Module 8 · L7 of 12~30 minMST-12-S2-10⚡ +75 XP available
Z-Scores
You scored 85 on a Maths test (mean 80, SD 5). Your friend scored 78 on English (mean 70, SD 8). Who performed better? Raw scores cannot answer this. The z-score converts any score into a universal measure, standard deviations from the mean, making fair comparison possible across any two distributions.
Today's hook, You score 72 on Test A (class mean 60, SD 8). Your friend scores 85 on Test B (class mean 75, SD 10). Who performed better relative to their class? Predict before you calculate.
0/5QUESTS
1
Get oriented
Set up your goals, the key formulas and the key terms for z-scores.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
You score 72 on Test A (class mean = 60, SD = 8). Your friend scores 85 on Test B (class mean = 75, SD = 10). Who performed better relative to their class?
Before calculating write your gut feeling. We will revisit this at the end.
This height is 1.25 standard deviations below the mean. The negative sign indicates below-average; the magnitude tells you by how much in SD units.
Sign check: If $x > \bar{x}$, the z-score is positive (above mean). If $x < \bar{x}$, the z-score is negative (below mean). If $x = \bar{x}$, then $z = 0$.
z-score formula: z = (x − μ) / σ. A z-score measures how many standard deviations a value x is from the mean. z > 0 means above average; z < 0 means below average; z = 0 means exactly at the mean.
Pause, copy z = (x − μ) / σ with definitions, the sign interpretation (z > 0: above mean; z < 0: below mean; z = 0: at the mean), and note the z-score gives the exact number of standard deviations from the mean into your book.
Quick check: A student scores 88 on a test where the mean is 80 and SD is 4. What is their z-score?
3
Compare across distributions
Use z-scores to compare results from two different tests.
06
Comparing across distributions, the power of z
core concept
The z-score formula z = (x − μ) / σ converts any raw value to the number of standard deviations it is from the mean. A positive z means above the mean; negative means below. Raw marks in different subjects (e.g., 75 in Maths and 70 in English) cannot be compared directly because the distributions have different means and spreads, z-scores remove this problem by expressing both results on the same standardised scale.
Z-scores allow fair comparison of performance on different tests, because they remove the effect of different means and spreads.
Example:
Student A, Maths: 72, mean = 60, SD = 8 → $z = (72-60)/8 = 1.5$
Student A performed better relative to their class ($z = 1.5 > 1.0$), even though their raw score is lower.
Converting back: $x = \bar{x} + z \cdot s$. For $z = 2.0$, mean = 70, SD = 12: $x = 70 + 2.0 \times 12 = 94$.
z-scores allow comparison across different distributions. If a student scores z = 1.5 in maths and z = 2.0 in English, their English performance is relatively better, even if their raw English mark was lower.
Pause, copy the cross-distribution comparison rule: to compare performance across different subjects or datasets, compare z-scores (not raw marks), the higher z-score indicates better relative performance regardless of the raw numbers into your book.
True or false: A student with a raw score of 90 always performed better than a student with a raw score of 75, regardless of the test.
4
Interpret the size of z
Read what a z-score of 0, 1, 2 or more actually tells you.
07
Interpreting z-score magnitude, what the numbers mean
core concept
Z-score
Position
Approx. percentile
−3
Far below average
0.15th
−2
Well below average
2.5th
−1
Below average
16th
0
Average
50th
+1
Above average
84th
+2
Well above average
97.5th
+3
Far above average
99.85th
z-scores standardise results across different distributions: a student with z = 2.0 in English performed better relative to their cohort than a student with z = 1.5 in Maths, regardless of raw marks. The magnitude of z also signals how unusual a result is: |z| < 2 is typical (within 95% of the distribution), |z| > 2 is unusual (outer 5%), and |z| > 3 is very rare (outer 0.3%).
z-score magnitude: |z| < 2 is typical (within 95% of data); |z| > 2 is unusual; |z| > 3 is very rare (outside 99.7%). Use these thresholds to classify individual values as typical, unusual, or exceptional in context.
Pause, copy the three z-score magnitude thresholds and their interpretations: |z| < 2 → typical (within 95% of data); |z| > 2 → unusual (outer 5%); |z| > 3 → very rare (outer 0.3%) into your book.
Fill the gap: A test has mean = 65 and SD = 10. A student with $z = -1.5$ has a raw score of $x =$ .
5
Shade the probability
Represent a probability as a shaded area under the normal curve.
Beyond the syllabus. Interpreting and comparing z-scores, and empirical-rule probabilities, are core. Deriving precise percentile cut-offs or using z-tables is extension: where an exam question needs a value like z = 1.28 for the top 10%, it will be supplied. Learn the empirical-rule percentages and how to use a supplied cut-off.
08
Drawing the probability, shading areas under the curve
core concept
A z-score is a position . A probability is an area . This whole topic gets easier the moment you stop trying to hold that distinction in your head and start sketching it, because the shape of the shading tells you which arithmetic to do before you do any of it.
The rule underneath every question of this kind: the total area under the curve is $1$ , that is $100\%$ of the data, and the curve is symmetric, so each half is exactly $0.5$ . The empirical rule then splits each half into pieces you can add.
Read the bands outwards from the centre and they are the empirical rule taken apart: the middle two make $68\%$ , adding the next pair out makes $95\%$ , and the pair after that makes $99.7\%$ .
Almost every question of this kind is one of three shapes. Sketch the curve, mark the z-score, shade what the question actually asks for, and the calculation follows from the picture:
The question asks
You shade
Worked from the bands
$P(z < 1)$
everything left of $z = 1$
$0.5 + 0.34 = 0.84$
$P(z > 1)$
everything right of $z = 1$
$1 - 0.84 = 0.16$
$P(-1 < z < 2)$
the strip between the two
$0.34 + 0.34 + 0.135 = 0.815$
The shading settles the confusion that costs the most marks. "More than one standard deviation above the mean" is $z > 1$ , one tail, $16\%$ . "More than one standard deviation away from the mean" is $z < -1$ or $z > 1$ , both tails, $32\%$ . The two phrases are nearly identical in English and the answers differ by a factor of two. Shaded, they are obviously different pictures, and you stop having to remember which is which.
The check that catches a wrong answer. A shaded area is a probability, so it can never be negative and never exceed $1$ . Better still, compare your number with your own sketch: if you have shaded visibly more than half the curve, the answer must be more than $0.5$ . Writing $0.16$ under a picture where most of the curve is shaded means you subtracted when you should not have.
A probability is an area under the normal curve, and the total area is 1. Sketch the curve, mark the z-score, then shade: to the left for "less than", to the right for "greater than", and the strip between two z-scores for "between". A greater-than area is 1 minus the matching less-than area.
Pause, sketch the three shading shapes with their z-scores marked, and copy the check that the shaded fraction of your curve must agree with the size of your answer.
Quick check: Heights are normally distributed. What proportion of people are more than $2$ standard deviations away from the mean, in either direction?
09
Drive it, two cohorts and the rule that only works on a bell
+XP for exploring
Everything above has assumed the marks are normally distributed, and every worked example has quietly relied on it. The simulator lets you take that assumption away and watch what breaks. Two cohorts sit side by side with their own mean and spread, the shading under each curve is the percentage, and one control swaps the underlying data for a skewed set with exactly the same mean and exactly the same standard deviation.
Interactive · Normal distribution and z-scores
Try this: drag a mark and watch the shaded area and the two percentage rows move together. Then press Skewed — the mean and the standard deviation do not change, and neither does any z-score, but the 68/95/99.7 table goes red. Finish with Predict before you look, which loads Kim's marks for you.
The number that matters is in the third row of the table. On the skewed data, the rule claims $68\%$ of values lie within one standard deviation of the mean and the actual figure is $86.5\%$ , and it claims $16\%$ lie below $\mu - \sigma$ where the actual figure is $0\%$ . The mean is right, the standard deviation is right, and the rule is still wrong, because $68/95/99.7$ was never a statement about a mean and a spread. It is a statement about a shape .
The 68/95/99.7 rule and any percentage read from a z-table apply only to NORMALLY distributed data. A z-score can always be calculated from a mean and a standard deviation, but the percentage of the cohort it corresponds to cannot be stated at all unless the distribution is normal.
Pause, copy the distinction the simulator draws: a z-score is a position and survives any shape; a percentage is an area and does not. Note the condition that has to be stated before any percentile question can be answered.
Use the simulator. Work the Predict before you look block so Kim's marks load, then switch Shape between Normal and Skewed with the Skew slider at its leftmost. Which Subject A readout is unchanged by that switch?
Use the simulator. Set Shape to Skewed with the Skew slider at its leftmost, then read the row of the 68/95/99.7 table for $\mu \pm 1\sigma$ . The rule claims $68\%$ ; this data actually has per cent. Type the number only, to one decimal place, without a per cent sign.
6
Drill and consolidate
Work the activities, then revisit your first answer.
Activities · two in-class tasks
1
Calculate z-scores for: (a) $x=85$, mean $=75$, SD $=10$; (b) $x=55$, mean $=70$, SD $=12$; (c) $x=120$, mean $=100$, SD $=15$. State whether each is unusual.
2
Three students sat different tests: Sarah, Maths: 78 (mean 70, SD 8); Tom, Science: 82 (mean 75, SD 6); Emma, History: 85 (mean 80, SD 12). Rank them by relative performance.
Match each z-score to its meaning:
z = 0
z = +2
z = −1
|z| > 3
Below average (16th percentile)
Very unusual value
At the mean
Well above average (97.5th percentile)
Top 3 list: Name THREE things z-scores allow you to do that raw scores alone cannot.
10
Revisit your thinking
Student A: $z = (72-60)/8 = 1.5$. Student B: $z = (85-75)/10 = 1.0$. Student A has the higher z-score and performed better relative to their class, even though their raw score (72) is lower than Student B's (85). This is exactly why z-scores exist, raw scores can be deeply misleading when scales differ.
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7
Practise z-scores
Answer the multiple-choice questions on calculating and interpreting z.
01
Multiple choice
+5 XP per correct · +25 XP all-correct
02
Short answer
ApplyBand 43 marks
SA 1. (a) Calculate z-scores for: (i) $x=92$, mean $=80$, SD $=8$; (ii) $x=65$, mean $=72$, SD $=6$; (iii) $x=110$, mean $=100$, SD $=15$. (b) Which value is most unusual? (c) Convert $z=-0.8$ back to a raw score with mean $=75$, SD $=10$. (3 marks)
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ApplyBand 43 marks
SA 2. Three students sat different tests: Ana, Maths: 78 (mean 70, SD 8); Ben, Science: 85 (mean 80, SD 5); Carla, English: 82 (mean 72, SD 10). (a) Calculate the z-score for each student. (b) Rank them by relative performance. (c) The top 10% receive an award (approximately $z > 1.28$). Who qualifies? (3 marks)
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Comprehensive answers (click to reveal)
SA 1: (i) $z=(92-80)/8=1.5$ (ii) $z=(65-72)/6=-1.17$ (iii) $z=(110-100)/15=0.67$. (b) $|z|=1.5$ is the largest, value (i) is most unusual. (c) $x=75+(-0.8\times10)=67$.
SA 2: Ana: $z=(78-70)/8=1.0$. Ben: $z=(85-80)/5=1.0$. Carla: $z=(82-72)/10=1.0$. (b) All have identical z-scores, equally strong relative to their class. (c) None qualify ($z=1.0 < 1.28$).
8
Finish the lesson
Take the module quiz, mark the lesson complete and continue.
01
Take the full module quiz
quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.