Track 2 · Describing Data · Lesson 7 of 11
z-scores and relative standing
Standardise values with z-scores, convert back from z to the original units, compare values measured on different scales, and use the empirical rule only where it applies.
A candidate scores 68 on a numerical reasoning test and 81 on a verbal test. Which is the stronger result? The raw numbers cannot say: the tests have different averages and different spreads. A z-score puts both on the same scale — how many standard deviations above or below average?
A z-score (standard score) re-expresses a value as its distance from the mean, measured in standard deviations. A z-score of means two SDs above the mean; means half an SD below; means exactly average. Because the units cancel, z-scores from different variables can be compared.
- the value being standardised
- the mean of the group it is compared with ( if you know the population mean)
- the standard deviation of that group ( for a population). This course uses the sample SD unless stated.
- the number of standard deviations lies above (positive) or below (negative) the mean; it has no units
Worked example
Two tests on different scales
Numerical test: mean 60, SD 5. Verbal test: mean 72, SD 12. A candidate scores 68 (numerical) and 81 (verbal). Which result is relatively stronger?
Numerical: — 1.6 SDs above average.
Verbal: — three quarters of an SD above average.
Compare: the numerical score is higher relative to other candidates, even though 81 > 68 in raw points. The verbal test is easier on average and more spread out, so 9 points above its mean is less remarkable.
The numerical result (z = 1.6) is the stronger one relative to the group.
Standardising a whole variable gives a new variable with mean 0 and standard deviation 1, but it does not change the shape: a skewed variable stays skewed. In the (synthetic) running_club data, the 500 finish times have mean 57.8 and SD 8.36 minutes; the slowest runner (86.3 minutes) has , and the fastest (35.3 minutes) .
The empirical rule. For a distribution that is roughly bell-shaped (close to normal), about 68% of values have , about 95% have and about 99.7% have . It is a rule of thumb for one particular shape — not a law for every dataset.
| Variable | Shape | Within 1 SD | Within 2 SD |
|---|---|---|---|
| Exam scores (student_scores) | roughly symmetric | 70.0% | 95.5% |
| 10 km finish times (running_club) | roughly symmetric | 66.0% | 96.6% |
| Commute times (commute) | right-skewed | 78.8% | 94.6% |
| Flight delays (flight_delays) | strongly right-skewed | 88.8% | 95.4% |
For the two roughly symmetric variables the rule works well. For the skewed flight delays it fails badly within 1 SD — the SD is inflated by the long tail, so most flights sit close to the mean in SD terms — and the most delayed flight has , which would be essentially impossible under a normal model. Check the shape (a histogram) before quoting 68–95–99.7.
Quick check
A delivery took 52 minutes when the mean is 40 and the SD is 8. What is its z-score, and is that unusual?
Exercises
The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.
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Exercise 1 · Guided, difficulty 1 of 3
Standardise a blood pressure reading
In a (simulated) health screening, systolic blood pressure has mean 130 mmHg and standard deviation 16 mmHg. A participant's reading is 141 mmHg.
What is the participant's z-score? Give it to 2 decimal places.
Exercise 2 · Practice, difficulty 1 of 3
From z back to the original units
A warehouse's daily number of orders shipped has mean 493 and standard deviation 77. Management wants an alert when a day is 2.5 standard deviations above the mean. Above how many orders should the alert fire?
Exercise 3 · Challenge, difficulty 2 of 3
Who did relatively better?
Two sales representatives work in different regions. Aisha closed 75 deals in a region where reps average 70 deals (SD 9). Omar closed 72 deals in a region where reps average 44 deals (SD 16).
Find each rep's z-score within their own region (2 decimal places).
Exercise 4 · Interpretation, difficulty 1 of 3
A negative z for a race time
Among the female finishers of a 10 km race, a runner's finish time has . What does this mean?
Exercise 5 · Application, difficulty 2 of 3
Standing among female runners
In the running_club dataset, a woman finishes the 10 km in 50 minutes. Calculate her z-score among the female finishers (sex = Female), using the sample mean and SD of their finish_min. Give it to 2 decimal places.
Data: running_club · Open in the lab to explore and calculate.
Exercise 6 · Application, difficulty 2 of 3
Does the empirical rule hold?
For exam_score in the student_scores dataset (400 students), what percentage of students have a score within one standard deviation of the mean, and what percentage within two standard deviations? Use the sample SD and count a score exactly on the boundary as within. Give both to 1 decimal place.
Data: student_scores · Open in the lab to explore and calculate.
Exercise 7 · Practice, difficulty 1 of 3
Write the conversion formula
A dashboard stores each value as a z-score. Write an expression that converts a z-score z back to the original units, given the mean m and standard deviation s.
Exercise 8 · Interpretation, difficulty 2 of 3
Interpret a z-score in context
Among the male finishers of a 10 km race, one runner's time has . The male finish times are roughly bell-shaped. In two or three sentences, interpret this z-score for a club newsletter.
What you can now do
- Calculate a z-score and convert a z-score back to the original units
- Compare values from different groups or scales with z-scores
- Translate the sign of z into words for the variable in question
- Use the 68–95–99.7 rule only for roughly bell-shaped data