Introduction to Statistics · Chapter 2 · Descriptive Statistics
Stem-and-Leaf Graphs, Line Graphs, and Bar Graphs
Three ways to picture a data set — split it, connect it, or separate it — and how to tell which one the data is asking for.
bookSHelf · Introduction to Statistics · §2.1 · a self-paced section
Learning objectives — by the end of this section you will be able to
Objectives
- Build a stem-and-leaf plot from a raw list and read the shape of the data off it stemplot
- Spot an outlier in a stemplot and say what makes it stand apart Def. 2.1.2
- Construct a side-by-side stemplot to compare two data sets on the same scale Def. 2.1.3
- Build a line graph from a frequency table and describe the pattern it traces Def. 2.1.4
- Build a bar graph for categorical data and choose between counts and percentages Def. 2.1.5
§2.1.1 — Stem-and-Leaf Graphs (Stemplots)
A stemplot splits each value into a stem and a leaf, so you can read the data's shape and its exact values off the same picture — a trade most graphs make you choose between.
It costs nothing but a pencil, and answers two questions at once: what shape is this data, and what are the actual values.
§2.1.1 — the definition
Stem-and-Leaf Graph (Stemplot)
Definition 2.1.1 — Stem-and-Leaf Graph (Stemplot)
A stemplot splits each observation into a stem (all digits but the last) and a leaf (the final digit). Stems run smallest to largest in a vertical column; each leaf sits beside its stem in increasing order.
23 → stem 2, leaf 3. 432 → stem 43, leaf 2. 9.3 → stem 9, leaf 3 — the leaf is always just the last significant digit.
Definition 2.1.1: Each score splits into a stem and a leaf, and the leaves stack into their stem's row.
Insight Note — the receipts are still there
A stemplot is a histogram that kept its receipts
Turn a stemplot on its side and the rows of leaves become bars — the same shape as a histogram. The difference is that a stemplot never throws the actual numbers away: every original value is still sitting there, ready to be read back.
§2.1.1 — a value that doesn't fit
Outlier
Definition 2.1.2 — Outlier
An outlier is an observation that does not fit the rest of the data — also called an extreme value. Graphed, it appears not to fit the pattern of the graph.
Some outliers are typos — writing 50 instead of 500. Others are the most interesting value in the whole data set. Telling which takes background information, covered later.
Definition 2.1.2: An outlier is defined by the empty stretch beside it, not by being large.
Context Pause — a question, not a verdict
An outlier is a question, not a verdict
Finding a value far from the others does not tell you to delete it. It tells you to go ask why it is there. Sometimes the answer is a typo; sometimes the answer is the most interesting thing in the whole data set.
Your turn — build the stems yourself
Try It Now 2.1.1
Try It Now 2.1.1 — Park City basketball scores, 30 games
Scores (smallest to largest): 32, 32, 33, 34, 38, 40, 42, 42, 43, 44, 46, 47, 47, 48, 48, 48, 49, 50, 50, 51, 52, 52, 52, 53, 54, 56, 57, 57, 60, 61. Construct a stem plot.
| Stem | Leaf |
|---|---|
| 3 | 2 2 3 4 8 |
| 4 | 0 2 2 3 4 6 7 7 8 8 8 9 |
| 5 | 0 0 1 2 2 2 3 4 6 7 7 |
| 6 | 0 1 |
Check: 5+12+11+2=30 ✓, matching the 30 games.
Answer: The team scored in the 40s and 50s most often, and nothing sits far away from the rest — no outlier here.
Try it in rāSHio — sorting before you deal leaves
Paste the thirty game scores straight into File → Delimited List… — the semicolons parse as-is, so you never retype a value. Once the scores are a column you can sort them, exactly the ordering a stemplot needs before you start dealing leaves into rows.
Figure 2.1.1
Loading a raw list into rāSHio
File → Delimited List…
Paste or type a list of values with any separator, and rāSHio lands them in a spreadsheet column — sorted and ready before you build a stemplot by hand.
Figure 2.1.1: Loading a raw list into rāSHio with File → Delimited List, the first step before building a stemplot by hand.
Worked Example 2.1.1
Exam Scores in a Pre-Calculus Class
Worked Example 2.1.1 — 31 first-exam scores, 33 to 100
Scores (smallest to largest): 33, 42, 49, 49, 53, 55, 55, 61, 63, 67, 68, 68, 69, 69, 72, 73, 74, 78, 80, 83, 88, 88, 88, 90, 92, 94, 94, 94, 94, 96, 100. Construct a stemplot and describe what it shows.
| Stem | Leaf |
|---|---|
| 3 | 3 |
| 4 | 2 9 9 |
| 5 | 3 5 5 |
| 6 | 1 3 7 8 8 9 9 |
| 7 | 2 3 4 8 |
| 8 | 0 3 8 8 8 |
| 9 | 0 2 4 4 4 4 6 |
| 10 | 0 |
Answer: Most scores fall in the 60s, 70s, 80s, and 90s. Eight of the 31 scores — about 26% — are 90 or above: a fairly high number of As.
Your turn — decimals split at the point
Try It Now 2.1.2
Try It Now 2.1.2 — distance from home to college (miles)
30 distances, 0.5 to 8.0 miles. Create a stem plot and identify any outliers.
| Stem | Leaf |
|---|---|
| 0 | 5 7 |
| 1 | 1 2 2 3 3 5 5 7 7 8 9 |
| 2 | 0 2 5 6 8 8 8 |
| 3 | 5 8 |
| 4 | 4 8 9 |
| 5 | 2 5 7 8 |
| 6 | |
| 7 | |
| 8 | 0 |
Answer: Stems 6 and 7 are empty, so 8.0 sits two full stems away from the next value (5.8) — a likely outlier. Most students live within 3 miles, concentrated between 1 and 2.
Context Pause — the decimal point is the split
With decimals, the decimal point is the split
Here the leaves are the digits to the right of the decimal, so 3.3 has stem 3 and leaf 3. The rule has not changed — the leaf is still the final significant digit — the split just happens to land on the decimal point.
Worked Example 2.1.2
Distances to Local Supermarkets
Worked Example 2.1.2 — 21 distances (km), 1.1 to 12.3
1.1, 1.5, 2.3, 2.5, 2.7, 3.2, 3.3, 3.3, 3.5, 3.8, 4.0, 4.2, 4.5, 4.5, 4.7, 4.8, 5.5, 5.6, 6.5, 6.7, 12.3. Create a stemplot — do the values concentrate anywhere?
| Stem | Leaf |
|---|---|
| 1 | 1 5 |
| 2 | 3 5 7 |
| 3 | 2 3 3 5 8 |
| 4 | 0 2 5 5 7 8 |
| 5 | 5 6 |
| 6 | 5 7 |
| 7–11 | (empty) |
| 12 | 3 |
Table 2.1.4 above. Stems 3 and 4 hold the most leaves; five empty rows (7–11) separate 6.7 from 12.3. Answer: 12.3 may be an outlier. Values concentrate at three and four kilometers.
§2.1.2 — Comparing Two Data Sets: Side-by-Side Stemplots
Put two data sets on one shared column of stems — one set's leaves growing left, the other's growing right — and compare shape, spread, and overlap without flipping between two pictures.
§2.1.2 — one ruler, two data sets
Side-by-Side Stem-and-Leaf Plot
Definition 2.1.3 — Side-by-Side Stem-and-Leaf Plot
Two data sets share the same stems in two columns — one set's leaves to the left of the stems, the other's to the right.
Only adjustment: left-side leaves run outward from the stem, increasing as you move toward the center.
Definition 2.1.3: Two data sets share one column of stems, with each set's leaves growing outward.
Insight Note — a shared spine
Two histograms sharing a spine
Think of the stem column as a shared ruler with one data set hanging off each side. Because both sides are measured against the same ruler, you can see at a glance which group sits higher, which is more spread out, and where they overlap.
Your turn — two columns, one stem
Try It Now 2.1.3
Try It Now 2.1.3 — Atlanta Hawks wins and losses, 42 seasons
Losses run 19–69; wins run 13–57. Build a side-by-side stemplot: losses left, wins right.
| Losses | Stem | Wins |
|---|---|---|
| 9 | 1 | 3 |
| 9 6 5 5 | 2 | 5 6 8 8 9 |
| 9 9 9 6 6 6 5 4 4 2 2 2 2 0 | 3 | 0 1 1 1 1 3 4 5 5 6 6 7 8 |
| 9 8 7 7 6 6 5 4 2 1 1 0 0 | 4 | 0 1 1 2 2 3 3 3 6 6 6 7 8 8 |
| 7 6 4 4 3 2 1 1 1 | 5 | 0 0 0 0 2 3 6 7 7 |
| 9 | 6 |
Answer: Near-mirror images — a fixed number of games per season means many losses automatically means few wins. The 69-loss season (left) and the 13-win season (right) are the same season, standing apart on both sides.
Worked Example 2.1.3
Presidential Ages at Inauguration and at Death
Worked Example 2.1.3 — inauguration ages (42–69) vs. ages at death (46–93)
Build a side-by-side stemplot: inauguration ages left, ages at death right.
| Ages at Inauguration | Stem | Ages at Death |
|---|---|---|
| 9 9 8 7 7 7 6 3 2 | 4 | 6 9 |
| 8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 4 2 2 1 1 1 1 1 0 | 5 | 3 6 6 7 7 8 |
| 9 8 5 4 4 2 1 1 1 0 | 6 | 0 0 3 3 4 4 5 6 7 7 7 8 |
| 7 | 0 1 1 1 2 3 4 7 8 8 9 | |
| 8 | 0 1 3 5 8 | |
| 9 | 0 0 3 3 |
Answer: Inauguration ages cluster tightly in the fifties; ages at death spread from the forties to the nineties, highest in the sixties through eighties — every president is inaugurated inside a narrow career window but can die at almost any age afterward.
§2.1.3 — Line Graphs
Plot each data value as a point against its frequency, then connect consecutive points. Your eye follows the line up to a peak and back down — that rise-and-fall is the shape of the distribution.
§2.1.3 — plotting frequencies as points
Line Graph
Definition 2.1.4 — Line Graph
A line graph plots each data value as a point and connects consecutive points with line segments. The x-axis holds data values; the y-axis holds frequency.
The connecting line is a reading aid, not a claim — the segment between two points doesn't assert anything happened in between.
Definition 2.1.4: Each frequency-table row becomes a point, and the connecting segments trace the distribution's shape.
Context Pause — the line is a reading aid
The connecting line is a reading aid, not a claim
In a graph like this the categories are whole counts — you cannot be reminded 2.5 times. The segment between two points does not assert that anything happened in between; it is there so your eye can follow the trend from one count to the next.
Your turn — plot the frequency table
Try It Now 2.1.4
Try It Now 2.1.4 — car repair visits per year, 40 people
| Times in shop | Frequency |
|---|---|
| 0 | 7 |
| 1 | 10 |
| 2 | 14 |
| 3 | 9 |
Construct a line graph.
Plot (0,7), (1,10), (2,14), (3,9); connect left to right. Check: 7+10+14+9=40 ✓.
Answer: The line peaks at two repair visits per year, falling off on both sides of the peak.
Try it in rāSHio — building the frequency table for you
Paste the raw repair counts into File → Delimited List…, then choose Graph → Frequency Table with Discrete values checked. rāSHio builds the value-and-frequency pairs for you, so you can check a frequency table you tallied yourself before you plot it.
Figure 2.1.2
Building a frequency table in rāSHio
Graph → Frequency Table
Check Discrete values and Calculate — rāSHio produces the value and frequency columns a line graph plots.
Figure 2.1.2: Building a frequency table in rāSHio with Graph → Frequency Table, the two columns a line graph plots.
Worked Example 2.1.4
How Often Teenagers Need Reminding
Worked Example 2.1.4 — 40 parents surveyed
| Times reminded | Frequency |
|---|---|
| 0 | 2 |
| 1 | 5 |
| 2 | 8 |
| 3 | 14 |
| 4 | 7 |
| 5 | 4 |
Plot the six points and connect them. Answer: three reminders per week is by far the most common response — a single clear peak.
Figure 2.1.3: Line graph of how many times per week a teenager must be reminded to do chores.
§2.1.4 — Bar Graphs
One bar per category, always separated from its neighbors — the gap is the signal that these categories don't flow into one another.
§2.2's histogram looks similar but pushes its bars together, because its categories really do run continuously into each other.
§2.1.4 — separated bars, one per category
Bar Graph
Definition 2.1.5 — Bar Graph
A bar graph displays data using rectangular bars separated from one another, one bar per category. Bar length is proportional to the value it represents.
The gap is not decoration — separated bars mean categories that are genuinely distinct, not intervals on a number line.
Definition 2.1.5: Bar graph bars are separated on purpose: the gaps mark the categories as distinct.
Insight Note — what the gap tells you
The gap tells you what kind of data you are looking at
Separated bars mean categories that are genuinely distinct. Touching bars mean a number line that has been chopped into intervals. Before you read any bar chart, check the gaps — they tell you whether the horizontal axis is a list or a ruler.
Your turn — one bar per category
Try It Now 2.1.5
Try It Now 2.1.5 — Park City population by age group
| Age group | People | Proportion |
|---|---|---|
| Children | 67,059 | 19% |
| Working-age adults | 152,198 | 43% |
| Retirees | 131,662 | 38% |
Three separated bars at 19%, 43%, 38%. Check: 19+43+38=100% ✓.
Answer: Working-age adults are the largest group, but retirees are close behind — together the two adult groups make up 81% of the town.
Worked Example 2.1.5
TikTok Users by Age Group
Worked Example 2.1.5 — U.S.-based TikTok users
| Age group | Proportion |
|---|---|
| 10–19 | 32.5% |
| 20–29 | 29.5% |
| 30–39 | 16.4% |
| 40–49 | 13.9% |
| 50+ | 7.1% |
Answer: TikTok's U.S. base skews young — nearly two-thirds of users are under 30, and each successive group is smaller than the one before it.
Figure 2.1.4: Bar graph of the proportion of U.S.-based TikTok users in each age group.
Your turn — six categories, one column each
Try It Now 2.1.6
Try It Now 2.1.6 — Park City registered voters by district
| District | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Registered voters | 15.5% | 12.2% | 9.8% | 17.4% | 22.8% | 22.3% |
Six separated bars at those heights. Check: sums to 100% ✓.
Answer: Districts 5 and 6 hold the largest shares of registered voters (22.8%, 22.3%); District 3 holds the smallest (9.8%).
Worked Example 2.1.6
Projected High School Graduates by Region
Worked Example 2.1.6 — projected 2030 U.S. graduates by region
| Region | Percentage |
|---|---|
| Northeast | 16.1% |
| Midwest | 21.6% |
| South | 39.0% |
| West | 23.3% |
Answer: The South is projected to produce 39.0% of the country's graduates — more than the next two regions combined, and nearly 2.5× the Northeast's share.
Figure 2.1.5: Bar graph of the percentage of projected 2030 high school graduates by U.S. geographic region.
Key Terminology
Key Terms
stemplot — splits each observation into a stem (all digits but the last) and a leaf (the final digit).
stem — the leading digits of an observation; the row label in a stemplot.
leaf — the final significant digit, written beside its stem.
outlier — an observation that does not fit the pattern of the rest of the data.
side-by-side stemplot — two data sets sharing one column of stems, leaves left and right.
line graph — plots each data value against its frequency, joined by segments.
bar graph — one separated bar per category, length proportional to its value.
The headline result
Match the graph to what the categories DO — split, connect, or separate.
Stemplots split values to preserve every number. Line graphs connect frequencies into a trend. Bar graphs separate categories that don't flow into each other.
The choice is never decoration — it is a claim about whether the horizontal axis is a list of distinct categories or a ruler of continuous values.
† The same discipline resurfaces in §2.2: a histogram is a bar graph whose bars touch, because its categories — intervals of a number line — really do run into each other.
§2.1 — conclusions
What §2.1 leaves you with
The core idea
A stemplot keeps every value while showing shape. A line graph traces a frequency trend across ordered counts. A bar graph compares separated categories by count or percentage — the gaps between bars are the tell.
Watch for
An outlier's stem gap is only visible if you write down the empty stems too. And touching bars claim a continuity that a categorical variable doesn't actually have — check the gaps before you trust the axis.
Next: §2.2 — Histograms, Frequency Polygons, and Time Series Graphs, where "touching bars" becomes the rule instead of the exception.