Standard Deviation - Worksheets, Questions and Revision

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H04 Standard Deviation

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A cafe manager records how long, in minutes, 6 customers wait to be served:

3, 5, 4, 6, 3, 9
(a)Work out the mean waiting time.(2)
(b)Work out the range of the waiting times.(1)
(Total for Question 1 is 3 marks)
2
For each statement about standard deviation, write down whether it is True or False.
(i)The standard deviation of a data set can never be negative.(1)
(ii)If every value in a data set is identical, the standard deviation is 0.(1)
(iii)A data set with a larger standard deviation always has a larger mean than a data set with a smaller standard deviation.(1)
(iv)The standard deviation of a data set is measured in the same units as the original data.(1)
(Total for Question 2 is 4 marks)
3
A hockey team scores the following number of goals in 5 matches:

2, 5, 3, 4, 6

The mean number of goals scored is 4.

Key formula: standard deviation = √sum of (x - mean)2 / n
(a)Show that the sum of (x - mean)2 for this data set is 10.(2)
(b)Hence work out the standard deviation of the number of goals scored, giving your answer correct to 3 significant figures.(2)
(Total for Question 3 is 4 marks)
4
5 friends compare how many text messages they sent yesterday:

8, 12, 10, 14, 6

An equivalent formula for standard deviation is:

standard deviation = √sum of x2 / n - mean2
(a)Show that the sum of x2 for this data set is 540.(2)
(b)Write down the mean number of texts sent.(1)
(c)Using the formula above, calculate the standard deviation of the number of texts sent, giving your answer correct to 3 significant figures.(2)
(Total for Question 4 is 5 marks)
5
Consider the evenly-spaced sequence of numbers:

5, 10, 15, 20, 25
(a)Using the symmetry of the sequence, write down its mean.(1)
(b)Calculate the standard deviation of the sequence, giving your answer correct to 3 significant figures.(3)
(Total for Question 5 is 4 marks)
6
The resistance, in ohms, of 5 electrical components is measured:

18, 22, 19, 21, 20
(a)Write down the formula that connects the variance of a data set to its standard deviation.(1)
(b)Calculate the variance of the 5 resistance readings.(2)
(c)Hence write down the standard deviation of the resistance readings, correct to 3 significant figures, stating its units.(2)
(d)Give one advantage of using the standard deviation, rather than the variance, to describe the spread of the resistance readings.(1)
(Total for Question 6 is 6 marks)
7
Two market stalls record their sales, in pounds, over the same 5 days.
Day12345
Stall A sales (£)4045384245
Stall B sales (£)2060157045
(a)Show that the mean daily sales for both Stall A and Stall B is £42.(2)
(b)Calculate the standard deviation of Stall A's daily sales, giving your answer correct to 3 significant figures.(3)
(c)Calculate the standard deviation of Stall B's daily sales, giving your answer correct to 3 significant figures.(3)
(d)Using the standard deviations, write down which stall has more consistent daily sales, giving a reason.(2)
(Total for Question 7 is 10 marks)
8
The bar chart shows the number of siblings for each of 20 pupils in a class survey.
Number of siblings01234
Frequency37631

Key formula for frequency data: standard deviation = √sum of fx2 / sum of f - mean2
0 1 2 3 4 5 6 7 8 0 1 2 3 4 Number of siblings Frequency
(a)Write down the modal number of siblings.(1)
(b)Show that the sum of fx for this data is 32.(2)
(c)Hence write down the mean number of siblings for this class.(1)
(d)Show that the sum of fx2 for this data is 74.(2)
(e)Using the formula above, calculate the standard deviation of the number of siblings, giving your answer correct to 3 significant figures.(3)
(Total for Question 8 is 9 marks)
9
A gardener records the height, h cm, of 30 plants in a grouped frequency table.
Height, h (cm)0 < h ≤ 1010 < h ≤ 2020 < h ≤ 3030 < h ≤ 40
Frequency49116
(a)Write down the midpoint of the class 20 < h ≤ 30.(1)
(b)Using midpoints 5, 15, 25 and 35, show that an estimate for the sum of fx is 640.(2)
(c)Hence write down an estimate for the mean height of the plants, correct to 3 significant figures.(1)
(d)Show that an estimate for the sum of fx2 is 16350.(2)
(e)Using standard deviation = √sum of fx2 / sum of f - mean2, calculate an estimate for the standard deviation of the plant heights, giving your answer correct to 3 significant figures.(3)
(Total for Question 9 is 9 marks)
10
The masses, in grams, of 5 packages are:

1024, 1031, 1019, 1028, 1022

The masses, x, are coded using y = x - 1020.
(a)Explain why coding the data using y = x - 1020 makes the values easier to work with.(1)
(b)Show that the mean of the coded data, y, is 4.8.(2)
(c)Hence write down the mean mass, x, of the original packages.(1)
(d)Calculate the standard deviation of the coded data, y, giving your answer correct to 3 significant figures.(3)
(e)Write down the standard deviation of the original masses, x, giving a reason for your answer.(1)
(Total for Question 10 is 8 marks)
11
A bakery manager wants to compare the consistency of loaf weights, in grams, produced by two ovens.
(a)Plan: Give one reason why the manager should weigh a sample of several loaves from each oven, rather than comparing just one loaf from each.(1)
(b)Collect: Give one step the manager should take when collecting the two samples to make the comparison fair.(1)
(c)Process: She weighs 5 loaves from each oven (grams):

Oven A: 798, 802, 805, 797, 803
Oven B: 780, 815, 790, 810, 805

Show that the mean loaf weight is 801 g for Oven A and 800 g for Oven B, and calculate the standard deviation of each sample, giving your answers correct to 3 significant figures.(8)
(d)Interpret: Using the standard deviations, write a conclusion comparing the consistency of loaf weights produced by the two ovens.(2)
(Total for Question 11 is 12 marks)
12
An athlete's first 4 javelin throws, in metres, are:

42, 45, 40, 41
(a)Find the mean of the original 4 throws.(1)
(b)Show that the standard deviation of the original 4 throws is 1.87 m, correct to 3 significant figures.(3)
(c)A fifth throw of 42 m is added to the data set. Work out the new standard deviation of all 5 throws, giving your answer correct to 3 significant figures.(3)
(d)Explain why adding this fifth throw decreased the standard deviation, even though it did not change the mean.(2)
(Total for Question 12 is 9 marks)
13
A student is asked to find the standard deviation of the data set 4, 8, 6, 10, 2, which has a mean of 6. Here is the student's working:

sum of (x - mean)2 = 40
standard deviation = √sum of (x - mean)2 = √40 = 6.32 (3 sf)
(a)Identify the error in the student's working.(1)
(b)Work out the correct value of the standard deviation, giving your answer correct to 3 significant figures.(2)
(Total for Question 13 is 3 marks)
14
A survey records 8 commute times, in minutes:

22, 25, 24, 23, 45, 26, 24, 23
(a)Write down the value in the sample that looks unusually different from the rest.(1)
(b)Show that the mean commute time is 26.5 minutes.(1)
(c)Show that the standard deviation of the commute times is 7.09 minutes, correct to 3 significant figures.(3)
(d)One rule for identifying an outlier is a value that is more than 2 standard deviations from the mean. Work out the boundary values for outliers using this rule.(2)
(e)Hence state which commute time in the sample should be classed as an outlier, giving a reason.(1)
(Total for Question 14 is 8 marks)
15
5 students' exam scores, out of 200, are:

120, 140, 160, 100, 180

The scores, x, are coded using y = (x - 100)/20.
(a)Using the coding y = (x - 100)/20, show that the coded values are 1, 2, 3, 0 and 4.(2)
(b)Write down the mean of the coded data, y.(1)
(c)Calculate the standard deviation of the coded data, y, giving your answer correct to 3 significant figures.(3)
(d)Hence write down the mean and the standard deviation of the original exam scores, x.(2)
(Total for Question 15 is 8 marks)
Mark scheme · H04 Standard Deviation

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15