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Displaying Distributions with Graphs | Study Mondo
Topics / Unit 1: Exploring One-Variable Data / Displaying Distributions with Graphs Displaying Distributions with Graphs Create and interpret histograms, dotplots, stemplots, bar graphs, and pie charts.
BC Written and reviewed by Brendan Cusack , Study Mondo Education Team • Last updated April 30, 2026
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Displaying Distributions with Graphs
Choosing a Display
Data Type Best Graph Why One quantitative variable, small n Dotplot shows individual values One quantitative variable, medium n Stemplot organizes and shows shape One quantitative variable, large n Histogram groups into bins One quantitative variable, compare groups Parallel boxplots side-by-side comparison Cumulative distribution Ogive shows percentiles Two quantitative variables Scatterplot shows relationship Categorical variable Bar chart / Pie chart shows proportions
Dotplot
Horizontal axis: values of variable
Dots stacked vertically for each value
One dot = one data point
Advantages: sees every individual value, shows gaps and clusters
Disadvantages: crowded with large datasets
Stemplot (Stem-and-Leaf Plot)
Stem: tens digit (left side)
Leaf: ones digit (right side)
Leaves ordered left-to-right
Example: Dataset 12, 15, 18, 21, 23, 25
Interpreting: stem = 1, leaf = 2 means 12
Back-to-back stemplot: compare two distributions
Group A | stem | Group B
8 5 2 | 1 | 3 4 7
1 0 | 2 | 2 5 8
Histogram
Bins (class intervals) on x-axis
Frequency (count) on y-axis
Bars touch (data is continuous)
Height = frequency (or relative frequency ÷ width)
Key choice: width of bins
Too wide: lose detail
Too narrow: too fragmented
Important: area of bar = relative frequency when using density scale
Boxplot
Box: from Q1 to Q3 (middle 50%)Line in box: median (Q2)Whiskers: extend to minimum/maximum (or 1.5·IQR rule)Dots: outliers beyond whiskers Formula for outlier detection:
Lower fence: \(Q1 - 1.5(IQR)\)
Upper fence: \(Q3 + 1.5(IQR)\)
Points outside fences are outliers
Ogive (Cumulative Distribution)
x-axis: values
y-axis: cumulative relative frequency (0 to 1 or 0% to 100%)
Points connected by line segments
Always increasing (non-decreasing)
Read up from x-value to curve, then left to y-axis
Or read left from y-axis to curve, then down to x-axis
Worked Example Data: Test scores for 20 students: 62, 68, 71, 74, 74, 75, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 92, 94, 98
6 | 2 8
7 | 1 4 4 5 7 8
8 | 0 1 2 3 5 6 7 8
9 | 0 2 4 8
Histogram (bins 60-70, 70-80, 80-90, 90-100):
[60, 70): 2 students
[70, 80): 6 students
[80, 90): 8 students
[90, 100]: 4 students
Boxplot: Q1 ≈ 74.5, Q2 ≈ 82.5, Q3 ≈ 87.5, range 62–98, no outliers
Common Mistakes
Histogram bars not touching: bars should touch (continuous data)Mislabeling x-axis in stemplot: leaves must be single digitIgnoring class width in histogram: density = frequency ÷ widthOutlier calculation: use 1.5·IQR rule (not "looks far away")Confusing relative vs. cumulative: ogive uses cumulative, histogram uses frequency
AP Exam Tip When asked to "display" data, choose the method that best shows the feature in question:
Comparing shapes of two distributions? → Parallel boxplots or back-to-back stemplots
Seeing exact values? → Dotplot or stemplot
Large dataset? → Histogram
Finding a percentile? → Ogive
Always label axes and title your graph.
📚 Practice Problems
1 Problem 1easy ❓ Question:A teacher collects quiz scores: 8, 9, 9, 10, 10, 10, 11, 12, 12, 13. Create a stemplot for this data.
💡 Show Solution Stemplot (Stem-and-Leaf Plot):
Stem ∣ Leaf \text{Stem} | \text{Leaf} Stem ∣ Leaf 0 ∣ (no leaves — no scores in 0-9 range except...) 0 | \text{(no leaves — no scores in 0-9 range except...)} 0∣ (no leaves — no scores in 0-9 range except...) or start at 8: \text{or start at 8:} or start at 8: 8 ∣ 8 | \text{ } 8∣ 9 ∣ 0 0 9 | 0 \; 0 9∣0 0 10 ∣ 0 0 0 10 | 0 \; 0 \; 0 10∣0 0 0 11 ∣ 0 11 | 0 11∣0 12 ∣ 0 0 12 | 0 \; 0 12∣0 0 13 ∣ 0 13 | 0 13∣0
Wait, let me recalculate. Scores are: 8, 9, 9, 10, 10, 10, 11, 12, 12, 13
Better notation:
8 = stem 0, leaf 8 (or stem 8, leaf nothing)
9, 9 = stem 0, leaves 9, 9
10, 10, 10 = stem 1, leaves 0, 0, 0
etc.
Stem ∣ Leaf \text{Stem} | \text{Leaf} Stem ∣ Leaf 0 ∣ 8 9 9 0 | 8 \; 9 \; 9 0∣8 9 9 1 ∣ 0 0 0 1 2 2 3 1 | 0 \; 0 \; 0 \; 1 \; 2 \; 2 \; 3 1∣0 0 0 1 2 2 3
Or in traditional format:
8 ∣ 8 | \text{ } 8∣ 9 ∣ 0 0 9 | 0 \; 0 9∣0 0 10 ∣ 0 0 0 10 | 0 \; 0 \; 0 10∣0 0 0 11 ∣ 0 11 | 0 11∣0 12 ∣ 0 0 12 | 0 \; 0
Interpretation : Most scores cluster at 10; shape is roughly symmetric with a slight left skew (tail toward lower scores). No outliers.
2 Problem 2medium ❓ Question:Explain when you would use a dotplot vs. a histogram. Give an example for each.
💡 Show Solution Dotplot: When to use
Small datasets (roughly < 20–30 values)You want to see each individual point
Data is discrete or you want to preserve exact values
Patterns and clusters matter more than overall frequency
Example : 10 students' test scores: 78, 82, 82, 85, 88, 90, 90, 92, 95, 98
Each dot placed above a number line shows exact scores, and you can see two students scored 82, two scored 90, etc.
Histogram: When to use
Large datasets (typically 30+ values)
3 Problem 3hard ❓ Question:A dataset has 500 values. How would you display it? Compare a histogram, boxplot, and dotplot in terms of what information each reveals.
💡 Show Solution Histogram:
Reveals : Full shape (symmetric, skewed, bimodal), exact frequency in each bin, where most data concentratesBest for : Overall distribution patternDrawback : Bin width choice can mislead; loses individual data points
Boxplot:
Reveals : Five-number summary (min, Q1, median, Q3, max), IQR (spread of middle 50%), identification of outliers, left/right skewBest for : Quick comparison of center and spread; identifying outliersDrawback : Hides the shape details (can't see if bimodal); loses frequency info
Explain using: 📝 Simple words 🔗 Analogy 🎨 Visual desc. 📐 Example 💡 Explain
⚠️ Common Mistakes: Displaying Distributions with GraphsAvoid these 3 frequent errors
1 Confusing correlation with causation
▾ 2 Using a z-test when the population standard deviation is unknown
▾ 3 Forgetting to check conditions for inference procedures
▾ 🧪 Practice Lab Interactive practice problems for Displaying Distributions with Graphs
▾ 📌 Related Topics in Unit 1: Exploring One-Variable Data❓ Frequently Asked QuestionsWhat is Displaying Distributions with Graphs?▾ Create and interpret histograms, dotplots, stemplots, bar graphs, and pie charts.
How can I study Displaying Distributions with Graphs effectively?▾ Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 3 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Displaying Distributions with Graphs study guide free?▾ Yes — all study notes, flashcards, and practice problems for Displaying Distributions with Graphs on Study Mondo are free to access. No account is needed.
What course covers Displaying Distributions with Graphs?▾ Displaying Distributions with Graphs is part of the AP Statistics course on Study Mondo, specifically in the Unit 1: Exploring One-Variable Data section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Displaying Distributions with Graphs?▾ Yes, this page includes 3 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.
💡 Study Tips✓ Work through examples step-by-step ✓ Practice with flashcards daily ✓ Review common mistakes 12∣0 0
Data is continuous or has many distinct values
You're interested in overall shape and frequency distribution , not individual points
You want to group values into intervals (bins) Example : Heights of 200 students
Instead of 200 individual dots, group heights into bins: 60–62", 62–64", 64–66", etc. Bars show how many students fall in each interval. The shape (symmetric, skewed, etc.) emerges clearly.
Key difference : Dotplots show individual data; histograms show distribution shape and patterns across groups.
Reveals : Each exact data point, clustering, individual valuesBest for : Small datasets or when precision mattersDrawback : With 500 points, it becomes unreadable — visual overload; impossible to see patterns Recommendation for 500 values:
Primary : Histogram for overall shape and distributionSecondary : Add boxplot alongside to highlight outliers and quartilesSkip dotplot — too many points obscure the view This combination gives complete picture: shape (histogram) + resistant summary (boxplot).