Neither the minimum nor the maximum is beyond a fence.

No outliers; whiskers extend to 4 and 31.

Exercise 4 – Outliers and whiskers

The star marks 35.

\(Q_1=9\), \(Q_3=15\), \(IQR=6\). Upper fence: \(15+9=24\).

35 is an outlier; the upper whisker ends at 16, the largest value inside the fence.

Exercise 5 – Quarters of the data

Each section holds about 25% of the data: \(0.25\times 48=12\).

About 12 students in each section.

Exercise 6 – Compare in context

Points per game.

Medians: A 38, B 40. IQRs: A \(46-30=16\), B \(55-28=27\). Ranges: A 40, B 70.

Team B has a slightly higher median (40 compared with 38 points) and is more variable (IQR 27 compared with 16, range 70 compared with 40).

Exercise 7 – Skew from a boxplot

The median is close to Q1 (2 above it) and far from Q3 (18 below it), and the upper whisker is long.

Skewed right.

Exercise 8 – Which has more spread?

By the IQR, Class 1 is more spread out in the middle half. By the range, Class 2 is more spread.

The IQR is resistant to extreme values, so Class 1 is more variable overall; the range of Class 2 is probably inflated by an extreme score.

Exercise 9 – Limits of a boxplot

A boxplot shows only five numbers. Different data can share those numbers while having one peak or two peaks.

No. A boxplot can hide gaps, clusters, and bimodality.

Exercise 10 – Percent above Q3

Above Q3: \(0.25\times 120=30\). Between the median and Q3: also 25%, which is 30.

About 30 students in each.

Exercises 11 and up

Exercise 11 – Build two boxplots

Class 1: 14, 17, 21, 25, 40. \(IQR=8\), upper fence 37, so 40 is an outlier.

Class 2: 15, 19, 22.5, 26, 30. \(IQR=7\), no outliers.

Minutes of homework.

Class 1: 14, 17, 21, 25, 40 (outlier 40). Class 2: 15, 19, 22.5, 26, 30.

Exercise 12 – Overlapping boxes

Overlap shows the middle halves share values. A boxplot alone cannot prove a difference; inference is needed.

No; overlap and a small difference in medians do not by themselves show a real difference.

Exercise 13 – Find a missing quartile

\(Q_1=52-18=34\). Lower fence: \(34-1.5(18)=7\). Upper fence: \(52+27=79\).

Q1 = 34; fences 7 and 79.

Exercise 14 – Delivery times

Median \(\frac{27+29}{2}=28\). Lower half 18-27: \(Q_1=23.5\). Upper half 29-70: \(Q_3=36\). \(IQR=12.5\), upper fence \(36+18.75=54.75\).

Summary 18, 23.5, 28, 36, 70; 70 is an outlier; the data are skewed right.

Exercise 15 – Salaries at two companies

Company X: 45, 49, 55, 62.5, 70. Company Y: 40, 53.5, 58, 63, 120.

IQR: X \(62.5-49=13.5\), Y \(63-53.5=9.5\).

Company Y has a higher median (58 compared with 55) and a smaller IQR, but it has an outlier at 120 thousand dollars, while Company X has no outliers.

Exercise 16 – Fix the comparison

A good comparison uses comparison words and context.

The median score in Class B (78) is higher than in Class A (72), but Class B scores are more variable (IQR 18 compared with 10).

Exercise 17 – Percentiles from the box

Her score 84 is above Q3 = 83.

She is in the top quarter, above the 75th percentile.

Exercise 18 – Multi-step

Only the maximum changes: now 51, 68, 76, 83, 90.

The range changes from \(97-51=46\) to \(90-51=39\), while the median, quartiles, and IQR \(83-68=15\) stay the same.

Only the maximum and range change.

Exercise 19 – Median position

A: the median is centered in the box (8 from each quartile) and the whiskers are 12 on each side.

B: the median is 3 above Q1 but 25 below Q3, and the upper whisker is long.

A is roughly symmetric; B is skewed right.

Exercise 20 – Interpret the IQR

\(IQR=79-61=18\).

The middle 50% of the quiz scores span 18 points, from 61 to 79.

Key facts behind this boxplots compare answers

Key facts

Five-number summary: minimum, Q1, median, Q3, maximum.
Each quarter of the ordered data holds about 25% of the values.
IQR = Q3 – Q1; outliers lie more than 1.5 IQR outside the box.
Compare center, spread, shape, and unusual features using comparison words.
Boxplots show skew and outliers but not the number of peaks.

How to use this boxplots compare answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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