Topic 1.9: Comparing Distributions of a Quantitative Variable
Comparing Distributions of a Quantitative Variable
Video Follow-Along Worksheet
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Learning Objectives:
Compare graphical representations for multiple sets of quantitative data.
Compare distributions using shape, center, variability, and unusual features.
Write complete comparison responses that use comparative words and context.
Use summary statistics such as median, range, and IQR to compare independent samples.
Essential Knowledge:
Graphs such as side-by-side box plots can be used to compare two or more independent samples.
When comparing quantitative distributions, discuss center, variability, clusters, gaps, outliers, and other relevant features.
Center can be compared with statistics such as the mean or median.
Variability can be compared with statistics such as the range, IQR, or standard deviation.
A complete written comparison should use comparative language and include context from the problem.
Key Vocabulary
Shape
The overall pattern of a distribution, such as symmetric or skewed.
Center
A typical value in a distribution, often described with the mean or median.
Variability
How spread out the data are.
Unusual Features
Features such as outliers, gaps, or clusters.
Skewed Right
A distribution with a longer tail on the right side.
Skewed Left
A distribution with a longer tail on the left side.
Symmetric
A distribution whose left and right sides are approximately balanced.
Range
The maximum minus the minimum.
IQR
The interquartile range, found by subtracting Q1 from Q3.
Standard Deviation
A measure of variability that describes typical distance from the mean.
Comparative Words
Words such as similar, greater, less, or same that directly compare distributions.
Context
The real-world variable being described, such as yearly salary or homework time.
Video 1: Comparing Distributions of a Quantitative Variable
[0:00 – 8:31]
1.[0:07 – 0:10] In this video, we are comparing distributions of a .
2.[0:14 – 0:24] The first big question asks what important to discuss when comparing distributions of quantitative data.
3.[0:27 – 0:32] The second big question asks what is needed for a complete when comparing distributions of quantitative data.
Review from the Previous Lesson[0:34 – 0:40] The lesson begins by reviewing how to describe a single quantitative distribution before comparing two distributions.
4.[0:43 – 0:47] There are important characteristics to describe or compare.
5.[0:47 – 0:56] The first characteristic is , and related vocabulary includes skewed left, skewed right, symmetric, unimodal, bimodal, and uniform.
6.[0:57 – 1:02] To address the center, we can use the or the .
7.[1:03 – 1:08] To address variability, we can use the , the , and the .
8.[1:09 – 1:14] Unusual features include , , and .
9.[1:24 – 1:40] When comparing two distributions, we still want to address the same important characteristics.
Released FRQ Context[1:47 – 2:41] The example comes from 2015 AP Exam Question 1. Two corporations hired entry-level accountants at the same starting salary in 2009, and side-by-side box plots summarize the 2014 salaries of employees who were still working there five years later.
10.[2:05 – 2:09] In 2009, the starting salary was a year at both corporations.
11.[2:12 – 2:20] Data were collected from employees at each corporation who were still employed years later.
12.[2:37 – 2:41] The prompt asks for a few sentences comparing the distributions of yearly .
Model Comparison of the Salary Box Plots[2:49 – 4:10]
Shape appears similar and fairly symmetric for both corporations
Median salary is approximately the same for both corporations
Range and IQR are greater for corporation A than corporation B
Corporation A has two high outliers, while corporation B has no outliers
13.[2:49 – 3:05] The shape appears to be fairly for both corporations A and B.
14.[3:08 – 3:19] With box plots, we say the distributions appear symmetric because we do not know how the values are distributed within each .
15.[3:24 – 3:35] The median salary is approximately the for both corporations.
16.[3:38 – 4:01] The and are greater for corporation than corporation B.
17.[4:04 – 4:10] The two highest salaries at corporation A are , while corporation B has outliers.
Complete Response Checklist[4:17 – 5:20]
Address all four characteristics: shape, center, variability, and unusual features
Use comparative words such as similar, same, greater, or less
Include context such as yearly salary
Be careful about shape claims from box plots and say appears when needed
18.[4:24 – 4:38] A complete response must address important characteristics.
19.[4:46 – 5:07] When comparing distributions, you must use such as similar, same, and greater than.
20.[5:12 – 5:20] You also need to include by referring to something specific like yearly salary.
21. Reflection: Explain how the yearly salary distributions for corporations A and B compare in shape, center, variability, and unusual features. Use comparative words and context in your answer.
Part (b) Reasoning[5:22 – 7:37] The follow-up question asks for one reason to choose corporation A and one reason to choose corporation B, using evidence that is visible in the box plots.
22.[5:46 – 5:55] Because the prompt says "based on the box plots," your response should mention a feature of the .
23.[6:03 – 6:13] At least out of , or , of the salaries at corporation A are greater than the maximum salary at corporation B.
24.[6:32 – 6:37] One reason to choose corporation A is the chance to earn a salary.
25.[6:57 – 7:09] The salary at corporation B is greater than the minimum at corporation A.
26.[7:14 – 7:26] At corporation A, some people may have never received a .
27. Reflection: Explain one reason someone might choose corporation A and one reason someone might choose corporation B based on the box plots. Mention a specific box-plot feature and explain why it matters.
Takeaways[7:40 – 8:22]
Compare distributions by discussing shape, center, variability, and unusual features
Use comparative words throughout the comparison
Include context from the problem so the response is about the actual variable
Exit Ticket
A school compared nightly homework times for students in two study halls using side-by-side box plots. Study Hall A had minimum 30, Q1 45, median 60, Q3 90, largest non-outlier 110, and one high outlier at 150. Study Hall B had minimum 45, Q1 54, median 60, Q3 66, maximum 75, and no outliers.
(a) Compare the centers and variabilities of the two distributions in context.
(b) Describe the unusual features and state what the shape appears to be for each distribution based on the box plots.
(c) Write one or two sentences that make the comparison complete by using comparative words and context.
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