Find the range
31, 32, 23, 10, 26
31, 32, 23, 10, 26
Worked Solution & Answer
(Range: Largest − smallest)
32 - 10 = 22 B1
32 - 10 = 22 B1
Answer: 22
Averages — Notes & Worked Examples Show / Hide
Explanation:
Memorise the four summary statistics: Mode = most common value; Median = middle value after putting the list in order from smallest to largest; Mean = Total of values ÷ Number of values; and Range = Biggest − Smallest (range measures spread, not average).
If there is an even number of values, there are two numbers in the middle — add those two middle numbers together and divide by 2 to find the median.
Example 1: Find the mode, median, mean, and range of:
8, 3, 5, 9, 3, 10, 4
Solution:
Order first: 3, 3, 4, 5, 8, 9, 10 (7 values)
• Mode: 3 (appears twice) • Median: 5 (4th value)
• Mean: 42 ÷ 7 = 6
• Range: 10 − 3 = 7
Example 2: Find the median, mean, and range of:
12, 7, 15, 9, 18, 11
Solution:
Order first: 7, 9, 11, 12, 15, 18 (6 values)
• Median: Halfway between 11 & 12 = 11 + 122 = 11.5
• Mean: 72 ÷ 6 = 12
• Range: 18 − 7 = 11
Find the median
6, 13, 10, 14, 19
6, 13, 10, 14, 19
Worked Solution & Answer
Ordered: 6, 10, 13, 14, 19
Median = 13 B1
Median = 13 B1
Answer: 13
Here are five cards:
[1] [5] [7] [9] [11]
[1] [5] [7] [9] [11]
One of the cards is removed.
The mean of the numbers on the remaining four cards is 6
Which card was removed?
You must show your working.
The mean of the numbers on the remaining four cards is 6
Which card was removed?
You must show your working.
Answer
Worked Solution & Answer
Sum of all five cards = 1 + 5 + 7 + 9 + 11 = 33 M1
Sum of remaining four cards = 4 × 6 = 24 M1
Removed card = 33 - 24 = 9 A1
Sum of remaining four cards = 4 × 6 = 24 M1
Removed card = 33 - 24 = 9 A1
Answer: Answer 9
Write down five positive whole numbers in the boxes below such that the numbers have:
• a range of 5 and
• a mean of 4 and
• a median of 3.
• a range of 5 and
• a mean of 4 and
• a median of 3.
Write down five positive whole numbers that satisfy the given conditions.
Worked Solution & Answer
Let the numbers in ascending order be a, b, c, d, e. M1
1. Median = 3, so middle number c = 3.
2. Mean = 4, so total sum = 5 × 4 = 20.
3. Range = 5, so e - a = 5
If a = 2, then e = 7 M1
Sum of remaining: b + d = 20 - (2 + 3 + 7) = 8
With 2 ≤ b ≤ 3 and 3 ≤ d ≤ 7:
Choose b = 2, d = 6 gives: 2, 2, 3, 6, 7.
Or choose b = 3, d = 5 gives: 2, 3, 3, 5, 7. A1
1. Median = 3, so middle number c = 3.
2. Mean = 4, so total sum = 5 × 4 = 20.
3. Range = 5, so e - a = 5
If a = 2, then e = 7 M1
Sum of remaining: b + d = 20 - (2 + 3 + 7) = 8
With 2 ≤ b ≤ 3 and 3 ≤ d ≤ 7:
Choose b = 2, d = 6 gives: 2, 2, 3, 6, 7.
Or choose b = 3, d = 5 gives: 2, 3, 3, 5, 7. A1
Answer: 2, 2, 3, 6, 7 (or 2, 3, 3, 5, 7)
This sign was in a doctor’s waiting room.
Work out the mean length of time for each missed appointment.
Give your answer in minutes.
Give your answer in minutes.
minutes
Worked Solution & Answer
(Mean = total of values ÷ number of values)
25.3 × 60 = 1518 minutes M2
1518 ÷ 115 = 13.2 minutes A1
25.3 × 60 = 1518 minutes M2
1518 ÷ 115 = 13.2 minutes A1
Answer: 13.2 minutes
Each number in a list is increased by 10
Tick **one** box for each statement.
@@TABULAR0@@
@@TABULAR0@@
Worked Solution & Answer
Adding 10 shifts every number up by 10, so the mode and median increase by 10, while the range (difference between highest and lowest) stays unchanged. M2 A1
Answer: Mode: True; Median: True; Range: False
(a) There are 4 rowers in a boat.
The mode and median of their shoe sizes are both 11.
The range of their shoe sizes is 4.
One of the rowers has size 13 shoes.
What is the shoe size of each rower?
(b) (i) How many teams does the rowing club have in total?
(b) (ii) Rona is a junior rower at the club.
What is the probability that Rona rows in a junior 8-seater boat?
The mode and median of their shoe sizes are both 11.
The range of their shoe sizes is 4.
One of the rowers has size 13 shoes.
What is the shoe size of each rower?
(b) (i) How many teams does the rowing club have in total?
(b) (ii) Rona is a junior rower at the club.
What is the probability that Rona rows in a junior 8-seater boat?
(a) There are 4 rowers in a boat.
The mode and median of their shoe sizes are both 11.
The range of their shoe sizes is 4.
One of the rowers has size 13 shoes.
What is the shoe size of each rower? [2 marks]
The mode and median of their shoe sizes are both 11.
The range of their shoe sizes is 4.
One of the rowers has size 13 shoes.
What is the shoe size of each rower? [2 marks]
(b) How many teams does the rowing club have in total? [1 mark]
(c) Rona is a junior rower at the club.
What is the probability that Rona rows in a junior 8-seater boat? [1 mark]
What is the probability that Rona rows in a junior 8-seater boat? [1 mark]
Worked Solution & Answer
(a)
Since one rower is 13 and the range is 4, the smallest size is 13 - 4 = 9 M1
With 4 rowers, the median is the average of the 2nd and 3rd sizes: (b + c) / 2 = 11
Since the mode is 11, the two middle values must both be 11
Hence the sizes in order are 9, 11, 11, 13. A1
With 4 rowers, the median is the average of the 2nd and 3rd sizes: (b + c) / 2 = 11
Since the mode is 11, the two middle values must both be 11
Hence the sizes in order are 9, 11, 11, 13. A1
Answer (a): 9, 11, 11, 13
(b)
2-seater: 3 + 2 + 1 = 6
4-seater: 5 + 1 + 3 = 9
8-seater: 2 + 1 + 0 = 3
Total = 6 + 9 + 3 = 18 B1
4-seater: 5 + 1 + 3 = 9
8-seater: 2 + 1 + 0 = 3
Total = 6 + 9 + 3 = 18 B1
Answer (b): 18
(c)
The bar chart shows there are 0 junior 8-seater boat teams, so the probability is 0 (or 0%). B1
Answer (c): 0
(a) What is the mode of the nine numbers below?
3 8 9 3 7 2 7 9 3
(b) What is the median of the seven numbers below?
8 3 4 9 2 8 1
(c) Work out the mean of the four numbers below.
3 12 10 7
3 8 9 3 7 2 7 9 3
(b) What is the median of the seven numbers below?
8 3 4 9 2 8 1
(c) Work out the mean of the four numbers below.
3 12 10 7
(a) What is the mode of the nine numbers below?
3 8 9 3 7 2 7 9 3 [1 mark]
3 8 9 3 7 2 7 9 3 [1 mark]
(b) What is the median of the seven numbers below?
8 3 4 9 2 8 1 [2 marks]
8 3 4 9 2 8 1 [2 marks]
(c) Work out the mean of the four numbers below.
3 12 10 7 [2 marks]
3 12 10 7 [2 marks]
Worked Solution & Answer
(a)
The number 3 occurs three times, which is more frequent than any other. B1
Answer (a): 3
(b)
Ordered list: 1, 2, 3, 4, 8, 8, 9. The middle (4th) value is 4. M1 A1
Answer (b): 4
(c)
Sum = 3 + 12 + 10 + 7 = 32 M1
Mean = 32 ÷ 4 = 8. A1
Mean = 32 ÷ 4 = 8. A1
Answer (c): 8
Sharifa keeps a record of the number of phone calls she makes each day. These are her results for one week:
7, 9, 6, 3, 7, 9, 6
7, 9, 6, 3, 7, 9, 6
(a) Why is the mode not suitable to use as the average number of calls made each day? [1 mark]
(b) Work out the range, and the median number of calls made each day. [2 marks]
(c) When Sharifa does not include the calls made on Saturday and Sunday, the new range is 4.
How many calls were made on Saturday? [1 mark]
How many calls were made on Saturday? [1 mark]
(d) What impact does this have on the median? [1 mark]
Worked Solution & Answer
(a)
When multiple values share the highest frequency, the mode fails to represent a central tendency. B1
Answer (a): There is no single mode (the data is multimodal: 6, 7, and 9 each occur twice)
(b)
Order the numbers: 3, 6, 6, 7, 7, 9, 9. M1
Range = 9 - 3 = 6
Median is the middle (4th) value = 7. A1
Range = 9 - 3 = 6
Median is the middle (4th) value = 7. A1
Answer (b): Range = 6, Median = 7
(c)
To obtain a new range of 4 from {3, 6, 6, 7, 7, 9, 9}, the values removed must be the two highest (9 and 9), giving remaining values {3, 6, 6, 7, 7} with range 7 - 3 = 4
Thus, 9 calls were made on Saturday. B1
Thus, 9 calls were made on Saturday. B1
Answer (c): 9
(d)
The remaining 5 numbers in order are 3, 6, 6, 7, 7. The new median is the 3rd value, 6, so it decreases by 1. B1
Answer (d): The median decreases from 7 to 6
Jeff and Ava have a pack of numbered cards.
(a) Jeff selects three cards: 3, 11, 6. [2 marks]
Jeff selects a fourth card.
The median of the numbers on his four cards is 7.5.
What number is on the fourth card?
The median of the numbers on his four cards is 7.5.
What number is on the fourth card?
(b) Ava selects three cards from the pack.
The mean of the numbers on these three cards is 8.
Ava selects a fourth card.
The mean of the numbers on her four cards is 9.5.
What number is on the fourth card? [4 marks]
The mean of the numbers on these three cards is 8.
Ava selects a fourth card.
The mean of the numbers on her four cards is 9.5.
What number is on the fourth card? [4 marks]
Worked Solution & Answer
(a)
Ordered given cards: 3, 6, 11 M1
For 4 cards with median 7.5, the sum of the two middle cards must be 7.5 × 2 = 15
Middle cards are 6 and the fourth card: 6 + x = 15 ⇒ x = 9
Ordered: 3, 6, 9, 11
median = (6 + 9)/2 = 7.5. A1
For 4 cards with median 7.5, the sum of the two middle cards must be 7.5 × 2 = 15
Middle cards are 6 and the fourth card: 6 + x = 15 ⇒ x = 9
Ordered: 3, 6, 9, 11
median = (6 + 9)/2 = 7.5. A1
Answer (a): 9
(b)
Total of 3 cards = 3 × 8 = 24 M2
Total of 4 cards = 4 × 9.5 = 38 M1
Fourth card = 38 - 24 = 14. A1
Total of 4 cards = 4 × 9.5 = 38 M1
Fourth card = 38 - 24 = 14. A1
Answer (b): 14