A stem-and-leaf plot is a compact way to organize numerical data while keeping the original values visible. It shows the shape of a distribution, including clusters, gaps, skew, and outliers, without hiding individual data points. This makes it especially useful when a data set is small or medium sized and you still want exact values.
Students use stem-and-leaf plots to connect raw lists of numbers to more visual summaries like histograms and box plots.
To build one, split each value into a stem and a leaf, usually by place value. For example, in a data set of two-digit numbers, 47 can be written with stem 4 and leaf 7. The stems are listed in order, and the leaves are placed beside them in increasing order so the data become easy to scan.
A back-to-back stem-and-leaf plot uses one shared stem column to compare two related groups, such as test scores from two classes.
Understanding Statistics: Stem-and-Leaf Plots in Depth
The most important choice is the scale of the plot. A stem does not always stand for tens. For heights measured in centimetres, a stem might represent the whole-number part and a leaf might represent tenths.
For data from 120 to 199, the first two digits can form the stem, leaving the final digit as the leaf. The key tells the reader exactly how to rebuild every number. Without a clear key, the picture can be misleading.
A leaf of 5 could mean 5, 50, or 0.5 depending on the chosen scale. Every stem must use the same place-value rule.
Repeated leaves are meaningful. If a stem has three leaves of 6, then that exact value occurred three times. Do not combine them into one leaf or write a count instead.
The repeated values show frequency while preserving the raw data. Include stems with no leaves when they help reveal a gap in the data. A blank row between occupied stems can show that no observations fell in that interval.
For a wide data set, one stem can be split into two rows. One row holds leaves from zero through four, and the other holds leaves from five through nine. This prevents one crowded row from becoming hard to read.
Once the plot is complete, read it as a distribution rather than just a list. Look for where most observations gather. This gives a sense of the typical region.
Look at how far the values spread from the smallest to the largest. A long tail toward larger values suggests right skew. A long tail toward smaller values suggests left skew.
A gap may mean the data come from two different groups, though it can happen by chance in a small sample. An isolated value deserves checking.
It might be a recording mistake, a rare event, or a real result that should not be ignored. The plot alone cannot explain the cause.
Students meet this kind of reasoning in test scores, daily temperatures, sports times, rainfall totals, and survey results. A back-to-back plot can compare two classes or two teams, but both sides must use the same stems and the same leaf scale. Otherwise the comparison is unfair.
Stem-and-leaf plots work best when there are enough values to show a pattern but not so many that the page becomes crowded. With hundreds of observations, a histogram or computer graph is usually clearer. When learning, practise reconstructing the original list from the leaves.
Count the values, check their order, and compare the smallest and largest entries with the raw data. These checks catch most errors.
Key Facts
- In 47, using tens as stems, the stem is 4 and the leaf is 7.
- A key explains place value, such as 4 | 7 = 47.
- Leaves should be ordered from least to greatest within each stem.
- Number of leaves = number of data values, unless a split-stem design is used and values are still counted once.
- Range = maximum value - minimum value.
- A stem-and-leaf plot preserves exact data values, while a histogram groups values into intervals.
Vocabulary
- Stem
- The stem is the leading digit or group of digits that organizes data values by place value.
- Leaf
- The leaf is the final digit or digits attached to a stem to complete each data value.
- Key
- The key tells how to read the stems and leaves as actual numbers.
- Back-to-back stem-and-leaf plot
- A back-to-back stem-and-leaf plot compares two data sets using a shared stem column with leaves extending in opposite directions.
- Distribution
- A distribution describes how data values are spread across the number line, including center, spread, clusters, and outliers.
Common Mistakes to Avoid
- Forgetting the key is wrong because readers may not know whether 6 | 2 means 62, 6.2, or another value.
- Leaving leaves unordered is wrong because it makes the plot harder to read and can hide the median, clusters, and gaps.
- Mixing place values in one plot is wrong because stems and leaves must follow one consistent rule for every value.
- Treating repeated leaves as duplicates to remove is wrong because repeated values are real data points and must appear every time they occur.
Practice Questions
- 1 Create a stem-and-leaf plot for the data set 42, 37, 45, 41, 39, 52, 48, 44, 37, 50. Use tens digits as stems and ones digits as leaves.
- 2 The plot has key 6 | 3 = 63 and rows 4 | 8 9, 5 | 1 1 6, 6 | 0 3 7, 7 | 2. List the original data values, then find the range.
- 3 Two classes take the same quiz. Class A scores are tightly clustered around the 80s, while Class B has scores spread from the 50s to the 90s. Explain how a back-to-back stem-and-leaf plot would help compare the two classes better than two separate lists.