Practice Maths

Stem-and-Leaf Plots and Dot Plots

Key Ideas

Key Terms

stem-and-leaf plot
displays numerical data by splitting each value into a stem (leading digit/s) and a leaf (last digit). Leaves are written in ascending order.
back-to-back stem-and-leaf plot
compares two data sets side by side sharing a common stem column. The left set’s leaves read outward (right to left).
dot plot
uses dots above a number line to represent individual data values. Each dot = one data value.
median
the middle value when data is ordered. For n values: position = (n+1)/2. If n is even, average the two middle values.
mode
the most frequently occurring value. In a stem-and-leaf plot: the most repeated leaf. In a dot plot: the tallest stack.
range
maximum value − minimum value.
outlier
a value that lies well apart from the rest of the data.

Reading a Stem-and-Leaf Plot

In the plot below, the stem is the tens digit and the leaf is the units digit.

  Stem | Leaf
     2 | 3 5 8
     3 | 1 4 4 9
     4 | 0 2 7
     5 | 6

This represents: 23, 25, 28, 31, 34, 34, 39, 40, 42, 47, 56

Hot Tip To find the median from a stem-and-leaf plot, first count the total number of leaves, then locate the middle position. Read the leaves in order from left to right (top to bottom). For a back-to-back plot, the left side reads from the stem outward (right to left).

Worked Example

Question: Find the median, mode, and range from this stem-and-leaf plot.

  Stem | Leaf
     1 | 2 5 7
     2 | 0 3 3 8
     3 | 1 5

Step 1 — List the values in order.
12, 15, 17, 20, 23, 23, 28, 31, 35.   Total = 9 values.

Step 2 — Median.
Position = (9+1)/2 = 5th value = 23.

Step 3 — Mode.
23 appears twice (most frequent). Mode = 23.

Step 4 — Range.
35 − 12 = 23.

What Is a Stem-and-Leaf Plot?

A stem-and-leaf plot is a clever way to display numerical data while keeping the actual values visible. Each number is split into two parts: the stem (usually the tens digit or leading digit) and the leaf (usually the units digit). The stems are listed in a column, and the leaves branch out to the right in order from smallest to largest.

For example, the numbers 23, 27, 31, 35, 35, 42 would be displayed as:

  Stem | Leaf
     2 | 3 7
     3 | 1 5 5
     4 | 2
  Key: 2 | 3 = 23

Always include a key so the reader knows what the stem and leaf represent. Leaves must be written in ascending order within each row.

Back-to-Back Stem-and-Leaf Plots

A back-to-back stem-and-leaf plot compares two data sets at once. One data set has its leaves going to the right (the usual way), and the other has its leaves going to the left. Both share a single stem column in the middle. This makes it easy to compare the shape, spread, and centre of the two groups.

Important: for the left-side data set, read the leaves from the stem outward (right to left). So if you see 8 5 2 | 4 |, the values are 42, 45, 48.

When comparing two groups, look at: which median is higher (who performed better on average?), which range is larger (who was more variable?), and whether the shapes look similar or different.

Reading Median and Mode from Displays

The median is the middle value when data is sorted in order. In a stem-and-leaf plot, the data is already ordered — just count in from both ends to find the middle. If there are n values, the median position is (n + 1) / 2. If that gives a .5 (e.g. 5.5), average the 5th and 6th values.

The mode is the value that appears most often. In a stem-and-leaf plot, look for the row with the most repeated leaves or the most leaves overall. In a dot plot, look for the tallest stack of dots.

The range = maximum value − minimum value. In a stem-and-leaf plot, the maximum is the last leaf on the highest stem, and the minimum is the first leaf on the lowest stem.

What Is a Dot Plot?

A dot plot uses a number line on the horizontal axis, and each data value is represented by a dot placed above its position on the number line. If the same value appears twice, a second dot is stacked above the first. Dot plots are best for small data sets and make it very easy to see clusters (where data is bunched together), gaps (where no data falls), and outliers (values far away from the rest).

Dot plots are especially useful for whole-number data like "number of siblings" or "quiz scores out of 10." They are less useful when data values are very spread out or when you have hundreds of data points.

Choosing the Right Display

Use a dot plot when: the data set is small (fewer than about 30 values), the values are whole numbers or close together, and you want to see the shape of the data clearly.

Use a stem-and-leaf plot when: values span a wider range (e.g. 20s to 80s), you need to keep the original data values visible, or you want to compare two groups with a back-to-back version.

Both displays preserve individual values (unlike a bar chart which just shows frequencies), which makes them great tools for calculating summary statistics by hand.

Key tip: Always order your leaves from smallest to largest within each stem row — unsorted leaves make it much harder to find the median and mode. When reading a back-to-back plot, remember the left side reads away from the stem, not toward it. Write out the full list of values if you are unsure, then find your statistics from the ordered list.

Mastery Practice

  1. Use this stem-and-leaf plot to answer the questions below. Fluency

    The plot shows the scores (out of 100) of 15 students on a maths test.

      Stem | Leaf
     4 | 2 8
     5 | 1 5 9
     6 | 0 3 3 7
     7 | 2 6 8
     8 | 4 9
     Key: 4 | 2 = 42
    1. List all the data values in order from smallest to largest.
    2. How many students scored in the 60s?
    3. What is the highest score?
    4. What is the lowest score?
    5. What is the range of scores?
    6. What is the mode?
    7. What is the median score?
    8. What fraction of students scored 70 or above?

  2. Use this dot plot to answer the questions. Fluency

    The dot plot shows the number of hours of TV watched per day by 18 students.

      .
  .   .   .
  .   .   .   .
  .   .   .   .   .
  ---------------------------------
  0   1   2   3   4   5
  Hours of TV per day

    (Reading bottom to top: 0 has 4 dots; 1 has 4 dots; 2 has 4 dots; 3 has 4 dots; 4 has 1 dot; 5 has 1 dot)

    1. How many students watch exactly 2 hours per day?
    2. What is the mode?
    3. How many students watch 3 or more hours per day?
    4. What is the range?
    5. What is the median?
    6. What percentage of students watch 0 or 1 hour per day?
    7. Describe the shape of the distribution (symmetric, skewed left, skewed right).

  3. Construct a stem-and-leaf plot for each data set. Then find the median, mode, and range. Fluency

    1. Heights (cm) of 12 Year 8 students: 152, 165, 148, 171, 163, 155, 168, 142, 157, 174, 160, 149
    2. Times (minutes) taken to complete a puzzle: 8, 14, 22, 9, 17, 22, 11, 25, 14, 19, 7, 20, 14, 28, 16
    3. Daily maximum temperatures (°C) for two weeks: 24, 31, 28, 27, 33, 29, 25, 30, 26, 31, 27, 34, 28, 29

  4. Use this back-to-back stem-and-leaf plot to answer the questions. Understanding

    The plot compares the number of goals scored per match by two football teams, the Eagles and the Sharks, over a 15-match season.

      Eagles         | Stem | Sharks
        9 8 5 2  |  0   | 1 3 4 6 9
          7 4 1  |  1   | 0 2 5 5 8
        8 6 3 2  |  2   | 1 4 7
              5  |  3   | 2

  Key: 2 | 0 | 1 means Eagles scored 2, Sharks scored 1
    1. List all Eagles’ scores in ascending order.
    2. List all Sharks’ scores in ascending order.
    3. What is the median score for the Eagles?
    4. What is the median score for the Sharks?
    5. What is the range for each team?
    6. What is the mode for each team?
    7. Which team had the higher median? What does this suggest about their performance?
    8. Compare the ranges. What does a larger range indicate about consistency?

  5. Create a back-to-back stem-and-leaf plot and compare the two data sets. Understanding

    The data shows the time (in minutes) two classes took to complete a science experiment.

    Class A: 32, 45, 38, 51, 44, 37, 52, 48, 41, 35, 49, 43, 36, 54, 40
    Class B: 28, 41, 35, 44, 39, 47, 33, 50, 42, 37, 46, 31, 48, 43, 38

    1. Construct the back-to-back stem-and-leaf plot.
    2. Find the median and range for each class.
    3. Compare the medians. Which class was generally faster?
    4. Compare the ranges. Which class had more consistent times?
    5. Are there any outliers in either data set? Explain how you identified them.

  6. Extended data display problems. Problem Solving

    1. A stem-and-leaf plot shows data with stems 2, 3, 4, 5. The median is 38 and there are 9 data values. The mode is 34. The range is 27. The only value in the 50s is 54. Construct a possible stem-and-leaf plot that fits all these conditions.
    2. Two athletes compare their race times (seconds) over 12 races:
      Athlete A: 58, 61, 57, 60, 63, 59, 62, 58, 65, 60, 57, 64
      Athlete B: 55, 68, 56, 70, 54, 66, 57, 71, 55, 63, 58, 69
      1. Create a back-to-back stem-and-leaf plot.
      2. Calculate the median and range for each athlete.
      3. Athlete A says she is more consistent. Athlete B says he is faster on his best days. Use statistics to support or challenge each claim.
    3. A teacher records student quiz scores (out of 20) in the dot plot below. The dot plot has a gap at 13 and a cluster between 15 and 18.
      1. Explain what a gap and a cluster tell you about the data.
      2. If the median is 16 and there are 20 students, what can you deduce about the number of students who scored 16 or below?
      3. If an outlier is defined as a value more than 5 units from the nearest other value, could a score of 7 be an outlier if the next lowest score is 14? Explain.

  7. Use the stem-and-leaf plot to answer all the questions below. Understanding

    The data shows the distances (in metres) thrown by 17 Year 8 students in a shot put event.

      Stem | Leaf
     5 | 2 4 8
     6 | 1 3 3 7 9
     7 | 0 2 5 5 8
     8 | 1 4 9
     9 | 3
     Key: 6 | 3 = 6.3 m
    1. List all data values from smallest to largest.
    2. Find the mean distance (round to 2 decimal places).
    3. Find the median distance.
    4. Find the mode(s).
    5. Find the range.
    6. What fraction of students threw 7 metres or more?
    7. Describe the shape of the distribution: is it symmetric, skewed left, or skewed right?

  8. The bar chart below shows the number of books read by students in a class over the holidays. Use it to answer the questions. Understanding

    Books read Frequency (students) Bar (█ = 1 student)
    03███
    17███████
    210██████████
    38████████
    44████
    52██
    1. How many students are in the class?
    2. What is the mode number of books read?
    3. Calculate the mean number of books read (to 1 decimal place).
    4. Find the median.
    5. What percentage of students read 3 or more books?
    6. Describe the shape of the distribution.

  9. A group of 12 athletes recorded their sprint times (seconds) for 100 m. Problem Solving

    Athlete Time (s) Athlete Time (s)
    113.2712.8
    214.1813.5
    312.5913.8
    413.21012.9
    513.61113.2
    614.51213.0
    1. Construct a stem-and-leaf plot for this data. (Use stems 12, 13, 14 where stem = units digit of the integer part.)
    2. Find the median, mode, and range.
    3. Are there any outliers? Explain how you decided.
    4. If the two slowest athletes are removed, how does the mean change? Calculate both means.

  10. A teacher gives a class quiz out of 10. The results are: 4, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 5, 6, 8, 9, 7, 6, 8, 10. Problem Solving

    1. Draw a dot plot for this data.
    2. Find the mean, median, mode, and range.
    3. The teacher says “Most students scored 8 or above.” Is this supported by the data? Show evidence.
    4. A new student scored 2. Draw a new dot plot including this value and describe how the mean and range change.
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