B. Probability Distributions · Continuous Distributions and the Normal Distribution

z-Scores and the Standard Normal Distribution

B2.8About 65 minutes

Learning goals

I can calculate a z-score and explain what it means.
I can convert between a value x and its z-score in both directions.
I can use z-scores to compare values from different distributions.
I can explain why every normal distribution can be turned into the standard normal distribution N(0, 1).

Words to know

z-score (standard score)
The number of standard deviations a value is above (positive) or below (negative) the mean: z = (x − μ)/σ.

standardizing
Changing a value x into its z-score. Standardizing every value of X ~ N(μ, σ²) gives Z ~ N(0, 1).

standard normal distribution
The normal distribution with mean 0 and standard deviation 1, written Z ~ N(0, 1).

relative standing
How a value compares with the rest of its own group, measured by its z-score (or percentile).

Which mark is better?

Noor got 78% on a math test and 85% on an English test. Which result is more impressive? You cannot tell from the marks alone. If the math class averaged 70% with marks tightly bunched, a 78 might be near the top. If the English class averaged 80% with marks widely spread, an 85 might be fairly ordinary.

To compare, we measure each mark in standard deviations from its own mean. That number is the z-score.

The z-score formula

For a value x from a distribution with mean μ and standard deviation σ:

z = (x − μ) / σ

z > 0: x is above the mean; z < 0: x is below the mean; z = 0: x is the mean.
z = 1.5 means "1.5 standard deviations above the mean".
z has no units: the units cancel.

For a sample, use z = (x − x̄)/s.

To go back from z to x, rearrange: x = μ + zσ.

Seedling heights N(32, 10.2²) with two scales. Each x-value has a matching z-value: 21.8 cm is z = −1, 42.2 cm is z = 1, and so on. A 24 cm seedling has z ≈ −0.78 and a 38 cm seedling has z ≈ 0.59.

Worked example

Example 1: Seedling z-scores

Heights of 16-month-old maple seedlings are N(32, 10.2²) cm. Find the z-scores of seedlings that are 24.0 cm and 38.0 cm tall, and describe each in words.

z = (24.0 − 32)/10.2 = −8/10.2 ≈ −0.784, which rounds to −0.78.
z = (38.0 − 32)/10.2 = 6/10.2 ≈ 0.588, which rounds to 0.59.

z ≈ −0.78 (0.78 SD below the mean) and z ≈ 0.59 (0.59 SD above the mean). We round z-scores to 2 decimal places, because that is what the z-table uses (next lesson).

Worked example

Example 2: Comparing marks from different tests

Noor's math mark is 78 (class: μ = 70, σ = 5). Her English mark is 85 (class: μ = 80, σ = 4). Relative to her classmates, which result is better?

Math: z = (78 − 70)/5 = 8/5 = 1.6.
English: z = (85 − 80)/4 = 5/4 = 1.25.
1.6 > 1.25, so the math mark is further above its class mean, measured in SDs.

Her math result is relatively better (z = 1.6 versus z = 1.25), even though the English mark is higher.

Noor's marks on their own class curves. The math mark (z = 1.6) is further into the upper tail than the English mark (z = 1.25), so a smaller share of the math class scored above her.

Worked example

Example 3: From z back to x

(a) Maple syrup volumes are N(500, 4²) mL. What volume has z = −1.5? (b) On a test with mean 70, a mark of 88 has z = 2.25. Find σ.

(a) x = μ + zσ = 500 + (−1.5)(4) = 500 − 6 = 494.
(b) z = (x − μ)/σ gives 2.25 = (88 − 70)/σ = 18/σ.
So σ = 18/2.25 = 8.

(a) 494 mL (b) σ = 8.

Why one table works for every normal distribution

If you standardize every value of X ~ N(μ, σ²):

subtracting μ slides the curve so its centre is at 0 (from the lesson Standard Deviation, adding or subtracting a constant does not change the SD);
dividing by σ rescales it so the SD becomes 1 (dividing every value by σ divides the SD by σ).

The result is the standard normal distribution Z ~ N(0, 1). The area to the left of x under the X-curve equals the area to the left of z under the Z-curve. So one table of areas for N(0, 1) works for every normal distribution. That table is the subject of the next lesson.

The empirical rule in z-language: about 68% of z-scores are between −1 and 1, 95% between −2 and 2, and 99.7% between −3 and 3.

Just for fun

Why are z-scores so calm at parties? They always know exactly where they stand compared with everyone else.

Hands-on activity

Family decathlon (well, biathlon)

You need: 3 or more people, a stopwatch, a list of 20 simple words you write in advance, paper and pencils, a calculator.

Event 1, memory: read the 20-word list aloud once. Each person writes as many words as they remember in 1 minute. Score = number correct.
Event 2, balance: each person stands on one foot with eyes closed (near a wall or chair for safety, with a grown-up spotting) and you time how long until the other foot touches down. Score = seconds.
For each event, find the mean and the population SD of the family's scores (the family is the whole group).
Find every person's z-score in both events. Who had the best relative performance in each event? Who was most balanced across both?

What to look for: a person can have the highest raw score in seconds but a lower z-score than someone else's memory score. z-scores let you compare events with completely different units.

Your turn

X ~ N(50, 8²). Find the z-score of x = 62.

X ~ N(50, 8²). Find the z-score of x = 44.

X ~ N(50, 8²). What value of x has z = −2.25?

Lengths of trout in a lake are N(38, 6²) cm. Find the z-score of a 47 cm trout and describe it.

Jamal ran 5 km in 22.5 min (club times: μ = 25, σ = 2). Priya swam 400 m in 7.0 min (club times: μ = 8.0, σ = 0.5). For times, lower is better. Whose result is more impressive?

A value has z = 0. What does that tell you?

Find the z-score of a seedling 50 cm tall (N(32, 10.2²)), rounded to 2 decimals.

More practice: warm-up

Write the formula for a z-score.

μ = 100, σ = 10, x = 120. Find z.

μ = 100, σ = 10, x = 85. Find z.

μ = 100, σ = 10, z = 0.4. Find x.

What are the mean and standard deviation of the standard normal distribution?

More practice: on your own

Birth masses are N(3.4, 0.5²) kg. Find the z-scores of babies of 2.6 kg and 4.3 kg.

A data value of 130 has z = 1.2 and a value of 70 has z = −1.8 in the same distribution. Find μ and σ.

About what percent of a normal distribution has z-scores between −2 and 1? (Use the empirical rule.)

Maya scored 31 on a test with x̄ = 25, s = 4. Ben scored 70 on a test with x̄ = 60, s = 8. Who did relatively better?

Every value in a data set is converted to a z-score. What are the mean and SD of the z-scores? Explain.

Challenge

A teacher wants to rescale a test with μ = 58, σ = 12 so the new mean is 70 and the new SD is 8, keeping every student's z-score. What new mark should a student with 76 get?

Show that if every value x is replaced by z = (x − μ)/σ, the new mean is 0.

Two seedlings from different nurseries are 40 cm (nursery A: N(32, 10.2²)) and 36 cm (nursery B: N(28, 6²)). Which is taller relative to its own nursery? By about how many standard deviations?

Check your understanding

Quiz

X ~ N(80, 5²). What is the z-score of x = 72?

A value has z = 2.5 in N(40, 4²). What is the value?

Which result is relatively best?

What is the distribution of Z if every value of X ~ N(μ, σ²) is standardized?

On a test, x = 64 has z = −0.5 and μ = 70. What is σ?

A z-score of −1.25 means the value is...

For parents and teachers

Big idea (B2.8): a z-score z = (x − μ)/σ says how many standard deviations a value is from its mean. z-scores let you compare results from different distributions and turn any normal distribution into the standard normal N(0, 1).

How to help: Use real comparisons: two siblings' marks in different courses (with made-up class means and SDs), or a hockey player's goals versus a basketball player's points. Ask: "Which is more impressive, relative to the group?"

Common difficulties:

Subtracting in the wrong order: it is x − μ, not μ − x.
Dividing by the variance instead of the standard deviation.
Forgetting that for times or golf scores, a lower (more negative) z can be the better result.
Rounding z too early. Round z to 2 decimals only at the end, for the table.