Standardised Scores - Worksheets, Questions and Revision

16 original exam-style questions - 13 pages of questions with a full mark scheme - free printable PDF.

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H05 Standardised Scores

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A standardised score, or z-score, tells you how many standard deviations a value lies above or below the mean of a data set.

z = (x - mean) / (standard deviation)

In a Year 11 statistics mock test, the mean mark was 60 and the standard deviation was 10.
(a)Ahmed scored 75 marks. Work out Ahmed's standardised (z) score.(2)
(b)Bella scored 51 marks. Work out Bella's standardised (z) score.(2)
(c)State, giving a reason, which of Ahmed and Bella performed better relative to the rest of the group.(1)
(Total for Question 1 is 5 marks)
2
In a national aptitude test, raw scores are standardised so that the mean is 100 and the standard deviation is 15.
(a)A candidate has a standardised score of z = 2. Work out the candidate's raw score.(2)
(b)Another candidate has a standardised score of z = -1.4. Work out the candidate's raw score.(2)
(c)Explain what a standardised score of z = 0 tells you about a candidate's raw score, without doing any further calculation.(1)
(Total for Question 2 is 5 marks)
3
For each statement about standardised (z) scores, write down whether it is True or False.
(i)A z-score of 0 means the raw value is exactly equal to the mean.(1)
(ii)A negative z-score always means the data value is a poor or unwanted result.(1)
(iii)The larger the size of a z-score, ignoring its sign, the further the value is from the mean.(1)
(iv)Two data sets with the same standard deviation will always give the same z-score for the same raw value.(1)
(Total for Question 3 is 4 marks)
4
Four students are asked to identify the correct formula for a standardised (z) score.
(a)Which of these is the correct formula for a standardised (z) score?(1)
  • A) z = (x - mean) / (standard deviation)
  • B) z = (mean - x) / (standard deviation)
  • C) z = (x - standard deviation) / mean
  • D) z = x / (mean x standard deviation)
(b)Explain why formula B would give the wrong sign for a raw value that is above the mean.(1)
(Total for Question 4 is 2 marks)
5
In her end-of-year exams, Priya's raw marks and her class's statistics were as follows.
SubjectClass meanClass standard deviationPriya's raw mark
Maths581282
Science60678
(a)Work out Priya's standardised (z) score in Maths.(2)
(b)Work out Priya's standardised (z) score in Science.(2)
(c)Priya scored a higher raw mark in Maths (82) than in Science (78). Use the z-scores from parts (a) and (b) to explain which subject actually shows Priya's stronger performance relative to her class.(2)
(Total for Question 5 is 6 marks)
6
A baby clinic records that, for babies of a certain age, the mean weight is 8.2 kg with a standard deviation of 0.6 kg. A baby is weighed at the clinic and found to weigh 9.4 kg.
(a)Work out the baby's standardised (z) score for weight.(2)
(b)Interpret what this z-score tells the clinic about this baby's weight compared with other babies of the same age.(2)
(c)Give one reason why the clinic uses a standardised score rather than just comparing babies' raw weights.(1)
(Total for Question 6 is 5 marks)
7
Five contestants' standardised (z) quiz scores are shown as points A to E on the number line below. All five results come from the same general-knowledge quiz, which has a mean score of 50 and a standard deviation of 8.
-3 -2 -1 0 1 2 3 Standardised score (z) A B C D E
(a)Write down the z-score shown at point D.(1)
(b)Work out the raw quiz score corresponding to point E.(2)
(c)Write down the letter of the point that represents a raw quiz score exactly equal to the mean.(1)
(d)Point E has the largest z-score of the five. Explain what this tells you about contestant E's quiz result compared with the other four contestants.(1)
(Total for Question 7 is 5 marks)
8
Three runners each complete a different long-distance race, so their times are standardised against their own race's mean and standard deviation. A lower time is a better result.
RunnerRace mean time (min)Race standard deviation (min)Runner's time (min)
A50544
B60451
C45340.5
(a)Work out the standardised (z) score for Runner A.(2)
(b)Work out the standardised (z) score for Runner B.(2)
(c)Work out the standardised (z) score for Runner C.(2)
(d)Rank the three runners from best to worst relative performance, explaining how you used the sign and size of each z-score.(2)
(Total for Question 8 is 8 marks)
9
A PE teacher wants to compare a pupil's performance in the 100 m sprint and the long jump fairly, even though the two events use completely different units and scales. She carries out a statistical enquiry in four stages: plan, collect, process and interpret.
EventClass meanClass standard deviationJake's result
100 m sprint15.0 s1.0 s13.8 s
Long jump3.60 m0.30 m4.02 m
(a)(Plan) Give one reason why the teacher decides to convert each pupil's raw result into a standardised (z) score before comparing the sprint and the long jump.(1)
(b)(Collect) Write down the mean and standard deviation the teacher needs from the table to standardise Jake's long jump result.(1)
(c)(Process) Work out Jake's standardised (z) score for the sprint and for the long jump.(2)
(d)(Interpret) A lower sprint time is better, and a longer jump distance is better. Using both z-scores, explain in which event Jake performed relatively better compared with his class.(2)
(Total for Question 9 is 6 marks)
10
For data that is roughly normally distributed, the empirical rule links standardised scores to percentages of the data, as shown on the diagram below.
-3 -2 -1 0 1 2 3 Standardised score (z) 68% 95% 99.7%
(a)Using the diagram, write down the percentage of values that lie within 1 standard deviation of the mean.(1)
(b)A data set has a mean of 500 and a standard deviation of 20. Work out the range of raw values that contains approximately 95% of the results.(2)
(c)A student in this data set has a raw score of 545. Using your answer to part (b), explain whether this score is unusual compared with the rest of the group.(2)
(Total for Question 10 is 5 marks)
11
Meera's raw marks and her class's statistics in three end-of-year exams were as follows.
SubjectClass meanClass standard deviationMeera's raw mark
Geography54870
Chemistry60571
French501073
(a)Work out Meera's standardised (z) score in Geography.(1)
(b)Work out Meera's standardised (z) score in Chemistry.(1)
(c)Work out Meera's standardised (z) score in French.(1)
(d)Rank the three subjects from Meera's strongest to weakest relative performance, and state which subject shows her strongest relative performance.(2)
(Total for Question 11 is 5 marks)
12
A student claims: 'If Priya has a higher raw score than Tom, then Priya must have a higher z-score.' Test this claim using the following data. Priya scored 66 in a test with mean 50 and standard deviation 20. Tom scored 60 in a different test with mean 40 and standard deviation 5.
(a)Work out Priya's standardised (z) score.(2)
(b)Work out Tom's standardised (z) score.(2)
(c)Write down whether this data supports or contradicts the student's claim, giving a reason.(1)
(Total for Question 12 is 5 marks)
13
In a national reading assessment, raw scores are standardised so that the mean is 100 and the standard deviation is 15. A pass certificate is awarded to anyone with a standardised score of at least z = 1.2.
(a)Work out the minimum raw score needed to be awarded a pass certificate.(2)
(b)A candidate scored 79 marks. Work out this candidate's z-score, and state whether they were awarded a pass certificate.(2)
(c)Give one reason why using a standardised score, rather than a fixed raw mark, to decide who passes might be considered fairer across different years of the assessment.(1)
(Total for Question 13 is 5 marks)
14
Adult resting heart rates in a large health survey are approximately normally distributed with a mean of 72 beats per minute (bpm) and a standard deviation of 8 bpm.
(a)Work out the standardised (z) score for a resting heart rate of 60 bpm.(2)
(b)Using the empirical rule (about 95% of values lie within 2 standard deviations of the mean), work out the range of heart rates that contains the middle 95% of adults in the survey.(2)
(c)A doctor becomes concerned when a patient's resting heart rate gives a z-score with a magnitude greater than 2 (i.e. z > 2 or z < -2). Explain, using the empirical rule, why the doctor uses this cut-off rather than simply checking whether the heart rate is above or below average.(2)
(Total for Question 14 is 6 marks)
15
In a diving competition, three divers are judged by three different panels, each with its own scoring mean and standard deviation for that event. A higher score is better.
DiverPanel meanPanel standard deviationDiver's score
Ana72681
Beth65474
Carla781089
(a)Work out the standardised (z) score for Ana.(1)
(b)Work out the standardised (z) score for Beth.(1)
(c)Work out the standardised (z) score for Carla.(1)
(d)Rank the three divers from best to worst relative performance, using their z-scores.(2)
(e)Carla has the highest raw score (89) of the three divers. Evaluate why it would not be fair to rank the divers using their raw scores alone, without standardising.(2)
(Total for Question 15 is 7 marks)
16
A college standardises its admission test scores so that the mean is 500 and the standard deviation is 100. A 'Distinction' flag is given to any candidate with a standardised score greater than z = 2. The college wants only about 100 candidates, out of a very large applicant pool of 4000, to receive this flag.
(a)Work out the minimum raw score needed for a Distinction flag.(2)
(b)Using the empirical rule (about 95% of values lie within 2 standard deviations of the mean), work out the approximate percentage of candidates expected to have a z-score greater than 2.(2)
(c)Evaluate whether the current cut-off of z > 2 achieves the college's target of about 100 Distinction flags out of 4000 candidates. Show your working.(2)
(Total for Question 16 is 6 marks)
Mark scheme · H05 Standardised Scores

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

Question 16