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Either paper 3 marks
The stem-and-leaf diagram shows the ages, in years, of the members of a chess club.
Stem Leaf 2 1 4 7 8 3 0 5 7 9 4 1 4 Key: 2 | 1 means 21 years
(a)How many members does the chess club have?
(b)State the range of the ages.
(c)How many members are aged from 30 to 39?
Show answer
(a) \(10\)
(b) \(23\ \text{years}\)
(c) \(4\)Every leaf is one member, so count the leaves: \(4 + 4 + 2 = 10\)
The youngest member is the first leaf on the top row, \(21\), and the oldest is the last leaf on the bottom row, \(44\).
\(44 - 21 = 23\) years
Those are the members on the stem 3. That row has 4 leaves, so 4 members are aged from 30 to 39.
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Either paper 4 marks
The stem-and-leaf diagram shows the number of minutes each of 11 people took to finish a puzzle.
Stem Leaf 3 1 4 9 4 1 1 4 6 8 5 1 6 7 Key: 3 | 1 means 31 minutes
(a)Find the median time.
(b)State the mode.
(c)Find the range of the times.
Show answer
(a) \(44\ \text{minutes}\)
(b) \(41\ \text{minutes}\)
(c) \(26\ \text{minutes}\)A stem-and-leaf diagram is already in order, so the values read off as
\(31,\ 34,\ 39,\ 41,\ 41,\ 44,\ 46,\ 48,\ 51,\ 56,\ 57\)
There are 11 values, so the median is the 6th one along:
\(44\) minutes
Look for a leaf that appears twice on the same stem. That is the 1 on stem 4, so two people took \(41\) minutes and the mode is \(41\) minutes.
\(57 - 31 = 26\) minutes
- (a) M1values in order \(31,\ 34,\ 39,\ 41,\ 41,\ 44,\ 46,\ 48,\ 51,\ 56,\ 57\), 6th of the 11 taken
- (a) A1median \(44\) minutes
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Either paper 4 marks
Here are the ages, in years, of nine people in a queue.
\(23,\ 39,\ 35,\ 41,\ 20,\ 45,\ 33,\ 43,\ 28\)
A stem-and-leaf diagram is to be drawn for these numbers, using the tens digit as the stem and the units digit as the leaf.
(a)State the number of leaves the diagram would have on the stem \(3\).
(b)State the leaves on the stem \(4\), in the order they would be written in the diagram.
(c)Find the median of the nine values.
Show answer
(a) \(3\)
(b) \(1,\ 3,\ 5\)
(c) \(35\ \text{years}\)The stem \(3\) holds every value from \(30\) to \(39\).
Looking down the list, there are \(3\) of them.
The values from \(40\) to \(49\) are the ones on that stem, and their units digits are the leaves.
A stem-and-leaf diagram is written in order, smallest first, so the \(3\) leaves go on as
\(1,\ 3,\ 5\)
Written in order the values are
\(20,\ 23,\ 28,\ 33,\ 35,\ 39,\ 41,\ 43,\ 45\)
With nine values the median is the \(5\)th: \(35\) years.
- (c) M1values in order \(20,\ 23,\ 28,\ 33,\ 35,\ 39,\ 41,\ 43,\ 45\)
- (c) A15th value, median \(35\) years
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Either paper 2 marks
The stem-and-leaf diagram shows the number of minutes some pupils spent on homework.
Stem Leaf 1 2 4 6 2 0 2 6 3 0 4 7 9 Key: 1 | 2 means 12 minutes
Find the fraction of the pupils who spent at least 30 minutes.
Give your answer in its simplest form.
Show answer
\(\frac{ 2 }{ 5 }\)Everything on the bottom row is at least 30, because that row has a stem of 3. That row has 4 leaves.
There are 10 pupils altogether, so the fraction is
\(\frac{ 4 }{ 10 }\), which in its simplest form is \(\frac{ 2 }{ 5 }\).
- M1\(4\) values on the bottom row out of \(10\) altogether
- A1fraction in simplest form \(\frac{ 2 }{ 5 }\)
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Either paper 3 marks
Here are the numbers of calories in 16 different snack bars.
142, 152, 143, 134, 157, 152, 132, 164, 132, 147, 167, 161, 142, 137, 161, 155
Show this information in a stem and leaf diagram. Remember to include a key.
Show answer
Key: 13 | 2 means 132 caloriesStem Leaf 13 2 2 4 7 14 2 2 3 7 15 2 2 5 7 16 1 1 4 7 Find the smallest value, 132, and the largest, 167. Their first two digits (hundreds and tens), 13 and 16, give the smallest and largest stems needed, so the stems for the diagram are 13, 14, 15 and 16. Sorted into order, smallest to largest, the 16 values are 132, 132, 134, 137, 142, 142, 143, 147, 152, 152, 155, 157, 161, 161, 164, 167 Splitting this ordered list by its first two digits (hundreds and tens) and writing each stem's units digits as its leaves, smallest first, gives
The key is read from the smallest value: 13 | 2 means 132 calories.Stem Leaf 13 2 2 4 7 14 2 2 3 7 15 2 2 5 7 16 1 1 4 7 - M1stems 13, 14, 15 and 16
- M1leaves in order, smallest first, with the repeated values shown
- B1key 13 | 2 means 132 calories
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Either paper 5 marks
The back-to-back stem-and-leaf diagram shows the times, in minutes, taken by nine adults and nine children to finish a puzzle.
Adults Stem Children 6 5 2 5 1 3 8 9 7 1 6 0 4 9 6 4 0 7 0 5 8 Key: on the adults side, 2 | 5 means 52 minutes. On the children side, 5 | 1 means 51 minutes. Both sides are read outwards from the stem.
(a)Calculate the median for the adults.
(b)Calculate the median for the children.
(c)Which group has the higher median?
Show answer
(a) \(67\ \text{minutes}\)
(b) \(64\ \text{minutes}\)
(c) AdultsOn the adults side the smallest leaf in each row is the one next to the stem, so reading each row from the stem outwards gives the values in order:
\(52,\ 55,\ 56,\ 61,\ 67,\ 69,\ 70,\ 74,\ 76\)
There are nine values, so the median is the \(5\)th: \(67\) minutes.
On the children side the leaves are already written smallest first, so in order the values are
\(51,\ 53,\ 58,\ 60,\ 64,\ 69,\ 70,\ 75,\ 78\)
The \(5\)th value is \(64\) minutes.
\(67\) minutes against \(64\) minutes, so the adults have the higher median.
Both sides share the same stems, which is what makes a back-to-back diagram worth drawing: the two sets of leaves line up row by row, so the rows can be compared directly without rewriting either list.
- (a) M1values in order \(52,\ 55,\ 56,\ 61,\ 67,\ 69,\ 70,\ 74,\ 76\), or identifying the 5th value
- (a) A1median \(67\) minutes
- (b) M1values in order \(51,\ 53,\ 58,\ 60,\ 64,\ 69,\ 70,\ 75,\ 78\), or identifying the 5th value
- (b) A1median \(64\) minutes
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Either paper 5 marks
The stem-and-leaf diagram shows the heights, in metres, of eleven young trees.
Stem Leaf 6 0 3 6 7 7 0 3 5 8 8 0 5 8 Key: 6 | 0 means 6.0 m
(a)Work out the range.
(b)Work out the median.
(c)State the number of trees that are under \(7\) m.
Show answer
(a) \(2.8\) m
(b) \(7.3\) m
(c) \(4\)The key says the leaf is the first decimal place, not a whole unit, so every value is a stem and a number of tenths.
The smallest is \(6.0\) m and the largest is \(8.8\) m.
\(8.8 - 6.0 = 2.8\) m
The diagram is already in order, so read the leaves left to right, row by row.
\(6.0,\ 6.3,\ 6.6,\ 6.7,\ 7.0,\ 7.3,\ 7.5,\ 7.8,\ 8.0,\ 8.5,\ 8.8\)
There are eleven trees, so the median is the \(6\)th value: \(7.3\) m.
Every value on a stem below \(7\) is under \(7\) m, and every value on the stem \(7\) itself is \(7\) m or more — the leaf only ever adds tenths.
Counting the leaves on the rows above that stem gives \(4\) trees.
- (a) M1largest \(8.8\) and smallest \(6.0\) read with the decimal key
- (a) A1range \(2.8\) m
- (b) M1values in order \(6.0,\ 6.3,\ 6.6,\ 6.7,\ 7.0,\ 7.3,\ 7.5,\ 7.8,\ 8.0,\ 8.5,\ 8.8\), or identifying the 6th value
- (b) A1median \(7.3\) m
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Either paper 3 marks
The stem-and-leaf diagram shows the number of emails eleven staff received in one hour.
The leaves on the middle row have not been put in order.
Stem Leaf 2 2 4 6 8 3 6 3 7 0 4 3 5 8 Key: 2 | 2 means 22 emails
(a)Write down the leaves on the middle row in order, smallest first.
(b)Calculate the median.
Show answer
(a) \(0,\ 3,\ 6,\ 7\)
(b) \(33\ \text{emails}\)Sorted smallest first, the middle row reads \(0,\ 3,\ 6,\ 7\).
There are \(11\) values, so the median is the 6th when they are in order.
The top row holds the first four. The next four are on stem 3, in order \(0,\ 3,\ 6,\ 7\), so the 6th value overall is the second of those — a leaf of 3 on stem 3.
The median is \(33\) emails.
- (b) M1middle row sorted \(0,\ 3,\ 6,\ 7\) and the 6th value counted to
- (b) A1median \(33\) emails
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Either paper 3 marks
The stem-and-leaf diagram shows the marks scored by ten pupils in a test.
Stem Leaf 1 0 4 8 2 2 4 7 8 3 0 6 8 Key: 1 | 0 means 10 marks
Work out the mean.
Show answer
\(24.7\ \text{marks}\)Read every value off the diagram, one leaf at a time:
\(10,\ 14,\ 18,\ 22,\ 24,\ 27,\ 28,\ 30,\ 36,\ 38\)
Row by row the totals are \(42\), \(101\) and \(104\), so everything adds to
\(247\)
There are \(10\) pupils, so the mean is
\(247 \div 10 = 24.7\) marks
- M1all values read from the diagram and added to give \(247\)
- M1total \(247\) divided by \(10\)
- A1mean \(24.7\) marks
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Either paper 3 marks
The stem-and-leaf diagram shows the marks of eleven pupils in a test.
Stem Leaf 5 1 4 6 9 6 1 3 5 9 7 0 5 9 Key: 5 | 1 means 51 marks
A twelfth pupil takes the test and scores \(62\) marks.
(a)Write down the stem and the leaf that would be used for \(62\).
(b)Find the median of all twelve values.
Show answer
(a) stem 6, leaf 2
(b) \(62.5\ \text{marks}\)The key says the stem is the tens digit and the leaf is the units digit, so \(62\) is written as a leaf of 2 on the stem 6.
With eleven values the median was the 6th, \(63\). Adding one more makes twelve, and the median of twelve values is halfway between the 6th and the 7th.
In order, the 6th is \(62\) and the 7th is \(63\).
\((62 + 63) \div 2 = 62.5\)
- (b) M16th and 7th values \(62\) and \(63\) with the new value in place
- (b) A1median \(62.5\) marks