A stem and a leaf
Eight players score these points in a game: 34, 23, 42, 31, 38, 27, 34 and 40. A stem-and-leaf diagram splits each number into two parts. The tens digit is the stem, and the ones digit is the leaf: 34 has a stem of 3 and a leaf of 4.
Write the stems once, in a column from smallest to largest, with a line to their right: 2, 3 and 4. Then go through the numbers and write each leaf in the row of its stem. The 34 puts a 4 in the row of 3, the 23 puts a 3 in the row of 2, and so on. The first time through, the row of 3 reads 4, 1, 8, 4, in the order the numbers came.
Every diagram needs a key, which says how a stem and a leaf make a number: 3 | 4 means 34. Without it, 3 | 4 could just as well mean 3.4 or 340.
Put the leaves in order
Now rewrite each row with its leaves in order from smallest to largest, evenly spaced. The row of 3 becomes 1, 4, 4, 8. This is an ordered stem-and-leaf diagram, and read from the top left it lists every value in order: 23, 27, 31, 34, 34, 38, 40 and 42.
Nothing is thrown away. Each leaf, read with its stem, gives back one exact value. A table that only said “2 values in the 20s” would lose the 23 and the 27; the diagram keeps them.
The row of 3 holds the four values 31, 34, 34 and 38. The key, 3 | 4 means 34, says the stem is the tens.
Each row is a bar on its side
Turn the diagram on its side and each row of leaves is like a bar: the longer the row, the more values it holds. The row of 3 is the longest, with 4 leaves, so half of the eight scores are in the 30s. The rows of 2 and of 4 have 2 leaves each.
This is why the leaves must be evenly spaced. When they are, the lengths of the rows can be compared at a glance, and the diagram shows the shape of the data as a bar chart would, while still keeping every value.
The longest row, the 30s, holds more values than any other. The two marked leaves are the two scores of 34, the 4th and 5th values in order.
The middle value and the spread
An ordered diagram makes it easy to count to the middle. The median is the middle value of a list in order. With an odd number of values n, it is the value in position . With an even number there are two middle values, and the median is halfway between them.
The 8 scores have two middle values, the 4th and the 5th. The row of 2 holds the 1st and 2nd values, so the 4th and 5th are the 2nd and 3rd leaves in the row of 3. Both are 34, so the median is 34.
The range measures how spread out the values are: the largest value minus the smallest. They are the last leaf and the first leaf, each read with its stem: 42 − 23 = 19 points. Using the stems alone, 40 − 20 = 20, ignores the leaves and gives the wrong range.
Leaves after a decimal point
The stem does not have to be the tens. For times such as 13.6 seconds, the whole seconds make the stems and the tenths make the leaves, and the key says so: 13 | 6 means 13.6 s. The diagram is built and read in exactly the same way.
Worked example: Sprint Times Put into an Ordered Stem-and-Leaf Diagram: The Median and the Range
Question The times, in seconds, of the 15 runners in a 100 m heat were 13.2, 12.4, 14.1, 12.9, 13.5, 15.0, 13.8, 12.7, 14.6, 13.2, 14.3, 13.9, 12.5, 14.8 and 13.6. (a) Draw an ordered stem-and-leaf diagram of the times and use it to find the median time. (b) Find the range of the times.
1.Use the whole seconds 12, 13, 14 and 15 as the stems. Go through the list once and write the tenths digit of each time beside its stem: 13.2 puts a 2 in the row of 13, and 12.4 puts a 4 in the row of 12.
The stem is the whole number of seconds and the leaf is the tenths digit. The leaves are written in the order of the list. 2.Put the leaves of each row in order and write the key, 12 | 4 means 12.4 s. The rows are 12 | 4 5 7 9, 13 | 2 2 5 6 8 9, 14 | 1 3 6 8 and 15 | 0. Check: 4 + 6 + 4 + 1 = 15 leaves, one for each runner.
Order the leaves in every row. There are 4 + 6 + 4 + 1 = 15 leaves. 3.With 15 values the median is the 15 + 12 = 8th value. The first row holds 4 values, so the 8th value is the 4th leaf in the row of 13, which is 6.
The median is the 8th of the 15 values. The first row holds 4, so it is the 4th leaf in the row of 13. 4.(a) The median time is 13.6 s. Seven runners were faster than this and seven were slower.
(a) The median time is 13.6 s. 5.(b) The range is the slowest time minus the fastest time, read from the last leaf and the first leaf: 15.0 − 12.4 = 2.6 s.
(b) The range is 15.0 − 12.4 = 2.6 s.
Answer: (a) the median time is 13.6 s; (b) the range is 2.6 s
Common mistakes
- Finding the median from the leaves in the order they were first written. In the row of 13 the leaves were first written as 2, 5, 8, 2, 9, 6, and counting along them gives a different value. The median is the middle value of the ordered list, so the leaves must be ordered first.
- Giving the range as 15 − 12 = 3 from the stems alone. The stems are only the whole seconds. The fastest time is 12.4 s, not 12 s, so the leaf has to be read with its stem.