Standardised Scores
A standardised score, or z-score, shows how many standard deviations a data value lies above or below the mean, calculated as (value - mean) divided by standard deviation. It allows values from different data sets, such as marks on two different tests, to be compared fairly on the same scale.
Before you start
Make sure you're comfortable with these topics first:
Method
- Find the mean and standard deviation of the data set that the value belongs to.
- Subtract the mean from the value you want to standardise.
- Divide the result by the standard deviation to get the standardised score (z-score).
- Interpret the sign: a positive z-score means the value is above the mean, and a negative z-score means it is below the mean.
- To compare two values from different data sets, calculate the standardised score for each and compare their sizes.
- Round the standardised score to an appropriate number of decimal places, usually 2, unless told otherwise.
Worked example
In a maths test, the mean mark was 60 with a standard deviation of 8. In an english test, the mean mark was 55 with a standard deviation of 5. Priya scored 72 in maths and 65 in english. Use standardised scores to determine in which subject Priya performed relatively better.
- Standardise the maths score: z = (72 - 60) / 8 = 12/8 = 1.5.
- Standardise the english score: z = (65 - 55) / 5 = 10/5 = 2.
- Compare the two z-scores: 2 is greater than 1.5.
- Since the english z-score is higher, Priya performed relatively better in english compared with the rest of her cohort.
- Final answer: Priya performed relatively better in english (z = 2) than in maths (z = 1.5).
Practice questions
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Q1A data set has mean 20 and standard deviation 4. Find the standardised score for a value of 28.Show answer
Answer: z = 2 ((28-20)/4).
Q2A data set has mean 50 and standard deviation 10. Find the standardised score for a value of 35.Show answer
Answer: z = -1.5 ((35-50)/10).
Q3A value has a standardised score of 0. What does this tell you about the value?Show answer
Answer: The value is equal to the mean.
Q4In a science test, the mean is 48 and the standard deviation is 6. Tom scored 60. Find his standardised score.Show answer
Answer: z = 2 ((60-48)/6).
Q5In a history test, mean = 70, standard deviation = 5; in a geography test, mean = 65, standard deviation = 8. Aisha scored 78 in history and 77 in geography. Use standardised scores to decide which subject she did relatively better in.Show answer
Answer: History (z = 1.6 vs geography z = 1.5), since 1.6 is greater than 1.5.
Q6A runner's time has a standardised score of -2 in a race where the mean time is 40 minutes and the standard deviation is 3 minutes. Find the runner's actual time.Show answer
Answer: 34 minutes (40 + (-2 x 3)).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A data set has mean 15 and standard deviation 3. Calculate the standardised score for a value of 21.
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In a spelling test the mean score was 30 with standard deviation 4. Ben's standardised score was 1.25. Find Ben's actual spelling test score.
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In a French test, the mean mark was 62 with standard deviation 6. In a German test, the mean mark was 58 with standard deviation 4. Leo scored 71 in French and 66 in German. (a) Calculate Leo's standardised score for each test. (b) State, with a reason, in which subject Leo performed relatively better.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the standardised scores worksheet pack - 18 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 2 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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