Grade by Grade
D. Statistical Analysis · One-Variable Data Analysis

z-Scores: Measuring Position in Standard Deviations

D1.2About 65 minutes
Learning goals
  • I can calculate a z-score and explain what its sign and size mean.
  • I can use z-scores to compare values from different data sets.
  • I can convert a z-score back into a data value with x = μ + zσ.
  • I can use z-scores to judge whether a value is unusual.

Words to know

z-score (standard score)
The number of standard deviations a value is above or below the mean: z = (x − μ) ÷ σ for a population, or z = (x − x̄) ÷ s for a sample.
standardizing
Converting values to z-scores. Standardized values have no units, a mean of 0 and a standard deviation of 1.
unusual value
For mound-shaped data, a value with |z| > 2 is often called unusual and |z| > 3 very unusual.

82 in Math or 88 in English?

Sam got 82% on a math test and 88% on an English test. Which result is better? The obvious answer is English, but the class did differently on each test. The math test had a class mean of 74 and standard deviation 6; the English test had mean 84 and standard deviation 5. Sam was 8 marks above the math mean, but only 4 marks above the English mean, and the English marks were bunched more tightly.

To compare fairly, measure each mark in standard deviations from its own mean. That measurement is the z-score, and it is the third tool the curriculum names for describing the position of a value, along with quartiles and percentiles.

The z-score

z = (x − μ) ÷ σ (population) or z = (x − x̄) ÷ s (sample)

  • z > 0: the value is above the mean. z < 0: below. z = 0: exactly at the mean.
  • |z| tells how many standard deviations away the value is.
  • z has no units: the units of x − μ and σ cancel. That is what lets you compare a mark with a time, or a height with a mass.

(If you studied the unit Continuous Distributions and the Normal Distribution, this is the same z-score you used there with normal curves; here it describes the position of a value in a data set.) Rearranging gives x = μ + zσ, which turns a z-score back into a data value.

Two scales for the same marks: μ = 74, σ = 6markz-score56−362−268−174080+186+292+382 ↔ z ≈ +1.33
Every mark lines up with a z-score. Each step of 6 marks is one standard deviation, so a mark of 82 is 8 ÷ 6 ≈ 1.33 standard deviations above the mean.
Worked example

Example 1: Comparing two tests

Sam scored 82 in Math (μ = 74, σ = 6) and 88 in English (μ = 84, σ = 5). On which test did Sam do better compared with the class?

  1. Math: z = (82 − 74) ÷ 6 = 8 ÷ 6 ≈ +1.33.
  2. English: z = (88 − 84) ÷ 5 = 4 ÷ 5 = +0.80.
  3. The math z-score is larger: Sam was 1.33 standard deviations above the math mean but only 0.8 above the English mean.

Relative to classmates, Sam did better in Math, even though the English mark is higher.

Which mark is more impressive?82: z = +1.33Mathμ 74, σ 65055606570758085909510088: z = +0.80Englishμ 84, σ 550556065707580859095100
The darker band is within 1 standard deviation of each mean, the lighter band within 2. Sam's math mark is further out in its distribution than the English mark.
Worked example

Example 2: z-scores for a whole data set

Mira ran 3, 5, 6, 8 and 13 km (μ = 7, σ ≈ 3.406 from lesson 6). Find each z-score, and the mean of the z-scores.

  1. z = (x − 7) ÷ 3.406. For 3: −4 ÷ 3.406 ≈ −1.17. For 5: ≈ −0.59. For 6: ≈ −0.29. For 8: ≈ +0.29. For 13: ≈ +1.76.
  2. Sum: −1.17 − 0.59 − 0.29 + 0.29 + 1.76 = 0.00, so the mean of the z-scores is 0.
  3. (It is always 0, because the deviations add to 0. And because z = (x − μ) ÷ σ is the transformation y = a + bx with b = 1/σ, lesson 7 tells us the z-scores have standard deviation σ × (1/σ) = 1.)

z ≈ −1.17, −0.59, −0.29, +0.29, +1.76. Standardized data always has mean 0 and standard deviation 1.

Worked example

Example 3: From z back to x

Scores on a university entrance test have μ = 500 and σ = 100. (a) What score has z = 1.5? (b) A scholarship requires a z-score of at least 2.2. What is the minimum score? (c) What z-score does a score of 430 have?

  1. (a) x = μ + zσ = 500 + 1.5(100) = 650.
  2. (b) x = 500 + 2.2(100) = 720.
  3. (c) z = (430 − 500) ÷ 100 = −0.7.

(a) 650, (b) at least 720, (c) z = −0.7 (0.7 standard deviations below the mean).

Is it unusual?

For data that is roughly mound-shaped and symmetric, very few values are more than 2 standard deviations from the mean, and almost none are more than 3 (lesson 11 makes this exact). So:

  • |z| > 2: often called unusual.
  • |z| > 3: very unusual; a possible outlier or error worth checking.

This is a second outlier rule, alongside the 1.5 × IQR rule from lesson 9. Be careful with skewed data: in a strongly skewed distribution, values with z > 2 are not rare at all, and the IQR rule is more trustworthy. Also, the mean and standard deviation are themselves pulled by outliers, which can hide them; the IQR rule does not have that problem.

Worked example

Example 4: Comparing across eras

In one hockey season, Player P scored 52 goals when full-time forwards in the league averaged 20 goals with standard deviation 10. Decades later, Player Q scored 65 goals when full-time forwards averaged 30 goals with standard deviation 14. (The players and numbers are invented for this example.) Whose season was more exceptional?

  1. Player P: z = (52 − 20) ÷ 10 = 3.2.
  2. Player Q: z = (65 − 30) ÷ 14 = 35 ÷ 14 = 2.5.
  3. Player P's season was 3.2 standard deviations above the mean of the era, Q's was 2.5.

Player P's season was more exceptional relative to the league at the time, even though Q scored more goals. Both are unusual (z > 2), and P's is very unusual (z > 3).

Just for fun

The highest compliment a statistician can give: "Your kindness has a z-score of at least 3." (That is very unusual, in the best possible way.)

Hands-on activity

Family fitness z-scores

You need: a stopwatch, a measuring tape, paper and a pencil, and at least 5 willing family members or friends (ages similar if possible; nobody should push past what is comfortable).

  1. Everyone does two quick tests: (a) the number of jumping jacks in 30 seconds, and (b) a standing long jump measured in cm (on a soft, non-slip surface like grass or a carpet).
  2. For each test, find the mean and the population standard deviation of the group.
  3. Find each person's two z-scores and their total z-score (sum of the two).
  4. Who is the best all-round athlete by total z-score? Is that the same person who won either event?
  5. Explain why adding the raw results (jumping jacks + centimetres) would be meaningless.

What to look for: z-scores with correct signs, a z-score mean of about 0 for each test (a good self-check), and an explanation that raw scores have different units and different spreads, while z-scores are unit-free. (Real multi-event competitions such as the decathlon use their own points tables, not z-scores, but the idea of putting different events on a common scale is the same.)

Your turn

1

Find z for x = 63 when μ = 70 and σ = 4.

2

A value has z = 2.5 in a population with μ = 40 and σ = 8. Find the value.

3

A sample has x̄ = 12.4 s and s = 1.6 s. Find the z-score of 10.0 s.

4

Ava ran 5 km in 24.0 min (her running club: μ = 27.0 min, σ = 2.0 min). Ben swam 400 m in 7.5 min (his swim club: μ = 8.4 min, σ = 0.5 min). Who did better relative to their club? (Lower times are better.)

5

What is the z-score of the mean?

6

A data set is mound-shaped with μ = 100 and σ = 15. Is a value of 140 unusual?

More practice: warm-up

1

If z is negative, where is the value?

2

μ = 50, σ = 5, x = 60. Find z.

3

μ = 50, σ = 5, z = −1. Find x.

4

What are the units of a z-score?

5

What are the mean and standard deviation of a full set of z-scores?

More practice: on your own

1

Two students apply to the same university program. Rhea has 91% from a school where the program's applicants average 86% with σ = 3%. Theo has 94% from a school where applicants average 90% with σ = 4%. Compare them with z-scores.

2

A value of 72 has z = 1.5 and a value of 48 has z = −1.5. Find μ and σ.

3

The masses of a sample of 40 apples have x̄ = 182 g and s = 14 g. Which apples have z-scores below −2?

4

A teacher converts all test marks to z-scores. What will the mean and standard deviation of the converted marks be? What happens to the shape of the distribution?

5

Explain why, in a strongly right-skewed data set, a value with z = 2.2 might not be unusual.

Challenge

1

Show that if every value of a data set is converted to y = 3x + 10, the z-score of each value does not change.

2

A data set of 5 values has z-scores −1.5, −0.5, 0, 0.5 and one unknown z. Find the unknown z-score. Then check whether these z-scores have standard deviation 1.

3

In a data set with μ = 60 and σ = 10, what is the largest possible percent of values with |z| ≥ 2? (Hint: Chebyshev's inequality says at most 1/k² of any data set lies k or more standard deviations from the mean.) How does this compare with mound-shaped data?

Check your understanding

Quiz

1.

A value has z = −0.5. What does that mean?

2.

μ = 120, σ = 15. What value has z = −2?

3.

Which student did best relative to their own class? (A: 78, μ 70, σ 4; B: 85, μ 80, σ 5; C: 90, μ 84, σ 6)

4.

Why can z-scores compare values measured in different units?

5.

For mound-shaped data, which value would usually be called very unusual?

6.

A full set of z-scores always has

For parents and teachers

Big idea (D1.2): a z-score, z = (x − μ) ÷ σ, gives a value's position in standard deviations from the mean. Because z-scores are unit-free, they compare results from different groups and different measurements fairly.

How to help: Use any two results your teen has (two test marks with class averages, two sports stats) and ask which was more impressive "compared with everyone else". The answer needs both the mean and the spread.

Common difficulties:

  • Reversing the subtraction (μ − x instead of x − μ), which flips the sign.
  • Dividing by the variance instead of the standard deviation.
  • Forgetting that for times (and other "lower is better" measures) a negative z-score is good.
  • Applying the |z| > 2 rule to strongly skewed data.