9.3Dot Plots & Histograms
Create, interpret, and compare dot plots and histograms. Describe the shape, center, spread, clusters, gaps, and outliers in data distributions.
Why This Matters
Dot plots and histograms are the most common tools for visualizing data distributions. You'll use them in AP Statistics, Biology, and any field that involves analyzing data — from sports analytics to public health.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do dot plots and histograms help us understand and interpret data distributions?
Lesson Overview
Data is most useful when we can see its shape. Dot plots and histograms are two of the most important tools for visualizing how data is distributed. A dot plot places one dot above a number line for each data value — perfect for small data sets where you want to see every individual value. A histogram groups data into intervals and shows the frequency of each group using bars — ideal for large data sets where individual values matter less than the overall pattern. Both graphs reveal the shape of a distribution (symmetric, skewed, uniform, bimodal), as well as clusters, gaps, and outliers.
Dot Plot vs. Histogram — Quick Reference
| Feature | Dot Plot | Histogram |
|---|---|---|
| Best for | Small data sets (n ≤ ~30) | Large data sets (n ≥ ~20) |
| Shows individual values | Yes — each dot = one value | No — values grouped into intervals |
| Bars/dots touch | Dots float above number line | Bars always touch (no gaps) |
| Reveals outliers | Very clearly — isolated dots | Less clearly — outlier bar may be small |
| Class intervals needed | No | Yes — must choose equal-width bins |
Dot Plot — Quiz Scores (out of 10)
Quiz Score
Cluster at 7–9. Score 5 is slightly separated. Roughly symmetric with a slight left tail.
Histogram — Student Heights (inches)
Height (inches)
Tallest bar: 66–70 (freq = 12). Roughly symmetric (bell-shaped). Most students are 62–74 inches tall.
Distribution Shapes
Symmetric
Mean ≈ Median
Skewed Right
Mean > Median
Skewed Left
Mean < Median
Uniform
Mean ≈ Median
Bimodal
Two peaks
With Outlier
Gap then isolated bar
Dot Plot — Clusters, Gaps, and Outliers
Worked Examples
Construct a dot plot for daily high temperatures (°F): 72, 75, 75, 78, 78, 78, 80, 80, 83, 85. Describe the distribution.
Draw a number line from 70 to 86.
Place one dot above each value: 72 (×1), 75 (×2), 78 (×3), 80 (×2), 83 (×1), 85 (×1).
Label the number line and give the dot plot a title.
Describe: cluster at 75–80, slight right tail toward 83–85. No gaps or outliers.
Read the dot plot: Scores are 4, 5, 5, 6, 6, 6, 7, 7, 8, 10. Find the mean, median, mode, range, and describe the shape.
Mean = (4+5+5+6+6+6+7+7+8+10) ÷ 10 = 64 ÷ 10 = 6.4.
Median: 10 values — average of 5th and 6th: (6+6) ÷ 2 = 6.
Mode = 6 (appears 3 times). Range = 10 − 4 = 6.
Shape: cluster at 5–7, isolated dot at 10 (possible outlier). Mean (6.4) > Median (6) → slightly skewed right.
Construct a histogram for test scores: 52, 58, 61, 65, 67, 70, 72, 74, 75, 78, 80, 82, 85, 88, 91, 94. Use class intervals of width 10 starting at 50.
Class intervals: [50–60), [60–70), [70–80), [80–90), [90–100).
Tally: [50–60): 52, 58 → 2. [60–70): 61, 65, 67 → 3. [70–80): 70, 72, 74, 75, 78 → 5. [80–90): 80, 82, 85, 88 → 4. [90–100): 91, 94 → 2.
Draw horizontal axis (class intervals), vertical axis (Frequency, 0–5).
Draw touching bars with heights 2, 3, 5, 4, 2. Label axes and title.
Choose appropriate class intervals for: 12, 18, 23, 27, 31, 35, 38, 42, 47, 51, 55, 60, 64, 68, 72
Range = 72 − 12 = 60.
15 values → aim for 5–6 intervals. 60 ÷ 6 = 10. Use width = 10.
Start below the minimum: 10. Intervals: [10–20), [20–30), [30–40), [40–50), [50–60), [60–70), [70–80).
Verify: every value falls in exactly one interval. ✓
Compare: Class A histogram — [60–70): 2, [70–80): 8, [80–90): 7, [90–100): 3. Class B histogram — [60–70): 5, [70–80): 5, [80–90): 5, [90–100): 5.
Both classes have 20 students.
Class A shape: bell-shaped / symmetric, peak at 70–80.
Class B shape: uniform — all intervals have equal frequency (5).
Class A: most students scored 70–90, less variability. Class B: scores evenly spread, more variability.
Guided Practice
Guided Practice Video: Dot Plots and Histograms
Review how to read and create dot plots and histograms — including shape, center, and spread — before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Construct a dot plot for: 3, 5, 5, 6, 6, 6, 7, 7, 8, 9. Describe the shape.
Hint: Draw a number line from 3 to 9. Place one dot above each value. Stack dots for repeated values. Then look at the overall shape.
From a dot plot, you read: 10, 12, 12, 14, 14, 14, 16, 16, 18, 20. Find the mean, median, and mode.
Hint: Add all values and divide by 10 for the mean. Find the middle two values for the median. The mode is the value that appears most often.
Tally these scores into class intervals of width 10 starting at 60: 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95
Hint: Intervals: [60–70), [70–80), [80–90), [90–100). Count how many values fall in each interval.
A histogram has bars: [0–4): 2, [4–8): 6, [8–12): 10, [12–16): 7, [16–20): 3. Describe the shape and identify the interval with the highest frequency.
Hint: Look at the bar heights from left to right. Does the distribution rise then fall? Is it symmetric? Which bar is tallest?
Dot plot data: 1, 2, 2, 3, 3, 3, 4, 4, 10. Identify any clusters, gaps, and outliers.
Hint: Where are the dots concentrated? Is there a region with no dots? Is any value far from the rest?
Key Vocabulary
Dot Plot
One dot per value above a number line. Best for small data sets. Also called a line plot.
Histogram
Bars show frequency per equal-width interval. Bars always touch. Best for large data sets.
Class Interval (Bin)
A range of values grouped together in a histogram. All bins must have equal width.
Cluster
A region where data values are concentrated closely together.
Gap
A region where no data values occur — separates clusters or indicates missing values.
Symmetric Distribution
Left and right sides mirror each other. Mean ≈ Median.
Skewed Right
Long tail on the right. Mean > Median. Peak is on the left.
Skewed Left
Long tail on the left. Mean < Median. Peak is on the right.
Uniform Distribution
All intervals have roughly equal frequency. Histogram looks like a flat rectangle.
Bimodal Distribution
Two distinct peaks. Suggests two different groups may be present in the data.
Interactive Practice — 5 Questions
A dot plot has dots at: 3, 5, 5, 6, 6, 6, 7, 7, 8. What is the mode?
A histogram has bars: [10–20): 8, [20–30): 5, [30–40): 3, [40–50): 2, [50–60): 1. What is the shape?
For a right-skewed distribution, which measure of center is more appropriate?
A student draws a histogram with class intervals [0–5), [5–15), [15–20). What is wrong?
A dot plot shows: 2, 3, 3, 4, 4, 4, 5, 5, 6, 20. What feature does the value 20 represent?
Independent Practice
Independent Practice
Construct a dot plot for: 8, 10, 10, 11, 11, 11, 12, 12, 13, 15. Describe the shape and identify any clusters, gaps, or outliers.
From a dot plot with values 5, 5, 6, 7, 7, 8, 8, 8, 9, 10, find the mean, median, mode, and range. Describe the shape of the distribution.
Tally these values into class intervals of width 5 starting at 20: 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43. Then describe the shape of the resulting histogram.
A histogram has bars: [0–10): 12, [10–20): 8, [20–30): 5, [30–40): 3, [40–50): 2. Describe the shape. Should you use mean or median as the measure of center? Explain.
Compare two distributions: Histogram A is bell-shaped centered at 50. Histogram B is right-skewed with most data at 20–40. Both have 30 values. Which has a higher mean? Which has a higher median? Explain your reasoning.
Common Mistakes
Leaving gaps between histogram bars as a formatting choice.
Histogram bars always touch — a gap between bars means no data in that interval, not a style decision.
Using unequal class interval widths in a histogram.
All bins must have equal width. Unequal widths make the graph misleading and hard to compare.
Saying a distribution is 'skewed right' because the peak is on the right.
Skewed right means the tail is on the right. The peak (bulk of data) is on the LEFT for a right-skewed distribution.
Placing a dot between two values on a dot plot instead of directly above the value.
Each dot must be stacked directly above the exact value it represents on the number line.
Math Tips
Choosing bin width: aim for 5–10 intervals. Divide the range by your target number of intervals and round to a convenient number.
Histogram bars always touch — a gap between bars means no data in that interval, not a formatting choice.
Skewed right = tail on the right, peak on the left. Mean > Median.
Skewed left = tail on the left, peak on the right. Mean < Median.
Symmetric distribution: mean ≈ median. Use mean as the measure of center.
Dot plots preserve every value — you can calculate mean, median, mode, and range directly from a dot plot.