Histograms and Stem-and-Leaf Plots
Histograms and stem-and-leaf plots are two ways to display numerical data so its pattern can be seen. This NSW Year 9 Core maths subtopic revises reading and using them from Stage 4 (NESA outcome MA4‑DAT‑C‑02), including back-to-back plots and grouped class intervals.
You will read bar heights, count scores and find the mode, mean and median from a histogram, including one with class intervals, and turn a stem-and-leaf plot back into a list to find its median, mode and range. You will build an ordered stem-and-leaf plot from raw data, read back-to-back plots from the stem outwards, as when comparing boys and girls in a writing task, choose a suitable display for a set of data, and see how one outlier changes the range. It is part of Year 9 Core Investigating Data.
Theory
A histogram shows how often each score occurs, and a stem-and-leaf plot does the same while keeping every score. This Year 9 maths page revises both displays from earlier years: reading the total and the mode, finding the mean with \(f \times x\), using class intervals, and finding the median, mode and range from ordinary and back-to-back stem-and-leaf plots.
A histogram puts the scores along the bottom and the frequency (how many times each score occurs) up the side. The bars touch, because the bottom axis is a number line. A column graph for categories, such as favourite sport, has gaps between its bars.
In a grouped histogram each bar covers a class interval, such as \(60\)–\(69\). The bar height is how many scores fall in that interval.
A stem-and-leaf plot splits each score in two. The stem is the tens digit and the leaf is the units digit, so \(6 \mid 4\) means \(64\). Every leaf is one score. The key says what a stem and leaf stand for. In some plots \(14 \mid 2\) means \(14.2\).
A back-to-back stem-and-leaf plot shows two groups on one stem. The left group's leaves are read from the stem outwards.
Numerical data with a small spread suit a dot plot or a histogram. A wider spread suits a stem-and-leaf plot, which keeps the actual values.
From a histogram, exactly as from a frequency table:
How to read a histogram or a stem-and-leaf plot
- Read the axes or the key. Check whether bars are single scores or class intervals, and what a stem and leaf mean.
- Count the scores: add the bar heights, or count the leaves.
- Find the measure you need: mode from the tallest bar or most repeated leaf, range as highest minus lowest, median by counting to the middle position.
- For the mean, multiply each score by its frequency, add, and divide by the number of scores.
- For back-to-back plots, read the left leaves outwards from the stem, and work out each group on its own before comparing.
Each bar height is a frequency, so add every height for the number of scores. The mode is the score under the tallest bar.
| \(\text{number of scores}\) | \(=\) | \(3 + 9 + 7 + 1\) |
| \(\) | \(=\) | \(\mathbf{20}\) |
| \(\text{tallest bar}\) | \(=\) | \(9\), above the score \(1\) |
| \(\text{mode}\) | \(=\) | \(\mathbf{1}\) |
The mode is the score \(1\), not the height \(9\).
A bar of height \(f\) above the score \(x\) stands for \(f\) scores of \(x\). Work out \(f \times x\) for each bar.
| \(3 \times 0\) | \(=\) | \(0\) |
| \(9 \times 1\) | \(=\) | \(9\) |
| \(7 \times 2\) | \(=\) | \(14\) |
| \(1 \times 3\) | \(=\) | \(3\) |
| \(\sum fx\) | \(=\) | \(26\) |
Divide by the number of scores, \(\sum f = 20\):
| \(\text{mean}\) | \(=\) | \(\dfrac{26}{20}\) |
| \(\) | \(=\) | \(\mathbf{1.3}\) |
| Stem | Leaf |
|---|---|
| 5 | 2 6 8 |
| 6 | 1 4 4 7 9 |
| 7 | 0 3 5 |
| 8 | 2 |
Key: \(5 \mid 2\) means \(52\)
The leaves are already in order, so read the plot from the top row to the bottom row.
| \(\text{mode}\) | \(=\) | \(64 \ (6 \mid 4 \ 4)\) |
| \(\text{range}\) | \(=\) | \(82 - 52 = \mathbf{30}\) |
Count the leaves: \(3 + 5 + 3 + 1 = 12\) scores. \(12\) is even, so average the \(6\)th and \(7\)th scores, \(64\) and \(67\).
| \(\text{median}\) | \(=\) | \(\dfrac{64 + 67}{2}\) |
| \(\) | \(=\) | \(\mathbf{65.5}\) |
Mode \(\mathbf{64}\), range \(\mathbf{30}\), median \(\mathbf{65.5}\).
| Class A | Stem | Class B |
|---|---|---|
| 8 5 | 3 | 2 |
| 7 4 1 | 4 | 0 6 9 |
| 6 3 | 5 | 1 2 5 8 |
| 0 | 6 | 3 7 |
Key: \(5 \mid 4 \mid 1\) means \(45\) for Class A and \(41\) for Class B
Class A's leaves are on the left. Read them from the stem outwards, so the leaf nearest the stem is the smallest.
Class A, \(8\) scores: \(35, 38, 41, 44, 47, 53, 56, 60\).
| \(\text{median A}\) | \(=\) | \(\dfrac{44 + 47}{2} = \mathbf{45.5}\) |
Class B, \(10\) scores: \(32, 40, 46, 49, 51, 52, 55, 58, 63, 67\).
| \(\text{median B}\) | \(=\) | \(\dfrac{51 + 52}{2} = \mathbf{51.5}\) |
Class B has the higher median, so Class B did better overall.
Common pitfalls
Frequently asked questions
How do you read a histogram?
Each bar's height is the frequency of the score under it. Add all the heights for the number of scores. The score under the tallest bar is the mode.
How do you find the mean from a histogram?
Multiply each score by its bar height, add the results, then divide by the total of the bar heights: \(\text{mean} = \dfrac{\sum fx}{\sum f}\).
How do you find the median of a stem-and-leaf plot?
Count the leaves to find how many scores there are. Find the middle position, \(\dfrac{n+1}{2}\). Count along the leaves, which are already in order, to that position. With an even number of scores, average the two middle ones.
How do you read a back-to-back stem-and-leaf plot?
The stem is in the middle. The right-hand leaves are read left to right as usual. The left-hand leaves are read from the stem outwards, so the leaf next to the stem is the smallest.
What is the difference between a histogram and a column graph?
A histogram shows numerical data on a number line, so its bars touch. A column graph shows categories, such as favourite sports, so its bars have gaps.