GCSE Maths: Stem-and-leaf diagrams (Higher)
Part of the GCSE Maths statistics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Stem-and-leaf diagrams
A stem-and-leaf diagram shows the data in order and keeps every single value, so it is better than a grouped table when we want to find the median or range. Each number is split into a stem (the first digit or digits, usually the tens) and a leaf (the last digit, usually the units). The stems go down the left in order. The leaves go along each row in order. The diagram on its side looks like a bar chart, so we can see the shape of the data.
| Stem | Leaf |
|---|---|
| 2 | 8 9 |
| 3 | 3 4 5 8 8 |
| 4 | 1 1 1 4 7 9 |
| 5 | 2 6 |
Always include a key that shows what a stem and leaf stand for, such as “3 | 4 means 34 marks”. A key is essential, because the same diagram could mean 3.4 or 340 if the key were different. To draw one: (1) write the stems in order; (2) go through the data and write each leaf on the right row; (3) redraw with the leaves in order; (4) add the key.
Back-to-back stem-and-leaf diagrams compare two sets of data using one set of stems in the middle. The left side leaves are written backwards so that the smallest leaf is nearest the stem on both sides. To find the median of ordered data, count to the middle value; the range is the biggest minus the smallest. The decimal version works the same way, with a key like “2 | 4 means 2.4 seconds”.
Key ideas
- Stem = leading digit(s). Leaf = last digit. The leaves must be in order.
- Always write a key: “3 | 4 means 34”.
- The diagram keeps all the data, so you can find the median, mode and range exactly.
- In a back-to-back diagram, the left leaves run outwards from the stem, with the smallest nearest the stem.
- Compare two data sets by their medians (average) and ranges (spread).
Method. Split each value into a stem (the tens) and a leaf (the units). Write the leaves next to the right stem, then put each row in order.
- Write the stems in order down the left. The stems are the tens digits. List every stem from the smallest to the largest, even if a stem has no leaves.2, 3, 4, 5
- Go through the data in the order given, writing each leaf next to its stem. The leaf is the units digit.2 | 8 9
3 | 4 5 8 3 8
4 | 1 7 1 4 1 9
5 | 2 6 - Rewrite each row with its leaves in order, smallest first. An ordered diagram is much easier to read, and you need it for the median.
Stem Leaf 2 8 9 3 3 4 5 8 8 4 1 1 1 4 7 9 5 2 6 Key: 3 | 4 means 34 marks - Add a key. The key explains what the numbers mean, so that anyone can read the diagram.3 | 4 means 34 marks
Check. Count the leaves: there should be 15 leaves in total, one for each value. ✓
Watch out. Write the leaves in order and keep them evenly spaced, and always include a key.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. The diagram is already in order, so we can read the values straight from it. First write them out as a list.
- Write the data as a list from the diagram. Read each row: stem then leaf.28, 29, 33, 34, 35, 38, 38, 41, 41, 41, 44, 47, 49, 52, 56
- (a) Median. There are 15 values, so the median is the (15 + 1) ÷ 2 = 8th value.8th value = 41
- (b) Mode. 41 appears three times, more than any other value.Mode = 41
- (c) Range. Biggest minus smallest.56 − 28 = 28
- (d) Count the values above 45. Look at the end of the list.47, 49, 52, 56: that is 4 students
Check. Count in from each end of the list: 7 values are below the median and 7 are above. ✓
Watch out. Count the position of the median in the list of values, not along the stems.
| Boys | Stem | Girls |
|---|---|---|
| 2 | 8 | |
| 8 5 2 | 3 | 1 3 6 8 9 |
| 7 5 4 1 | 4 | 2 3 6 9 |
| 5 2 | 5 |
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the median) and the spread (the range) for the two groups.
- Boys: median. There are 9 values, so the median is the 5th value.Median = 44
- Girls: median. There are 10 values, so the median is halfway between the 5th and 6th.(38 + 39) ÷ 2 = 38.5
- Boys: range. 55 − 32 = 23
- Girls: range. 49 − 28 = 21
- Write a comparison. Use the medians for the average and the ranges for the spread, and write in context.On average the boys scored higher (median 44 compared with 38.5). The spread is similar (range 23 for boys and 21 for girls).
Check. The boys’ leaves are in higher rows (stems 4 and 5) more often than the girls’. ✓
Watch out. Do not read the left side of a back-to-back diagram from the right. The smallest leaf is nearest the stem.
Method. Split each value into a stem (the tens) and a leaf (the units). Write the leaves next to the right stem, then put each row in order.
- Write the stems in order down the left. The stems are the tens digits. List every stem from the smallest to the largest, even if a stem has no leaves.2, 3
- Go through the data in the order given, writing each leaf next to its stem. The leaf is the units digit.2 | 1 4 7 4
3 | 0 2 5 - Rewrite each row with its leaves in order, smallest first. An ordered diagram is much easier to read, and you need it for the median.
Stem Leaf 2 1 4 4 7 3 0 2 5 Key: 2 | 4 means 2.4 seconds - Add a key. The key explains what the numbers mean, so that anyone can read the diagram.2 | 4 means 2.4 seconds
Check. Count the leaves: there should be 7 leaves in total, one for each value. ✓
Watch out. Write the leaves in order and keep them evenly spaced, and always include a key.
Watch out
Do not forget the key, and put the leaves in order. Also, keep the leaves evenly spaced, like columns, so that the diagram is a fair picture of the data.Exercise 24I Stem-and-leaf diagrams
Non-calculator| Stem | Leaf |
|---|---|
| 1 | 9 |
| 2 | 2 3 7 9 |
| 3 | 1 1 5 8 |
| 4 | 2 5 |
| 5 | 1 |
Method. Split each value into a stem (the tens) and a leaf (the units). Write the leaves next to the right stem, then put each row in order.
- Write the stems in order down the left. The stems are the tens digits. List every stem from the smallest to the largest, even if a stem has no leaves.1, 2, 3, 4, 5
- Go through the data in the order given, writing each leaf next to its stem. The leaf is the units digit.1 | 9
2 | 3 7 2 9
3 | 1 5 1 8
4 | 2 5
5 | 1 - Rewrite each row with its leaves in order, smallest first. An ordered diagram is much easier to read, and you need it for the median.
Stem Leaf 1 9 2 2 3 7 9 3 1 1 5 8 4 2 5 5 1 Key: 2 | 3 means 23 years - Add a key. The key explains what the numbers mean, so that anyone can read the diagram.2 | 3 means 23 years
Check. Count the leaves: there should be 12 leaves in total, one for each value. ✓
Watch out. Write the leaves in order and keep them evenly spaced, and always include a key.
| Stem | Leaf |
|---|---|
| 14 | 3 7 8 9 |
| 15 | 0 2 4 5 7 8 |
| 16 | 0 1 3 6 |
| 17 | 1 |
Method. Split each value into a stem (the tens) and a leaf (the units). Write the leaves next to the right stem, then put each row in order.
- Write the stems in order down the left. The stems are the tens digits. List every stem from the smallest to the largest, even if a stem has no leaves.14, 15, 16, 17
- Go through the data in the order given, writing each leaf next to its stem. The leaf is the units digit.14 | 7 3 9 8
15 | 2 8 0 5 7 4
16 | 1 6 3 0
17 | 1 - Rewrite each row with its leaves in order, smallest first. An ordered diagram is much easier to read, and you need it for the median.
Stem Leaf 14 3 7 8 9 15 0 2 4 5 7 8 16 0 1 3 6 17 1 Key: 15 | 2 means 152 cm - Add a key. The key explains what the numbers mean, so that anyone can read the diagram.15 | 2 means 152 cm
Check. Count the leaves: there should be 15 leaves in total, one for each value. ✓
Watch out. Write the leaves in order and keep them evenly spaced, and always include a key.
| Stem | Leaf |
|---|---|
| 1 | 2 5 7 |
| 2 | 1 3 3 6 8 |
| 3 | 0 1 4 5 8 |
| 4 | 1 5 |
Method. “More than 30” does not include 30, so count the values after 30.
- Look at the stems 3 and 4. The row with stem 3 is 30, 31, 34, 35, 38; the stem 4 row is 41, 45.30, 31, 34, 35, 38, 41, 45
- Leave out 30 and count. More than 30 means bigger than 30.31, 34, 35, 38, 41, 45: 6 values
Check. Count the leaves in the stem 3 row (5) and stem 4 row (2) and subtract the 30: 5 + 2 − 1 = 6. ✓
Watch out. “More than 30” does not include 30.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. The mode is the value that appears most often.
- Write the values from the diagram as a list. Read each row: the stem then each leaf in order.12, 15, 17, 21, 23, 23, 26, 28, 30, 31, 34, 35, 38, 41, 45
- Count how many times each value appears. The mode is the value that occurs most often.12 appears 1 time
15 appears 1 time
17 appears 1 time
21 appears 1 time
23 appears 2 times
26 appears 1 time
28 appears 1 time
30 appears 1 time
31 appears 1 time
34 appears 1 time
35 appears 1 time
38 appears 1 time
41 appears 1 time
45 appears 1 time - Find the value that appears most often. 23 appears 2 times, more than any other value.Mode = 23
Check. Look back at the list: no other value appears more times. ✓
Watch out. The mode is the value, not how many times it appears.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. The median is the middle value when the data is in order.
- Write the values from the diagram as a list. Read each row: the stem then each leaf in order.12, 15, 17, 21, 23, 23, 26, 28, 30, 31, 34, 35, 38, 41, 45
- Write the values in order, smallest to largest. The median is the middle value, so the data must be in order first.12, 15, 17, 21, 23, 23, 26, 28, 30, 31, 34, 35, 38, 41, 45
- Count the values. We need to know where the middle is.There are 15 values.
- Find the middle position. With an odd number of values, the middle one has the same number of values on each side. Position = (n + 1) ÷ 2.(15 + 1) ÷ 2 = 8, so the median is the 8th value.
- Pick out that value. Count along the ordered list to the 8th value.12, 15, 17, 21, 23, 23, 26, 28, 30, 31, 34, 35, 38, 41, 45
Median = 28
Check. There are 7 values below 28 and 7 values above it. ✓
Watch out. Put the numbers in order first. The middle of an unordered list is not the median.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Range = biggest value − smallest value.
- Write the values from the diagram as a list. Read each row: the stem then each leaf in order.12, 15, 17, 21, 23, 23, 26, 28, 30, 31, 34, 35, 38, 41, 45
- Find the biggest value. The range measures how spread out the data is, from one end to the other.Biggest = 45
- Find the smallest value. Smallest = 12
- Subtract: biggest minus smallest. The range is the gap between the two ends.45 − 12 = 33
Check. Count on from 12 by 33 and you reach 45. ✓
Watch out. The range is one number (the difference), not the pair “from 12 to 45”.
| Stem | Leaf |
|---|---|
| 1 | 2 5 8 |
| 2 | 1 3 3 6 |
| 3 | 0 4 |
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. The median is the middle value when the data is in order.
- Write the values from the diagram as a list. The key says 1 | 2 is 1.2 kg.1.2, 1.5, 1.8, 2.1, 2.3, 2.3, 2.6, 3, 3.4
- Write the values in order, smallest to largest. The median is the middle value, so the data must be in order first.1.2, 1.5, 1.8, 2.1, 2.3, 2.3, 2.6, 3, 3.4
- Count the values. We need to know where the middle is.There are 9 values.
- Find the middle position. With an odd number of values, the middle one has the same number of values on each side. Position = (n + 1) ÷ 2.(9 + 1) ÷ 2 = 5, so the median is the 5th value.
- Pick out that value. Count along the ordered list to the 5th value.1.2, 1.5, 1.8, 2.1, 2.3, 2.3, 2.6, 3, 3.4
Median = 2.3 kg
Check. There are 4 values below 2.3 and 4 values above it. ✓
Watch out. Put the numbers in order first. The middle of an unordered list is not the median.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. The mode is the value that appears most often.
- Write the values from the diagram as a list. The key says 1 | 2 is 1.2 kg.1.2, 1.5, 1.8, 2.1, 2.3, 2.3, 2.6, 3, 3.4
- Count how many times each value appears. The mode is the value that occurs most often.1.2 appears 1 time
1.5 appears 1 time
1.8 appears 1 time
2.1 appears 1 time
2.3 appears 2 times
2.6 appears 1 time
3 appears 1 time
3.4 appears 1 time - Find the value that appears most often. 2.3 appears 2 times, more than any other value.Mode = 2.3 kg
Check. Look back at the list: no other value appears more times. ✓
Watch out. The mode is the value, not how many times it appears.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Range = biggest value − smallest value.
- Write the values from the diagram as a list. The key says 1 | 2 is 1.2 kg.1.2, 1.5, 1.8, 2.1, 2.3, 2.3, 2.6, 3, 3.4
- Find the biggest value. The range measures how spread out the data is, from one end to the other.Biggest = 3.4 kg
- Find the smallest value. Smallest = 1.2 kg
- Subtract: biggest minus smallest. The range is the gap between the two ends.3.4 − 1.2 = 2.2 kg
Check. Count on from 1.2 by 2.2 and you reach 3.4. ✓
Watch out. The range is one number (the difference), not the pair “from 1.2 to 3.4”.
| Boys | Stem | Girls |
|---|---|---|
| 2 | 8 | |
| 8 5 2 | 3 | 1 3 6 8 9 |
| 7 5 4 1 | 4 | 2 3 6 9 |
| 5 2 | 5 |
Method. Count to the middle of each group’s ordered data.
- Boys. 9 values, so the median is the 5th value.44
- Girls. 10 values, so halfway between the 5th and 6th values.(38 + 39) ÷ 2 = 38.5
Check. The boys have 4 values below their median and 4 above. ✓
Watch out. With 10 values there are two middle values.
Method. Biggest minus smallest for each group.
- Boys. 55 − 32 = 23
- Girls. 49 − 28 = 21
Check. Each range is the gap between the outer values of the diagram. ✓
Watch out. Use the first and last leaves of each side.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Use the medians and the ranges.
- Compare the averages. The median of the boys is higher.44 > 38.5
- Compare the spreads. The ranges are close.23 and 21
- Write the comparison in context. The boys scored higher on average, but the scores of the two groups were about equally spread out.
Check. Both statements refer to the numbers. ✓
Watch out. Always compare an average and a spread, and write the answer in context.
Method. Check the rules for a stem-and-leaf diagram one at a time.
- Check the order of the leaves. The leaves in each row must be written smallest first. In the row 2 | 8 3 1 the leaves are not in order.Not ordered: should be 2 | 1 3 8 and 3 | 2 5
- Check for a key. Without a key, we cannot tell whether 2 | 8 means 28, 2.8 or 280.No key.
- State the two faults. The leaves are not in order, and there is no key.
Check. A correct diagram would be 2 | 1 3 8; 3 | 2 5; 4 | 9, with a key such as “2 | 1 means 21 years”. ✓
Watch out. Leaves in order and a key are marked in exam questions.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. To find the mean, add up all the values and divide by how many values there are.
- Add up all the values. The mean uses the total of all the data.23 + 31 + 19 + 42 + 35 + 27 + 31 + 45 + 22 + 38 + 29 + 51 = 393
- Count how many values there are. The mean shares the total equally between all the values, so we need to know how many there are.There are 12 values.
- Divide the total by the number of values. mean = total ÷ number of values.393 ÷ 12 = 32.7500… = 32.8 to 1 decimal place
Check. The mean must lie between the smallest (19) and the largest (51) value, and 32.8 does. Also 32.8 × 12 = 393.6, which is close to the total 393. ✓
Watch out. The mean is not the middle value. Add every value, and divide by how many values there are (not by the largest value).
| Stem | Leaf |
|---|---|
| 6 | 1 4 8 |
| 7 | 0 3 3 9 |
| 8 | 5 |
Method. Read each row: put the stem and then each leaf together.
- Read the row for stem 6. The key says 6 | 1 means 61.61, 64, 68
- Read the row for stem 7. 70, 73, 73, 79
- Read the row for stem 8. 85
- Write the list. 61, 64, 68, 70, 73, 73, 79, 85
Check. There are 8 leaves in the diagram and 8 numbers in the list. ✓
Watch out. Put the stem in front of each leaf: stem 7 and leaf 3 gives 73.
| Stem | Leaf |
|---|---|
| 1 | 2 5 7 |
| 2 | 1 3 3 6 8 |
| 3 | 0 1 4 5 8 |
| 4 | 1 5 |
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.12, 15, 17, 21, 23, 23, 26, 28, 30, 31, 34, 35, 38, 41, 45
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 15, median = 28 (the middle value)
- Split into two halves, leaving out the median. With an odd number of values the median belongs to neither half.Lower half: 12, 15, 17, 21, 23, 23, 26
Upper half: 30, 31, 34, 35, 38, 41, 45 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 21
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 35
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.35 − 21 = 14
Check. About a quarter of the values are below Q1 and a quarter are above Q3. Here 3 are below 21 and 3 are above 35. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Method. Compare what each display keeps.
- Stem-and-leaf. Every value is still shown (stem and leaf together make the value).All the data is kept.
- Grouped table. Only the number of values in each class is shown.The exact values are lost.
- Conclude. The exact median can be found from the ordered diagram but not from the grouped table.Maya is right.
Check. From the grouped table we could only name the median class. ✓
Watch out. Grouped tables lose the detail that stem-and-leaf diagrams keep.
| Stem | Leaf |
|---|---|
| 1 | 0 2 5 8 |
| 2 | 0 1 6 9 |
| 3 | 3 6 |
| 4 | 4 |
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Work out the mean and the median, then compare them.
- Median. There are 11 values, so the median is the 6th value.Median = 21
- Total. Add the values.10 + 12 + 15 + 18 + 20 + 21 + 26 + 29 + 33 + 36 + 44 = 264
- Mean. Divide by 11.264 ÷ 11 = 24
- Compare. 24 > 21, so the mean is greater than the median.
Check. The larger values 36 and 40 are far above the median and pull the mean up. ✓
Watch out. State both values and the comparison.
| Group X | Stem | Group Y |
|---|---|---|
| 7 2 | 2 | 5 9 |
| 8 5 1 | 3 | 2 3 |
| 6 4 | 4 | 0 1 8 |
| 5 | 0 |
Method. Use one set of stems down the middle. Group X’s leaves go on the left, written outwards from the stem, and group Y’s go on the right.
- Write the stems in the middle. The scores are from 20 to 50, so the stems are 2, 3, 4 and 5.2, 3, 4, 5
- Put the leaves of Group X on the left. Write each row with the smallest leaf nearest the stem.X: 2 | 2 7; 3 | 1 5 8; 4 | 4 6
- Put the leaves of Group Y on the right. Again with the smallest leaf nearest the stem.Y: 2 | 5 9; 3 | 2 3; 4 | 0 1 8; 5 | 0
- Draw the diagram and add a key.
Group X Stem Group Y 7 2 2 5 9 8 5 1 3 2 3 6 4 4 0 1 8 5 0 Key: for Group X 3 | 1 means 31; for Group Y 3 | 2 means 32
Check. Count the leaves: 7 for Group X and 8 for Group Y. ✓
Watch out. On the left side the leaves run backwards, so the smallest is next to the stem.
| Stem | Leaf |
|---|---|
| 1 | 2 8 |
| 2 | 1 3 6 9 |
| 3 | 4 5 |
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. The median is the middle value when the data is in order.
- Write the lengths as a list. The key says 1 | 2 is 1.2 metres, so each leaf is a tenth of a metre.1.2, 1.8, 2.1, 2.3, 2.6, 2.9, 3.4, 3.5
- Write the values in order, smallest to largest. The median is the middle value, so the data must be in order first.1.2, 1.8, 2.1, 2.3, 2.6, 2.9, 3.4, 3.5
- Count the values. We need to know where the middle is.There are 8 values.
- Find the two middle positions. With an even number of values there are two middle values. They are at positions n ÷ 2 and n ÷ 2 + 1.8 ÷ 2 = 4, so the 4th and 5th values.
- Pick out the two middle values. Count along the ordered list.1.2, 1.8, 2.1, 2.3, 2.6, 2.9, 3.4, 3.5
- Find the number halfway between them. Add the two middle values and divide by 2 (their mean).(2.3 + 2.6) ÷ 2 = 4.9 ÷ 2 = 2.45 m
Check. 4 values are below 2.45 and 4 values are above it. ✓
Watch out. Put the numbers in order first. The middle of an unordered list is not the median.