z-scores
Standardise a data value using ; interpret a z-score as the number of standard deviations from the mean; find original values, means or standard deviations given a z-score; compare values from different distributions.
Worked examples
Calculating a z-score
Straightforward
Problem
In a state-wide exam, scores have a mean of 65 and a standard deviation of 12. A student scored 89. Calculate their z-score and interpret it.
1
Write the z-score formula and identify the values.
, ,
2
Substitute and calculate.
3
Interpret the z-score.
A z-score of 2 means this student scored 2 standard deviations above the mean. Fewer than 2.5% of students scored higher.
Answer
The z-score is 2. The student performed significantly above average.
Finding the original value from a z-score
Moderate
Problem
Reaction times in a study are normally distributed with a mean of 250 ms and standard deviation of 30 ms. A participant's reaction time has a z-score of . Find their actual reaction time.
1
Write the formula and rearrange to find .
From , rearrange to:
2
Substitute the known values.
3
Interpret the result.
A negative z-score means a value below the mean. This participant's reaction time of 208 ms is faster than average (lower is faster in reaction-time studies).
Answer
The participant's reaction time is 208 ms.
Comparing performance across two distributions
Challenging
Problem
Two students sit different end-of-year exams. Asha scores 82 in Science (mean 70, standard deviation 8). Ben scores 90 in History (mean 75, standard deviation 12). Who performed better relative to their respective cohorts?
1
Calculate Asha's z-score in Science.
2
Calculate Ben's z-score in History.
3
Compare the z-scores.
Asha: — scored 1.5 standard deviations above her Science cohort.
Ben: — scored 1.25 standard deviations above his History cohort.
Asha has the higher z-score, so she performed better relative to her peers despite scoring a lower raw mark.
Ben: — scored 1.25 standard deviations above his History cohort.
Asha has the higher z-score, so she performed better relative to her peers despite scoring a lower raw mark.
Answer
Asha performed better relative to her cohort. Her z-score of 1.5 is higher than Ben's z-score of 1.25.
Practise
Q1·Straightforward
In a class test, scores are normally distributed with a mean of 60 and a standard deviation of 10. A student scores 75. Calculate their z-score.
Explanation
This student scored 1.5 standard deviations above the class mean.
Q2·Straightforward
A data set has a mean of 50 and a standard deviation of 8. A value has a z-score of 1.5. Find the original value .
Explanation
Q3·Straightforward
Student A scores 72 in an exam where the class mean is 65 and standard deviation is 7. Calculate Student A's z-score, rounded to two decimal places.
Explanation
Student A scored exactly 1 standard deviation above the mean.
Q4·Straightforward
Heights in a group have a mean of 168 cm and a standard deviation of 9 cm. A person is 150 cm tall. Calculate their z-score.
Explanation
This person's height is 2 standard deviations below the mean.
Q5·Moderate
A quality-control test produces a z-score of 2.4 for a measurement of 88, given that the population mean is 70. Find the standard deviation .
Explanation
Q6·Moderate
In a distribution with standard deviation 10, the value 95 has a z-score of 1.6. Find the mean .
Explanation
Q7·Moderate
Daily maximum temperatures in a city have a mean of 22°C and a standard deviation of 3°C. Calculate the z-score for a temperature of 28°C.
Explanation
A temperature of 28°C is 2 standard deviations above the average maximum.
Q8·Moderate
Emma scores 78 in Maths (class mean 66, standard deviation 6) and 84 in English (class mean 72, standard deviation 8). What is Emma's z-score in Maths?
Explanation
For comparison:
Emma performed better relative to her class in Maths — 2 standard deviations above the mean versus 1.5 in English.
Q9·Moderate
A machine produces bolts with a mean diameter of 50 mm and standard deviation of 2 mm. A bolt measures 53.8 mm. Calculate its z-score.
Explanation
This bolt's diameter is 1.9 standard deviations above the target mean.
Q10·Challenging
A distribution has mean and standard deviation . Find the value that corresponds to a z-score of 2.5.
Explanation
Q11·Challenging
A distribution has mean 120 and standard deviation 15. Find the value whose z-score is .
Explanation
Q12·Challenging
In a normal distribution, the value 62 has a z-score of 0.8, and the value 50 has a z-score of . Find the mean of the distribution.
Explanation
From , write two equations:
From (1):
Substitute into (2):
Multiply through by 0.8:
Verification: ; ✓; ✓
From (1):
Substitute into (2):
Multiply through by 0.8:
Verification: ; ✓; ✓
Open Math
z-scores
Statistical Analysis · MS-S5
Name:
Date:
Q1Straightforward
In a class test, scores are normally distributed with a mean of 60 and a standard deviation of 10. A student scores 75. Calculate their z-score.
Q2Straightforward
A data set has a mean of 50 and a standard deviation of 8. A value has a z-score of 1.5. Find the original value .
Q3Straightforward
Student A scores 72 in an exam where the class mean is 65 and standard deviation is 7. Calculate Student A's z-score, rounded to two decimal places.
Q4Straightforward
Heights in a group have a mean of 168 cm and a standard deviation of 9 cm. A person is 150 cm tall. Calculate their z-score.
Q5Moderate
A quality-control test produces a z-score of 2.4 for a measurement of 88, given that the population mean is 70. Find the standard deviation .
Q6Moderate
In a distribution with standard deviation 10, the value 95 has a z-score of 1.6. Find the mean .
Q7Moderate
Daily maximum temperatures in a city have a mean of 22°C and a standard deviation of 3°C. Calculate the z-score for a temperature of 28°C.
Q8Moderate
Emma scores 78 in Maths (class mean 66, standard deviation 6) and 84 in English (class mean 72, standard deviation 8). What is Emma's z-score in Maths?
Q9Moderate
A machine produces bolts with a mean diameter of 50 mm and standard deviation of 2 mm. A bolt measures 53.8 mm. Calculate its z-score.
Q10Challenging
A distribution has mean and standard deviation . Find the value that corresponds to a z-score of 2.5.
Q11Challenging
A distribution has mean 120 and standard deviation 15. Find the value whose z-score is .
Q12Challenging
In a normal distribution, the value 62 has a z-score of 0.8, and the value 50 has a z-score of . Find the mean of the distribution.
Worked solutions and answers at openmath.au/year-12/standard-2/the-normal-distribution/z-scores