Stem and Leaf Plots

A **stem-and-leaf plot** shows every value of a data set while also showing its shape. Each number is split into two parts: - The **stem** is every digit except the last one (for two-digit numbers, the tens digit) - The **leaf** is the last digit (the ones digit) Stems are written once, in a column from smallest to largest. The leaves of each stem are written in a row to the right, **in order from smallest to largest**. A **key** tells the reader how to read one entry. Here are the quiz scores of 15 students: 52, 58, 61, 63, 67, 67, 70, 74, 75, 78, 81, 84, 86, 90, 95. $$\begin{array}{r|l} \text{Stem} & \text{Leaves} \\ \hline 5 & 2~8 \\ 6 & 1~3~7~7 \\ 7 & 0~4~5~8 \\ 8 & 1~4~6 \\ 9 & 0~5 \end{array}$$ **Key:** $5 \mid 2 = 52$ Every leaf is one data value, so the number of leaves is the number of data values (here 15). Turned on its side, the plot looks like a histogram with intervals 50-59, 60-69 and so on, but unlike a histogram you can still read every single score.

Building a Stem-and-Leaf Plot

Make a stem-and-leaf plot of these quiz scores: 74, 58, 86, 61, 90, 67, 52, 78, 70, 84, 63, 95, 67, 75, 81.

Choose the stems: The smallest score is 52 and the largest is 95, so the tens digits go from 5 to 9 = Stems: 5, 6, 7, 8, 9

Write each ones digit as a leaf next to its stem: 74 gives leaf 4 on stem 7, 58 gives leaf 8 on stem 5, and so on for all 15 scores = 15 leaves written

Put the leaves of every row in order: $$\begin{array}{r|l} 5 & 2~8 \\ 6 & 1~3~7~7 \\ 7 & 0~4~5~8 \\ 8 & 1~4~6 \\ 9 & 0~5 \end{array}$$ = Ordered stem-and-leaf plot

Add a key and check the count: Key: $5 \mid 2 = 52$. Count the leaves: $2 + 4 + 4 + 3 + 2 = 15$ = 15 leaves for 15 scores

Answer: The plot has stems 5 to 9 with ordered leaves 2 8 / 1 3 7 7 / 0 4 5 8 / 1 4 6 / 0 5 and the key $5 \mid 2 = 52$.

Reading Values from the Plot

Use the quiz plot from Example 1. What are the lowest and the highest scores, and how many students scored 80 or more?

Find the lowest score: First stem, first leaf: stem 5, leaf 2 = Lowest score = 52

Find the highest score: Last stem, last leaf: stem 9, leaf 5 = Highest score = 95

Count the scores of 80 or more: Those are the leaves on stems 8 and 9: $3 + 2 = 5$ = 5 students

Answer: The lowest score is 52, the highest is 95, and 5 students scored 80 or more.

Median, Mode and Range

Find the median, the mode and the range of the 15 quiz scores in the plot from Example 1.

Find the position of the median: With $n = 15$ values the median is value number $\frac{15 + 1}{2} = 8$ = The 8th value

Count leaves from the top: Stem 5 has 2 leaves, stem 6 has 4 (that makes 6). On stem 7: 70 is the 7th value and 74 is the 8th = Median = 74

Find the mode: Look for a leaf that repeats in one row: stem 6 has the leaf 7 twice = Mode = 67

Find the range: $95 - 52 = 43$ = Range = 43

Answer: Median = 74, mode = 67, range = 43.

Comparing Two Classes with a Back-to-Back Plot

Two classes took the same test. In a back-to-back plot the stems are in the middle, Class A's leaves go to the left and Class B's leaves to the right: $$\begin{array}{r|c|l} \text{Class A} & \text{Stem} & \text{Class B} \\ \hline 8 & 4 & \\ 9~5~3 & 5 & 7 \\ 4~4~2 & 6 & 3~8 \\ 3~1 & 7 & 2~4~6~9 \\ 0 & 8 & 4~8 \\ & 9 & 1 \end{array}$$ Key: $3 \mid 7 = 73$ for Class A, $7 \mid 2 = 72$ for Class B. Compare the medians and the ranges.

Read Class A from the stem outward: Left leaves are read from the stem to the left: 48, 53, 55, 59, 62, 64, 64, 71, 73, 80 = Class A: 10 scores

Read Class B: 57, 63, 68, 72, 74, 76, 79, 84, 88, 91 = Class B: 10 scores

Find both medians: 10 values: the median is the mean of the 5th and 6th values. A: $\frac{62 + 64}{2} = 63$, B: $\frac{74 + 76}{2} = 75$ = Medians 63 and 75

Find both ranges: A: $80 - 48 = 32$, B: $91 - 57 = 34$ = Ranges 32 and 34

Answer: Class B did better: its median (75) is 12 points higher than Class A's (63), while the spread is similar (ranges 32 and 34).

Split Stems

The commute times (in minutes) of 14 workers are 21, 23, 24, 26, 27, 27, 28, 29, 31, 32, 34, 35, 36, 38. Make a plot with split stems and find the median.

See why ordinary stems are not enough: Only the stems 2 and 3 would be used, so the plot would have just two long rows and hide the shape = Split every stem into two rows

Give each stem two rows: The first row of a stem takes the leaves 0 to 4, the second row the leaves 5 to 9 = Rows: 2 (0-4), 2 (5-9), 3 (0-4), 3 (5-9)

Place the ordered leaves: $$\begin{array}{r|l} 2 & 1~3~4 \\ 2 & 6~7~7~8~9 \\ 3 & 1~2~4 \\ 3 & 5~6~8 \end{array}$$ = Key: $2 \mid 1 = 21$ minutes

Find the median: 14 values: mean of the 7th and 8th values, $\frac{28 + 29}{2} = 28.5$ = Median = 28.5 minutes

Answer: With split stems the plot has four rows (3, 5, 3 and 3 leaves) and the median commute time is 28.5 minutes.

Stem-and-leaf plots are a quick way to organize a small or medium data set by hand: - **No data is lost**: unlike a histogram or a box plot, every original value can be read back from the plot - **Shape at a glance**: the length of each row shows where the data is crowded and where it is thin - **Ordering for free**: once the leaves are in order, the data is sorted, so the median, the mode and the range are easy to find - **Comparing groups**: a back-to-back plot puts two data sets on the same stems, for example the test scores of two classes or the temperatures of two towns Sports reporters use them for points per game, doctors for pulse rates, and timetables in many countries still list departure minutes next to each hour, which is exactly the stem-and-leaf idea.

Mistake: Writing the leaves in the order the data was given (for example 6 | 7 1 7 3)

Why: The plot still shows the shape, but counting to the median or spotting repeated values only works when each row is in order.

Correct: Always rewrite each row with the leaves from smallest to largest: 6 | 1 3 7 7.

Mistake: Leaving out the key or ignoring it

Why: The same entry $4 \mid 7$ can mean 47, 4.7 or 470. Without the key the reader has to guess.

Correct: Always write a key such as $4 \mid 7 = 4.7$ seconds and read every value with it.

Mistake: Skipping a stem that has no leaves

Why: If no value is between 30 and 39 and the row for stem 3 is left out, the gap in the data disappears and the shape looks wrong.

Correct: Write every stem from the smallest to the largest, even when a row stays empty.

Mistake: Reading the left side of a back-to-back plot in the wrong direction (reading $3 \mid 7$ as 37)

Why: On the left side the leaf stands before the stem, but the stem is still the tens digit.

Correct: Read left leaves outward from the stem: leaf 3 next to stem 7 is 73. The key of a back-to-back plot shows this.

Points per Game in Basketball

Coaches and sports reporters use stem-and-leaf plots to show a player's scoring over a season without losing any game.

A player's points in 12 games: $$\begin{array}{r|l} 0 & 8~9 \\ 1 & 2~4~4~7 \\ 2 & 0~1~5~6 \\ 3 & 3~8 \end{array}$$ Key: $1 \mid 2 = 12$ points (the stem 0 holds the one-digit scores 8 and 9).

Comparing Temperatures of Two Towns

Weather services compare towns with back-to-back plots: the same stems in the middle, one town on each side.

Daily high temperatures in °C: $$\begin{array}{r|c|l} \text{Town A} & \text{Stem} & \text{Town B} \\ \hline 9~6 & 0 & \\ 8~5~3~1 & 1 & 4~7 \\ 2~0 & 2 & 0~3~5~7~8 \\ & 3 & 1~2~4 \end{array}$$ Key: $6 \mid 0 = 6$ °C for Town A, $1 \mid 4 = 14$ °C for Town B. Town A has 8 days, Town B has 10 days.

Resting Pulse Rates

School nurses and doctors record pulse rates in stem-and-leaf plots to see at once which values are typical and which are unusual.

Resting pulse rates of 12 students in beats per minute: $$\begin{array}{r|l} 6 & 2~5~8 \\ 7 & 0~1~4~4~9 \\ 8 & 3~6 \\ 9 & 0~7 \end{array}$$ Key: $7 \mid 4 = 74$ beats per minute.

A stem-and-leaf plot splits each value into a stem (all digits but the last) and a leaf (the last digit)

Stems go in a column from smallest to largest, leaves in each row are written in order, and every plot needs a key such as $5 \mid 2 = 52$

Each leaf is one data value, so the number of leaves is the number of values and every value can be read back

Because the data is in order, the median is found by counting leaves, the mode is a repeated leaf in one row and the range is last value minus first value

A back-to-back plot shares the stems between two groups: leaves on the left are read outward from the stem

Split stems (leaves 0-4 and 5-9 in separate rows) show the shape better when the data uses only a few stems

Q: What is the stem of a one-digit number like 7?

A: The stem is 0 and the leaf is 7, so 7 is written as $0 \mid 7$. That way one-digit values fit in the same plot as two-digit values.

Q: How do I show decimals or three-digit numbers?

A: The last digit is still the leaf and the key says what it means. For 4.7 seconds use $4 \mid 7 = 4.7$; for 125 grams use $12 \mid 5 = 125$.

Q: What is the difference between a stem-and-leaf plot and a histogram?

A: Both show how many values fall in each interval. A histogram shows only the counts, while a stem-and-leaf plot keeps every single value, so you can still find the exact median, mode and range.

Q: When should I use split stems?

A: When nearly all the data shares only two or three stems, the rows get very long. Splitting each stem into leaves 0-4 and 5-9 doubles the number of rows and shows the shape more clearly.

Exit Presentation

Stem and Leaf Plots

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Statistics / Statistical Graphs

Stem and Leaf Plots

Learn to build and read stem-and-leaf plots, find the median, mode and range from them, and compare two groups with a back-to-back plot.

Intermediate25 min

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