U1.5 Quantitative Graphs (Dotplot, Stemplot, Histogram)
Master AP Statistics 1.5: build and read dotplots, stemplots, and histograms, and pick smart bin widths for quantitative data.
What you'll do in this lesson
A voice-first session with the Crimsora tutor on U1.5 Quantitative Graphs (Dotplot, Stemplot, Histogram), then targeted practice and FRQs — with the tutor adapting to where you get stuck.
What this lesson covers
Every distribution starts with a picture. Before you can describe shape, center, and spread (that's the next lesson), you need graphs that display a quantitative variable honestly. In topic 1.5 you learn three workhorse graphs — the dotplot, the stemplot, and the histogram — and how to choose bin widths that reveal the true story rather than hiding or exaggerating it.
This guide shows you how to construct each graph from raw data, how to read frequencies and intervals correctly, and the small technical details (like frequency versus relative frequency, and boundary rules) that AP graders reward. Get these right and Units 1.6 through 1.10 become much easier.
This guide shows you how to construct each graph from raw data, how to read frequencies and intervals correctly, and the small technical details (like frequency versus relative frequency, and boundary rules) that AP graders reward. Get these right and Units 1.6 through 1.10 become much easier.
Three graphs for quantitative data
A quantitative variable takes numerical values you can meaningfully order and do arithmetic with (heights, test scores, wait times). Each of the three graphs displays these values differently.
A dotplot places one dot above a number line for each observation. Repeated values stack, so the height of a stack equals the frequency of that value. Dotplots keep every individual data value visible, which makes them ideal for small data sets.
A stemplot (stem-and-leaf plot) splits each number into a stem (the leading digit or digits) and a leaf (usually the final digit). Leaves are written in increasing order to the right of each stem. Like a dotplot, it preserves the actual data values while also showing shape.
A histogram groups values into equal-width intervals called bins and draws a bar whose height is the count (frequency) or proportion (relative frequency) in that bin. Histograms handle large data sets and continuous variables where individual values would clutter the picture.
All three let you judge shape and spot outliers, but only the dotplot and stemplot let you recover the exact original data.
A dotplot places one dot above a number line for each observation. Repeated values stack, so the height of a stack equals the frequency of that value. Dotplots keep every individual data value visible, which makes them ideal for small data sets.
A stemplot (stem-and-leaf plot) splits each number into a stem (the leading digit or digits) and a leaf (usually the final digit). Leaves are written in increasing order to the right of each stem. Like a dotplot, it preserves the actual data values while also showing shape.
A histogram groups values into equal-width intervals called bins and draws a bar whose height is the count (frequency) or proportion (relative frequency) in that bin. Histograms handle large data sets and continuous variables where individual values would clutter the picture.
| Graph | Shows individual values? | Best for |
|---|---|---|
| Dotplot | Yes | Small data sets |
| Stemplot | Yes | Small/medium, comparing two groups |
| Histogram | No (grouped) | Large data sets |
Building a stemplot correctly
To make a stemplot, decide what counts as the stem. For two-digit numbers like 47, 52, and 58, the tens digit is the stem and the ones digit is the leaf. So 47 becomes stem 4, leaf 7.
List stems vertically in order, including any stems with no leaves (gaps matter — they show the shape). Then write each leaf next to its stem in increasing order. Always include a key, such as , so a reader knows the scale.
When data cluster too tightly, use a split stem: each stem appears twice, once for leaves 0–4 and once for leaves 5–9. This spreads out the display and reveals shape that a compressed plot would hide.
For comparing two groups, a back-to-back stemplot shares one column of stems, with one group's leaves growing to the left and the other's to the right. Reading the left leaves right-to-left keeps them in order.
A common student error is forgetting to include empty stems, which distorts the visible gaps and spread. Another is writing leaves out of order. Graders check that leaves are sorted and that a key is present.
List stems vertically in order, including any stems with no leaves (gaps matter — they show the shape). Then write each leaf next to its stem in increasing order. Always include a key, such as , so a reader knows the scale.
When data cluster too tightly, use a split stem: each stem appears twice, once for leaves 0–4 and once for leaves 5–9. This spreads out the display and reveals shape that a compressed plot would hide.
For comparing two groups, a back-to-back stemplot shares one column of stems, with one group's leaves growing to the left and the other's to the right. Reading the left leaves right-to-left keeps them in order.
A common student error is forgetting to include empty stems, which distorts the visible gaps and spread. Another is writing leaves out of order. Graders check that leaves are sorted and that a key is present.
Histograms and choosing bin width
A histogram's appearance depends heavily on the bin width you choose. Too few, very wide bins oversmooth the data and hide features like gaps or multiple peaks. Too many, very narrow bins produce a jagged plot where random noise looks meaningful. A good bin width shows the overall shape clearly, usually giving roughly 5 to 12 bins depending on sample size.
Bins must have equal width and must not overlap. You need a consistent boundary rule: a value that lands exactly on a boundary always goes into the same side (for example, the interval includes 10 but not 20). Every observation belongs to exactly one bin.
Height can be frequency (raw count) or relative frequency (proportion of the total). Relative frequency histograms are useful for comparing data sets of different sizes because the total of all bar proportions equals 1.
Unlike bar charts for categorical data, histogram bars touch because the horizontal axis is a continuous number line. Leaving gaps only where a bin genuinely has zero observations is correct.
Bins must have equal width and must not overlap. You need a consistent boundary rule: a value that lands exactly on a boundary always goes into the same side (for example, the interval includes 10 but not 20). Every observation belongs to exactly one bin.
Height can be frequency (raw count) or relative frequency (proportion of the total). Relative frequency histograms are useful for comparing data sets of different sizes because the total of all bar proportions equals 1.
| Bin width | Number of bins | Risk |
|---|---|---|
| Too wide | Too few | Hides gaps, peaks, and clusters |
| Too narrow | Too many | Exaggerates random noise |
| Balanced | About 5–12 | Reveals true shape |
How the AP exam tests topic 1.5
The exam rarely asks you to build a large histogram from scratch under time pressure, but it frequently asks you to read one. Expect questions where you compute the number or proportion of observations in a range by adding bar heights, or determine which bin contains the median or a given percentile.
A classic multiple-choice trap involves boundary values: if a bar spans with height 8, the value 40 is not counted in that bar. Read axis labels carefully — confusing a frequency axis with a relative frequency axis changes every answer.
On free-response questions you may be asked to describe what a dotplot or stemplot reveals, or to explain how changing the bin width would change the impression the histogram gives. Strong answers reference the data in context, not generic phrases.
A frequent misconception is treating a histogram like a scatterplot or connecting bar tops as if the horizontal axis were time. Remember: the horizontal axis is the quantitative variable's values, and the vertical axis is how often those values occur. Also, do not confuse a histogram (quantitative, bars touch) with a bar chart (categorical, bars separated), which belongs to topic 1.1.
A classic multiple-choice trap involves boundary values: if a bar spans with height 8, the value 40 is not counted in that bar. Read axis labels carefully — confusing a frequency axis with a relative frequency axis changes every answer.
On free-response questions you may be asked to describe what a dotplot or stemplot reveals, or to explain how changing the bin width would change the impression the histogram gives. Strong answers reference the data in context, not generic phrases.
A frequent misconception is treating a histogram like a scatterplot or connecting bar tops as if the horizontal axis were time. Remember: the horizontal axis is the quantitative variable's values, and the vertical axis is how often those values occur. Also, do not confuse a histogram (quantitative, bars touch) with a bar chart (categorical, bars separated), which belongs to topic 1.1.
Key terms
- Quantitative variable.
- A variable whose values are numbers that can be ordered and used in arithmetic, such as heights or scores.
- Dotplot.
- A graph placing one dot per observation above a number line; stacked dots show how many share a value.
- Stemplot.
- A stem-and-leaf display that splits each value into a leading stem and a final-digit leaf, preserving actual data.
- Histogram.
- A graph of equal-width bins with bars whose heights show the count or proportion of values in each interval.
- Bin (class).
- One of the equal-width intervals used to group data in a histogram; each observation falls in exactly one bin.
- Frequency vs. relative frequency.
- Frequency is the raw count in a bin; relative frequency is that count divided by the total, giving a proportion.
- Split stem.
- A stemplot technique using each stem twice (leaves 0–4 and 5–9) to spread out tightly clustered data.
- Key.
- A note on a stemplot showing how to read a stem and leaf, for example .
Worked example
The number of text messages 15 students sent in one hour were: 12, 15, 15, 18, 22, 23, 23, 23, 27, 31, 34, 34, 42, 45, 48. (a) Construct a stemplot with a key. (b) Build a frequency histogram using bins of width 10 starting at 10. (c) State how many students sent at least 30 messages.
Part (a): Use the tens digit as the stem and the ones digit as the leaf. Sort leaves within each stem.
Key: messages. Notice every stem 1 through 4 appears; none are empty here.
Part (b): Bins of width 10 starting at 10 give intervals , , , . Use the rule that a value on the left boundary belongs to that bin.
: 12, 15, 15, 18 = 4 students. : 22, 23, 23, 23, 27 = 5 students. : 31, 34, 34 = 3 students. : 42, 45, 48 = 3 students.
The histogram has four touching bars of heights 4, 5, 3, and 3. The counts total , confirming every student was counted once.
Part (c): 'At least 30' means the and bins: students sent 30 or more messages.
Key: messages. Notice every stem 1 through 4 appears; none are empty here.
Part (b): Bins of width 10 starting at 10 give intervals , , , . Use the rule that a value on the left boundary belongs to that bin.
: 12, 15, 15, 18 = 4 students. : 22, 23, 23, 23, 27 = 5 students. : 31, 34, 34 = 3 students. : 42, 45, 48 = 3 students.
The histogram has four touching bars of heights 4, 5, 3, and 3. The counts total , confirming every student was counted once.
Part (c): 'At least 30' means the and bins: students sent 30 or more messages.
Practice questions
A histogram uses bins , , , and with frequencies 6, 11, 8, and 5. How many observations are less than 20?
- 6
- 11
- 17
- 25
Answer: 17
'Less than 20' includes the and bins because the interval stops just below 20. Add their frequencies: . A common mistake is including part of the bin, but 20 and above are not less than 20.
A researcher makes a histogram of 200 reaction times using a bin width of 0.005 seconds, producing about 60 very narrow bars, many with height 0 or 1. Explain why this bin width is a poor choice and what changing it would accomplish.
Answer: The bins are too narrow, so the histogram is jagged and dominated by random noise instead of showing the overall shape.
With 60 bins for 200 values, each bar holds very few observations, so tiny random differences create spikes and gaps that do not reflect real structure. Widening the bins (giving roughly 8 to 12 bins) would smooth the display so that the true center, spread, and shape — such as skewness or a single peak — become visible. This tests the understanding that bin width controls how much detail versus noise the histogram reveals.
Which graph would let you recover every original data value AND is well suited to comparing two groups side by side?
- Frequency histogram
- Relative frequency histogram
- Back-to-back stemplot
- Dotplot with jitter
Answer: Back-to-back stemplot
Histograms group data into bins, so individual values cannot be recovered from them. A stemplot keeps every value as a leaf, and the back-to-back form shares one stem column to compare two groups directly. A single dotplot shows one group, not two side by side.
FAQ
- What is the difference between a histogram and a bar chart?
- A histogram displays a quantitative variable, so its bars touch along a continuous number line and represent intervals of values. A bar chart displays a categorical variable, so its bars are separated and each represents a distinct category. Mixing them up is a common AP error.
- How do I choose a good bin width for a histogram?
- Aim for roughly 5 to 12 bins so the picture shows real shape without exaggerating noise. Bins must be equal width and non-overlapping. If the plot looks jagged and spiky, widen the bins; if it looks like one or two giant blocks hiding detail, narrow them.
- When a value lands exactly on a bin boundary, which bin does it go in?
- You must pick a consistent rule and apply it to every value. The standard convention is that a value equal to the left boundary is included and the right boundary is excluded, so the interval is written like containing 10 but not 20. State your rule if it is not obvious.
- Should I use frequency or relative frequency on the vertical axis?
- Use frequency (raw counts) when you care about how many observations fall in each bin. Use relative frequency (proportions) when comparing data sets of different sizes, since proportions sum to 1 and make the two distributions directly comparable regardless of sample size.
Learn this with a teacher, not a page
The Crimsora tutor teaches U1.5 Quantitative Graphs (Dotplot, Stemplot, Histogram) live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.