The times, t minutes, taken by 40 runners to complete a 5 km park run are summarised by
Σt = 1260 Σt2 = 41 200
Calculate the mean time.
Calculate the standard deviation of the times.
One more runner completes the course in exactly the mean time. State, giving a reason, the effect on the standard deviation of including this runner.
4 marks
Show answer
a) 31.5 minutesb) 6.14 minutes (3 s.f.)c) the standard deviation decreases: the new time is 0 from the mean, so the sum of squared deviations is unchanged but n increases
3
The table shows the number of goals, x, scored by a hockey team in each of 30 matches.
Goals, x
0
1
2
3
4
5
Frequency
4
8
9
5
3
1
Write down the mode.
Find the median.
Calculate the mean number of goals.
Calculate the standard deviation of the number of goals.
5 marks
Show answer
a) 2b) 2c) 1.93 (3 s.f.)d) 1.29 (3 s.f.)
4
The masses, m grams, of 80 apples are summarised in the table.
Mass, m (grams)
Frequency
100 ≤ m < 120
6
120 ≤ m < 130
14
130 ≤ m < 140
25
140 ≤ m < 150
21
150 ≤ m < 170
14
Use linear interpolation to estimate the median mass.
Estimate the mean and the standard deviation of the masses.
Explain why your answers to part (b) are estimates.
7 marks
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a) 138 gb) mean 138 g (3 s.f.), standard deviation 13.7 g (3 s.f.)c) the exact masses are not known; each apple is assumed to have the midpoint mass of its class
5
A machine fills bottles with olive oil. The volume, x ml, in each of 20 bottles is measured and the data are coded using
y = x − 5002
The coded data are summarised by Σy = 36 and Σy2 = 410
Find the mean of x.
Find the standard deviation of x.
Find the value of Σx2.
6 marks
Show answer
a) 503.6 mlb) 8.31 ml (3 s.f.)c) 5 073 640
6
A helpline records the time, t minutes, that each of 200 callers waits before their call is answered.
Waiting time, t (minutes)
Frequency
0 ≤ t < 2
18
2 ≤ t < 4
46
4 ≤ t < 6
62
6 ≤ t < 10
48
10 ≤ t < 15
20
15 ≤ t < 25
6
Use linear interpolation to estimate the median waiting time.
Use linear interpolation to estimate the 10th to 90th interpercentile range.
Estimate the number of callers who waited more than 8 minutes.
Find the mean and the standard deviation of the marks of all 30 students.
6 marks
Show answer
mean 61, standard deviation 6.74 (3 s.f.)
8
Marek measured the lengths, in cm, of 15 fish caught in a lake. He found the mean length was 52 cm and the standard deviation was 8 cm.
He then noticed that one length had been written down as 48 cm when it should have been 84 cm.
Find the correct mean and the correct standard deviation of the lengths.
5 marks
Show answer
mean 54.4 cm, standard deviation 11.2 cm
9
The prices, in thousands of pounds, of the nine houses sold in a village last year were
185 210 196 224 202 1150 215 190 207
Calculate the mean price.
Find the median price.
An outlier is a value greater than Q3 + 1.5 × (Q3 − Q1) or less than Q1 − 1.5 × (Q3 − Q1). Show that 1150 is an outlier.
State, giving a reason, whether the mean or the median gives a better average price for a house in this village.
6 marks
Show answer
a) £309 thousand (3 s.f.)b) £207 thousandc) Q1 = 196, Q3 = 215, upper limit 243.5 < 1150, so 1150 is an outlierd) the median, because the mean is pulled up by the outlier 1150
10
A group of people were asked how many times, x, they went to the cinema last month. The results are shown in the table, where f is a positive integer.
x
0
1
2
3
4
Frequency
7
12
f
6
3
The mean number of visits is 1.6
Show that f = 7
Calculate the standard deviation of x.
Find the median of x.
6 marks
Show answer
a) (42 + 2f)/(28 + f) = 1.6 gives 0.4f = 2.8, so f = 7b) 1.22 (3 s.f.)c) 1
11
Anya records the daily maximum temperature, x °C, at a weather station in Scotland for the 31 days of July. She codes the data using
y = 2x − 30
and finds Σy = 155 and Σy2 = 1520
Find the mean daily maximum temperature.
Find the standard deviation of the daily maximum temperatures.
For the same 31 days at a weather station in southern England the mean daily maximum temperature was 21.3 °C and the standard deviation was 4.1 °C.
Compare the daily maximum temperatures at the two weather stations in July.
7 marks
Show answer
a) 17.5 °Cb) 2.45 °C (3 s.f.)c) on average it was warmer in southern England (mean 21.3 °C against 17.5 °C), and temperatures there varied more (standard deviation 4.1 °C against 2.45 °C)
12
Eighteen students took a quiz. Their scores had a mean of 12 and a standard deviation of 2
Two more students, Ola and Kwame, then took the quiz. Ola scored a and Kwame scored b, where a < b.
For all 20 students the mean score was 13 and the standard deviation was √13
Find the value of a and the value of b.
8 marks
Show answer
a = 20, b = 24
Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.