Statistics · Distributions
Z-Scores and the Normal Distribution: A Plain-English Guide
The 68–95–99.7 rule, what a z-score really measures, and how to answer “what percentage scored above 85?” without fear.
The normal distribution — that famous bell curve — is everywhere: heights, test scores, measurement errors, sample means. And the z-score is the key that unlocks it, turning any normal distribution into one universal ruler.
Get comfortable with these two ideas and a whole category of exam questions becomes mechanical. Let’s build them from the ground up.
What a z-score actually is
🔑 Key term
Z-score — the number of standard deviations a value sits above or below the mean. A z-score of +2 means “two standard deviations above average.” A z-score of −1.5 means “one and a half standard deviations below.”
📐 Formula
z = (x − μ) / σ
Take your value, subtract the mean, divide by the standard deviation. That’s it. The subtraction re-centres everything on zero; the division rescales everything into standard-deviation units.
💡 Insight — the universal ruler
Here’s why z-scores are so powerful. A test score of 85 means nothing on its own — is that good? It depends on the average and the spread. But a z-score of +2 means the same thing everywhere: better than about 97.5% of people, whether it’s a maths exam, a height, or a blood pressure reading.
Standardising strips away the units and the scale, letting you compare apples and oranges. A z of +1.5 in the SATs and a z of +1.5 in a swimming time are equally impressive relative to their fields. This is why z-scores are the common language of the normal distribution.
The 68–95–99.7 rule
For any normal distribution, a fixed percentage of values falls within each band of standard deviations from the mean. Memorise these three numbers and you can answer a huge range of questions in your head.
| Within… | …of the mean lies |
|---|---|
| ±1 standard deviation (z between −1 and +1) | about 68% of values |
| ±2 standard deviations | about 95% of values |
| ±3 standard deviations | about 99.7% of values |
So if IQ is normal with mean 100 and SD 15, then about 95% of people score between 70 and 130 (that’s ±2 SD), and a score above 130 puts you in the top 2.5%. No calculator needed — just the rule.
Answering “what percentage scored above X?”
This is the workhorse question, and it’s always the same three steps:
Step 1 — Standardise. Convert your value X into a z-score with z = (x − μ)/σ.
Step 2 — Look up the probability. Use a z-table or software (in R: pnorm(z)) to find the area to the left of that z. That’s the proportion below X.
Step 3 — Adjust for the direction. Want “above X”? Subtract from 1. Want “between two values”? Find both and subtract.
Step 2 — Look up the probability. Use a z-table or software (in R: pnorm(z)) to find the area to the left of that z. That’s the proportion below X.
Step 3 — Adjust for the direction. Want “above X”? Subtract from 1. Want “between two values”? Find both and subtract.
Worked example. Exam scores are normal, mean 70, SD 10. What percentage scored above 85?
z = (85 − 70)/10 = 1.5. The area to the left of z = 1.5 is about 0.933. So the area to the right (above 85) is 1 − 0.933 = 0.067, or about 6.7%. Done.
⚠ Common error — forgetting which side of the curve you want
The z-table almost always gives the area to the left (below your value). If the question asks “what proportion is above?” and you report the table value directly, you’ve answered the opposite question. Always sketch the bell curve, shade the region you actually want, and check whether your number should be more or less than half. A quick drawing catches this every time.
💡 Insight — the empirical rule is your sanity check
Even when you use exact tables, keep the 68–95–99.7 rule in mind as a reality check. If you calculate that 40% of values lie above a point that’s two standard deviations above the mean, something is wrong — only about 2.5% should be up there. The rule catches arithmetic slips and z-table misreadings before they cost you the question.
Practice questions
Q1. Heights are normal, mean 170cm, SD 8cm. What’s the z-score of someone 186cm tall? Roughly what percentage are taller?
Q2. Using the 68–95–99.7 rule, what percentage of a normal distribution lies below one standard deviation above the mean?
Q3. Two students: Ana scored 78 on a test with mean 70, SD 4. Ben scored 88 on a different test with mean 80, SD 10. Who did better relative to their class?
Q4. A value has a z-score of −2.5. Is it common or unusual? Roughly what proportion of values are more extreme (further from the mean in either direction)?
Worked answers
A1. z = (186 − 170)/8 = 16/8 = +2. Two SDs above the mean. By the empirical rule, about 95% lie within ±2 SD, leaving 5% in the two tails combined, so about 2.5% are taller than 186cm.
A2. About 84%. Here’s the logic: 68% lie within ±1 SD, so 34% sit between the mean and +1 SD. Add the 50% below the mean: 50% + 34% = 84% lie below +1 SD.
A3. Standardise both. Ana: z = (78 − 70)/4 = 2.0. Ben: z = (88 − 80)/10 = 0.8. Ana did better — she’s 2 SDs above her class mean (top ~2.5%), while Ben is 0.8 SDs above his (top ~21%). Ben’s raw score is higher, but relative to his field Ana’s performance is far more exceptional. This is exactly what z-scores are for.
A4. Unusual — a z of −2.5 is well out in the tail. By the empirical rule, about 99% lie within ±2.5 SD (a bit more than the 95% for ±2), so roughly 1% of values are more extreme than ±2.5 in the two tails combined. A value this far out is genuinely rare.
The short version
• z = (x − μ)/σ — how many SDs from the mean.
• z-scores are a universal ruler: they let you compare across different scales.
• 68–95–99.7: the proportions within ±1, ±2, ±3 SD.
• “Above X”: standardise, look up the left area, subtract from 1.
• Always sketch the curve and shade what you want. It stops the direction error.
• z-scores are a universal ruler: they let you compare across different scales.
• 68–95–99.7: the proportions within ±1, ±2, ±3 SD.
• “Above X”: standardise, look up the left area, subtract from 1.
• Always sketch the curve and shade what you want. It stops the direction error.
The normal distribution is also why the Central Limit Theorem matters — sample means become normal, so you can z-score them too.
References
1. Moore, D.S., McCabe, G.P. & Craig, B.A. (2021) Introduction to the Practice of Statistics. 10th edn. New York: W.H. Freeman.
2. Freedman, D., Pisani, R. & Purves, R. (2007) Statistics. 4th edn. New York: W.W. Norton.
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