Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Year 7, Unit 6, lesson 3) Tally 4, 2, 4, 5, 4, 2. What is the frequency of 4?
Answer
3. The 4 appears three times; the frequencies 2, 3 and 1 add to the 6 values. - (Year 7, Unit 1, lesson 1) Put 0.6, 0.45, 0.5 and 0.09 in order, smallest first.
Answer
0.09, 0.45, 0.5, 0.6. Compare the tenths first, then the hundredths. - (Year 7, Unit 5, lesson 3) What number is halfway between 18 and 26?
Answer
22. Add them and halve: 44 ÷ 2 = 22.
Words used in this lesson
- stem
- the first digits of a value, the tens in this lesson. For example: in 47 the stem is 4.
- leaf
- the last digit of a value, the units in this lesson. For example: in 47 the leaf is 7.
- key
- a note that says what one entry in the plot means. For example: 4 | 7 means 47 kg.
- mode
- the value that appears most often. For example: in 2, 5, 5, 9 the mode is 5.
- median
- the middle value when the data is in order. For example: in 2, 5, 9 the median is 5.
- data set
- all the values collected for one question. For example: 21 waiting times.
What does “typical” mean?
A sign at a minibus stop in Kisumu says “Typical wait: about 25 minutes.” Twenty-one passengers each timed how long they waited, and wrote the time in minutes. A stem-and-leaf plot keeps every one of those times and also shows their shape. Predict first: will the most common wait turn out to be shorter or longer than 25 minutes? Then place the waits one at a time, and press the button to put the leaves in order.
5 waits placed, leaves in the order they arrived. Press Put the leaves in order to see the middle value.
Show me the steps
- 5 waits placed (23, 31, 18, 25, 42): every value appears once, so there is no mode yet. In order they are 18, 23, 25, 31, 42, so the median is 25.
- 12 waits placed, leaves in order: 23 appears 3 times, more than any other value, so the mode is 23. With 12 values the median is the mean of the 6th and 7th, (25 + 27) ÷ 2 = 26.
- 21 waits placed, leaves in order: stem 1 has 2 4 8 9, stem 2 has 3 3 3 3 5 6 7 8 9, stem 3 has 1 1 3 6 7 8, stem 4 has 2 5. The mode is 23 (4 times) and the median is the 11th value, 27.
A stem-and-leaf plot splits each value into a stem and a leaf. For two-digit values the stem is the tens and the leaf is the units. The key says what one entry means, such as 2 | 3 = 23 minutes. Put the leaves in order, smallest nearest the stem, and the plot lists the whole ordered data set.
The mode is the value that appears most often: the longest run of the same leaf in one row. There can be two modes, or none when every value appears once.
The median is the middle value of the ordered data. With n values it is the ((n + 1) ÷ 2)th value when n is odd, and the mean of the two middle values when n is even. It splits the data so that half of the values are at or below it and half are at or above it.
Why order first: the middle only means something once the values are in order.
A common mistake: “the median is the middle of the list as it was written.” The first five waits were written 23, 31, 18, 25, 42, and the third one written is 18. In order they are 18, 23, 25, 31, 42, and the middle of those is 25. The list has to be in order before you count to the middle.
Worked example: build a plot, then read it
Heights of 11 seedlings, in cm: 41, 28, 35, 42, 28, 39, 31, 47, 35, 28, 44. Make a stem-and-leaf plot and find the mode and the median.
- The stems are the tens digits: 2, 3 and 4. Write each leaf in its row: stem 2 has 8, 8, 8; stem 3 has 5, 9, 1, 5; stem 4 has 1, 2, 7, 4.
- Put each row in order and add the key. 2 | 8 8 8, 3 | 1 5 5 9, 4 | 1 2 4 7, key 2 | 8 means 28 cm. The row counts 3 + 4 + 4 = 11 match the 11 seedlings.
- The longest run of one leaf is 8 in stem 2: mode 28 cm, three times.
- With 11 values the median is the (11 + 1) ÷ 2 = 6th value. Count: 3 in stem 2, then 1, 5, 5 in stem 3. Median 35 cm.
Two plots back to back, and an even number of values
Two football clubs list the ages of 7 players each. A back-to-back plot shares one column of stems: one club's leaves read outwards to the left and the other's to the right, each with the smallest leaf nearest the stem.
| Club X leaves | Stem | Club Y leaves |
|---|---|---|
| 5 5 2 | 2 | 0 3 9 |
| 8 4 1 | 3 | 0 6 6 |
| 1 | 4 | 4 |
Key: a leaf 5 beside stem 2 means 25 years, on either side. Read the left side from the stem outwards.
Find each club's median age.
- Club X, in order: 22, 25, 25, 31, 34, 38, 41. Seven values, so the median is the 4th: 31.
- Club Y, in order: 20, 23, 29, 30, 36, 36, 44. The 4th is 30.
- The clubs' middle ages differ by only 1 year. If a club had 8 players, the median would be the mean of the 4th and 5th values, halfway between them.
Your go. The first step is given. The number of messages 10 students sent in a day: 24, 19, 32, 19, 27, 41, 19, 35, 27, 22. Make an ordered stem-and-leaf plot and find the mode and the median.
- The stems are 1, 2, 3 and 4. Stem 1 has the leaves 9, 9, 9 (the three 19s).
- Now fill in stems 2, 3 and 4 in order, add the key, then find the mode and the median, then check.
Check the answer
1 | 9 9 9, 2 | 2 4 7 7, 3 | 2 5, 4 | 1, key 1 | 9 means 19 messages. The mode is 19. With 10 values the median is the mean of the 5th and 6th, 24 and 27: 25.5.
Check before you go on
1. Hana writes down the number of goals in five matches in this order: 6, 2, 9, 4, 7. What is the median?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. The plot shows the lengths in cm of 13 leaves. Key: 3 | 1 means 31 cm.
1 | 6 8
2 | 0 3 3 5 9
3 | 1 4 4 4 7
4 | 2
What is the mode, in cm?
3. A shoe shop sold these sizes in one day: 5, 8, 5, 6, 9, 5, 7. It will restock the size it sold most often. Which size?
Level 2 · Building
4. Use the leaf-length plot from question 2 (13 values). What is the median, in cm?
5. Find the median of these six numbers: 14, 20, 11, 20, 16, 9.
6. Standing long jumps, in metres, of 7 students. Key: 2 | 3 means 2.3 m.
1 | 2 5 5 8
2 | 0 3 4
What is the median jump, in metres?
Level 3 · Stretch
7. Six values are in order, and the median is 14. The 3rd value is 11. What is the 4th value?
8. Marks for two classes of 9 students. Key: a leaf 4 beside stem 6 means 64, on either side.
Class P leaves | stem | Class Q leaves
7 2 | 5 | 5 8
8 4 4 1 | 6 | 2 6 9
9 5 3 | 7 | 1 4 7
(none) | 8 | 5
By how many marks is class Q's median higher than class P's?
Something to think about
Can the median be a number that is not in the data set? Can the mode be the smallest value in the set while the median is much larger? Make up a small example for each.
One way to think about it
Yes to both. 2, 4, 6, 8 has median 5, which is not in the list, because with an even number of values the median is halfway between the two middle ones. 1, 1, 5, 7, 9 has mode 1, the smallest value, while its median is 5.
Exit ticket
9. The number of push-ups 8 students did in one minute. Key: 4 | 1 means 41 push-ups.
3 | 5 8 8
4 | 1 2 6
5 | 0 4
What is the median?