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Stem and leaf: worked examples and method
Foundation · Grade 2 · Non-calculator
This page sets out the method for stem and leaf in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Stem and leaf topic page.
Key facts
A key such as 2∣3 means 23 tells the reader how to put a stem and leaf back together.
Leaves must be in order from smallest to largest along each row in an ordered diagram.
Every value gets one leaf, even repeats — so 35 and 35 give two 5s in the 3 row.
The median is the middle leaf when you count through all the leaves in order.
Range = largest value − smallest value (use the last leaf of the bottom row and the first leaf of the top row).
The method, step by step
Method: drawing an ordered diagram
When you see a list of raw data and 'draw a stem and leaf diagram', do an unordered version first, then tidy it.
Write the stems (tens digits) in a column from smallest to largest, with a vertical line to their right.
Go through the data in the order given, writing each units digit as a leaf in the correct row. Cross values off as you use them.
Redraw the diagram with the leaves in each row in size order.
Add a key, e.g. 2∣3 means 23.
Count the leaves to check they match the number of data values.
Method: reading statistics off the diagram
When you see an ordered stem and leaf diagram and 'find the median/mode/range', count leaves rather than rewriting the data.
Count the total number of leaves, n.
For the median, find the middle position: the 2n+1th leaf. If n is even, average the two middle leaves.
Count through the leaves row by row to reach that position, then rebuild the value with its stem.
Mode: the leaf that repeats most within one row. Range: last value − first value.
How the lesson explains it
Stem and leaf
The stem is the rest of the number. The leaf is the last digit.
27 has stem 2 and leaf 7.
Ordered leaves
Leaves on a stem are written in order, smallest first.
An unordered row hides the median.
The key
A key such as 2 | 7 = 27 tells you the place value.
Without a key, stem 2 leaf 7 could be 2.7.
Median from the diagram
Count how many numbers there are, then find the middle position.
Walk through the leaves in order until you reach it.
Range
The range is the largest value minus the smallest value.
In an ordered diagram these are the last leaf and the first leaf. Use the key to read them as real numbers.
Back-to-back
Two sets can share a stem, one each side.
Compare the shape, then read values with the key.
The key and the place value
Rebuild one number from the key before you rebuild the rest. If the key says 3 mid 2 means 32, then stem 3 and leaf 4 is 34, and stem 1 and leaf 5 is 15. The stem is not the whole answer. A stem of 3 on its own is 30 only if the leaf is 0, and the key still has to say these are whole numbers.
A leaf of 0 is a real value. Stem 2 and leaf 0 is 20, not "nothing". Dropping the zeros shortens the list and moves the median. Count every leaf, including repeats and zeros. Each leaf is one person, one score, or one plant.
Method: 1. Read the key and write one full number from it. 2. Use the same place value for every other leaf. 3. Keep a leaf of 0. It is a real measurement. 4. Count the leaves. That count is n, the number of data values.
Median and mode
Count the leaves to get n. The median position in an ordered list is (n+1)/2. For 5 numbers the position is 3, so the median is the 3rd value. For 7 numbers it is the 4th. You walk through the leaves in order until you reach that place. Do not pick the middle stem and hope. The middle of the stems is not the middle of the leaves.
When n is even, (n+1)/2 is a half. For 10 numbers the position is 5.5, so the median is the mean of the 5th and 6th values. Add those two full numbers and divide by 2. Choosing only the 5th, or only the 6th, is a method error. The mode, if the question asks for it, is still the full number that occurs most often. A list can have more than one mode, or no mode if every number appears once.
Method: 1. Count n from the leaves. 2. Find the median place from (n + 1) divided by 2. 3. If n is odd, read that one value from the ordered diagram. 4. If n is even, average the two values either side of the halfway position.
The range
The range measures the spread from the smallest value to the largest. On an ordered diagram those are the first leaf and the last leaf, rebuilt with the key. Subtract. A diagram that starts at 16 and ends at 35 has range 35 - 16 = 19. Subtracting the stems, 3 minus 1, gives 2, which is not the range of the data.
The range uses only the two ends, so one unusual value moves it a long way. That is why a later topic, the interquartile range, ignores the ends. For this diagram you still want the range when it is asked for. Write both end numbers in full before you subtract, so a leaf of 0 cannot be lost.
Method: 1. Rebuild the first leaf as a full number. 2. Rebuild the last leaf as a full number. 3. Subtract the smallest from the largest. 4. Do not subtract the stems and call that the range.
Drawing a stem and leaf diagram from raw data
A question that says "draw a stem and leaf diagram" gives you a jumble of numbers. Work in two passes. First look at the numbers and decide the stems, which are the tens digits. Write the stems down the page in order. Then go through the data once, in the order given, and write each units digit next to the right stem.
Draw a second, ordered version with the leaves in each row going from smallest to largest. Repeated values must be repeated: if 25 appears twice, the leaf 5 appears twice in the row. Before you finish, count the leaves and check the number matches the number of data values. Add a key, such as 2 mid 5 means 25. A diagram with no key does not earn the last mark.
Method: 1. Find the smallest and largest values and write the stems in order down the page. 2. Go through the data once and put each leaf on its stem. 3. Redraw with the leaves in each row in order, smallest first, and keep repeats. 4. Count the leaves, check the total, and write a key.
Tip: If a stem has no data, still write the stem with an empty row. A missing stem hides a gap in the data.
Keys with three digits or decimals
The stem does not have to be a single digit. For heights such as 173 cm and 207 cm the stems can be 17, 18, 19, 20 and the key says 17 mid 3 means 173 cm. The leaf is always the last digit, whatever the stem is. Read the key every time, because the same row can mean 17 3, 1.73 or 173 depending on the key.
Decimals work in the same way. If the key says 5 mid 3 means 5.3 seconds, then the row 6 mid 0 2 2 7 stands for 6.0, 6.2, 6.2 and 6.7 seconds. Write each value in full on a spare line before you use it for a median or range. A range of 6.7 and 5.3 is 6.7 - 5.3 = 1.4, with the unit seconds, not 1.4 on its own.
Method: 1. Read the key and write what one example row means in full. 2. Rewrite the values you need in full, including decimal points and units. 3. Do the median, range or mode on the full values. 4. Give the answer in the same units as the key.
Finding the median by counting
The leaves are already in order in an ordered diagram, so the median is found by counting. Count all the leaves to find n. If n is odd, the median is the value in position (n + 1)/2. If n is even, the median is half way between the values in positions n/2 and n/2 + 1. For 10 values, take the 5th and 6th, add them and halve.
Count along the rows in order. Tick off each leaf as you go, starting at the first stem. Do not pick the middle stem or the middle of the longest row. The median is about the position of a value in the whole list, and a short row at the top counts just as much as a long row in the middle. State the answer as the full value, using the key.
Method: 1. Count the leaves to find the number of values. 2. Work out the position of the median. 3. Count along the rows in order until you reach that position. 4. Write the value in full using the key.
Comparing two diagrams and using the counts
To compare two sets of data, give a comment on the average and a comment on the spread, using the context. For heights: "The median height of the boys is 160 cm, which is higher than 158 cm for the girls, so the boys are taller on average. The range for the girls is smaller, so their heights are more consistent." Quote both numbers.
A stem and leaf diagram can also be used for probability. Count the total number of leaves, then count the leaves that meet the condition. If 5 of 13 students scored 45 or more, the probability that a student chosen at random did so is 5/13. Count carefully across rows, especially when a boundary value such as 45 itself is included.
Method: 1. Write the median and the range for each diagram. 2. Compare medians for the average and ranges for the spread, and say what each means. 3. For probability, count all the values, then count those that fit. 4. Write the answer as a fraction in its simplest form.