Dot Plots, Stem and Leaf Plots
Dot plots and stem-and-leaf plots keep every value in order, so statistics can be read straight off them. This NSW Year 8 maths subtopic (NESA outcomes MA4‑DAT‑C‑01 and MA4‑DAT‑C‑02) starts with the key, because \(7 \mid 4\) might mean 74 or 7.4.
You will read split stems and back-to-back plots, find the median, mode, range and mean, and describe the shape of the data as symmetrical or skewed, with any clusters, gaps and outliers explained in context. Comparisons include two classes’ quiz marks, two brands of phone battery and the fish caught from two boats on a school camp. It is part of Year 8 Investigating Data.
Every question with a fully worked solution.
- Dot Plots, Stem and Leaf Plots - Video - Dot Plots, Stem and Leaf Plots 2 Watch
Theory
A dot plot and a stem-and-leaf plot keep every value in order, so the range, mode, median and shape can be read straight off. Always read the key first: \(7 \mid 4\) may mean \(74\) or \(7.4\). This Year 8 maths page shows how.
The key is the scale. \(24 \mid 3\) can mean \(243\), and \(7 \mid 4\) can mean \(7.4\); the stem is not always the tens digit.
Dot plots. One dot per value above a labelled number line. Check the scale: it may run below zero or go up in halves.
Split stems. When one row holds too many leaves, split each stem in two: \(4\) takes the leaves \(0\) to \(4\) and \(4^{*}\) takes \(5\) to \(9\), with a key such as \(4^{*} \mid 6 = 46\).
Back-to-back plots compare two groups on one column of stems. The left leaves read outwards: the leaf nearest the stem is the smallest in that row.
Measures. Count the leaves or dots for \(n\), then read the middle position for the median (halve the sum of the two middle values when \(n\) is even). Range \(=\) last \(-\) first, and the mode appears most often.
Shape. Symmetrical: the halves roughly match. Positively skewed: the tail stretches to the high values. Negatively skewed: the tail stretches to the low values. Look also for clusters, gaps and outliers.
Split stems and a back-to-back plot:
| Stem | Leaf |
|---|---|
| 3 | 1 2 3 4 |
| 3* | 5 5 6 7 7 8 9 |
| 4 | 0 1 2 4 |
| 4* | 5 6 8 |
| 5 | 1 3 |
Key: \(4 \mid 1 = 41\), \(4^{*} \mid 6 = 46\)
| Monday | Stem | Friday |
|---|---|---|
| 7 4 | 2 | 3 6 9 |
| 9 6 3 1 | 3 | 0 2 5 |
| 5 | 4 | 1 8 |
Key: \(4 \mid 2 \mid 3\) means \(24\) on Monday and \(23\) on Friday
Reading a plot
- Read the key (or the scale of the dot plot).
- Count the leaves or dots for \(n\).
- Find the range, the mode and the median (middle position), and the mean if asked.
- Describe the shape: symmetrical or skewed, with any clusters, gaps or outliers.
Comparing two groups
- Find each group’s median (typical value) and range (consistency).
- Compare in a sentence using both measures.
(i) the range
(ii) the mode
(iii) the median
(iv) the mean.
| Stem | Leaf |
|---|---|
| 5 | 1 4 7 |
| 6 | 3 3 5 6 |
| 7 | 7 8 8 8 9 9 |
| 8 | 5 6 6 8 |
| 9 | 1 2 2 |
Key: \(5 \mid 1 = 51\)
There are \(20\) scores. The 10th and 11th are both \(78\).
| \(\text{Range}\) | \(=\) | \(92 - 51 = 41\) |
| \(\text{Median}\) | \(=\) | \(\dfrac{78 + 78}{2} = 78\) |
| \(\text{Mean}\) | \(=\) | \(\dfrac{1508}{20} = 75.4\) |
(i) \(41\)
(ii) \(78\)
(iii) \(78\)
(iv) \(75.4\)
(i) the range
(ii) the mode
(iii) the median
(iv) the mean.
Running totals of the dots: \(1, 3, 6, 10\), so the 10th value is \(6\) and the 11th is \(7\).
| \(\text{Range}\) | \(=\) | \(10 - 3 = 7\) |
| \(\text{Median}\) | \(=\) | \(\dfrac{6 + 7}{2} = 6.5\) |
| \(\text{Mean}\) | \(=\) | \(\dfrac{128}{20} = 6.4\) |
(i) \(7\)
(ii) \(7\) (five dots)
(iii) \(6.5\)
(iv) \(6.4\)
(i) How many jumps were at least \(3.5\) m?
(ii) Find the range.
| Stem | Leaf |
|---|---|
| 2* | 8 9 |
| 3 | 0 2 3 4 |
| 3* | 5 6 8 9 |
| 4 | 1 3 |
| 4* | 7 |
| \(4 + 2 + 1\) | \(=\) | \(7\) |
| \(\text{Range}\) | \(=\) | \(4.7 - 2.8 = 1.9\) |
(i) \(7\) jumps (the row \(3^{*}\) and below it)
(ii) \(1.9\) m
(i) Find the median.
(ii) Name the shape.
\(14\) dots, so the 7th and 8th values. Counting the columns from the left gives \(1\), then \(3\), then \(7\): the 7th is \(1\) and the 8th is \(1.5\).
| \(\text{Median}\) | \(=\) | \(\dfrac{1 + 1.5}{2} = 1.25\) |
(i) \(1.25\) hours
(ii) Positively skewed: the tallest column is at \(1\) hour and the tail runs to the right.
| Magpies | Stem | Tigers |
|---|---|---|
| 8 | 5 | |
| 3 1 | 6 | 6 8 |
| 6 4 2 0 | 7 | 0 1 2 3 5 6 8 |
| 1 | 8 | |
| 5 | 9 |
Key: \(3 \mid 6 \mid 6\) means \(63\) and \(66\) runs
Nine totals each, so the median is the 5th.
Magpies: \(58, 61, 63, 70, 72, 74, 76, 81, 95\).
Tigers: \(66, 68, 70, 71, 72, 73, 75, 76, 78\).
| \(\text{Medians}\) | \(=\) | \(72 \text{ and } 72\) |
| \(\text{Magpies range}\) | \(=\) | \(95 - 58 = 37\) |
| \(\text{Tigers range}\) | \(=\) | \(78 - 66 = 12\) |
A typical match is the same for both (\(72\) runs), but the Tigers are more consistent.
(i) Name the shape.
(ii) Find the median.
(iii) Find the mean, to \(1\) decimal place.
(iv) Which measure is better?
| Stem | Leaf |
|---|---|
| 1 | 0 1 2 4 5 7 8 |
| 2 | 0 2 4 |
| 3 | |
| 4 | |
| 5 | 5 |
Key: \(1 \mid 2 = 12\) seedlings
| \(\text{Median}\) | \(=\) | \(\text{6th value} = 17\) |
| \(\text{Total}\) | \(=\) | \(97 + 66 + 55 = 218\) |
| \(\text{Mean}\) | \(=\) | \(218 \div 11 \approx 19.8\) |
(i) Positively skewed, with \(55\) an outlier after a gap.
(ii) \(17\)
(iii) \(19.8\)
(iv) The median: the mean sits above \(7\) of the \(11\) values.
Common pitfalls
Frequently asked questions
What is a stem-and-leaf plot?
A display that splits each value into a stem (the leading digits) and a leaf (the last digit), with the leaves in order and a key.
What is a split stem?
Each stem is used twice: once for the leaves 0 to 4 and once (marked with a star) for the leaves 5 to 9.
How do you read a back-to-back stem-and-leaf plot?
The stems are in the middle. Read the right leaves from the stem out to the right, and the left leaves from the stem out to the left.
What does positively skewed mean?
Most of the data are low and the tail stretches out to the high values.
How do I find the median from a dot plot?
Count the dots, find the middle position, then count dots column by column from the left until you reach it.