Dot Plots, Histograms, and Distribution Shape
A free Statistics and Data Analysis lesson from the “Data Foundations” unit, with a worked example and practice problems including step-by-step solutions.
Quantitative data has shape. Dot plots show individual values, while histograms group values into intervals. Both help reveal whether a distribution is symmetric, skewed, clustered, or unusual. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Choose a display for quantitative data
- Describe shape, cluster, gap, and outlier patterns
- Read intervals from a histogram
Worked example
Problem. Which variable is quantitative?
- Worked Example: First identify exactly what the question is asking: Which variable is quantitative?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Quantitative variables measure amounts.
- Time in minutes is numerical and can be averaged.
Answer: Time in minutes
Practice problems
1. Practice case A: Which variable is quantitative?
Choices: Time in minutes · Eye color · Lunch choice · Device brand
Show solution
- Warm-up: First identify exactly what the question is asking: Which variable is quantitative?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Quantitative variables measure amounts.
- Time in minutes is numerical and can be averaged.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Time in minutes
2. Practice case B: A frequency table shows 7 first-choice votes, 12 second-choice votes, and 5 third-choice votes. How many responses are in the table?
Show solution
- Warm-up: First identify exactly what the question is asking: A frequency table shows 7 first-choice votes, 12 second-choice votes, and 5 third-choice votes. How many responses are in the table?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Add all category counts: 7 + 12 + 5.
- The total is 24.
- Check the result by substituting or estimating: the response should match 24 and make sense in the original problem.
Answer: 24
3. Practice case C: A bar chart has 8 votes for A out of 27 total votes. About what percent chose A? Enter the nearest whole percent.
Show solution
- Warm-up: First identify exactly what the question is asking: A bar chart has 8 votes for A out of 27 total votes. About what percent chose A? Enter the nearest whole percent.
- For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
- Relative frequency is count divided by total.
- 8 / 27 is about 30%.
- Check the result by substituting or estimating: the response should match 30 and make sense in the original problem.
Answer: 30
4. Practice case D: Which display is best for comparing counts across categories?
Choices: Residual plot · Bar chart · Histogram · Box plot
Show solution
- Warm-up: First identify exactly what the question is asking: Which display is best for comparing counts across categories?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A bar chart compares category counts.
- Histograms and box plots are for quantitative distributions.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Bar chart
5. Practice case E: A histogram is used mainly to show:
Choices: the shape of quantitative data · the exact value of a population parameter · the cause of an association · the treatment in an experiment
Show solution
- Warm-up: First identify exactly what the question is asking: A histogram is used mainly to show:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Histograms group quantitative values into intervals.
- That reveals distribution shape.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: the shape of quantitative data
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