The normal distribution and z-scores, explained
Many real measurements, from heights to test scores, follow a bell-shaped normal distribution. Z-scores let you compare any value with the rest of the data and find probabilities.
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What a normal distribution is
A normal distribution is a symmetric, bell-shaped distribution described by two numbers: its mean μ, which sets the centre, and its standard deviation σ, which sets the spread. We write X ~ N(μ, σ) in AP Statistics. (Some textbooks and A level courses write N(μ, σ²), using the variance, so always check which is meant.) The total area under the curve is 1, and the area between two values is the probability that a randomly chosen value falls between them.
Key features: the mean, median and mode are all equal and sit at the centre; the curve is symmetric about the mean; it never touches the horizontal axis; and the points where the curve changes from bending down to bending up are exactly one standard deviation from the mean.
Why the normal distribution appears so often
Many measurements are the combined result of lots of small, independent effects. A person's height, for example, depends on many genes and environmental factors. When many small effects add up, the total tends to follow a normal distribution, which is what the Central Limit Theorem describes. That is why the bell curve appears in so many different settings.
Z-scores
A z-score tells you how many standard deviations a value is from the mean:
z = (x − μ) / σ
A positive z means above the mean; a negative z means below. Converting to z-scores turns any normal distribution into the standard normal distribution, N(0, 1), so one table or calculator function covers every normal problem. Z-scores are also useful for comparing values from different distributions, such as a score on one test against a score on another.
The empirical rule
In any normal distribution, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. Use it for quick estimates and to judge whether a value is unusual. For exact answers, use a table or calculator.
Worked examples
Example 1: computing a z-score
IQ scores are N(100, 15). Find the z-score for an IQ of 130.
- z = (130 − 100)/15 = 2. The score is two standard deviations above the mean.
Example 2: comparing values
Amira scored 82 on a test with mean 70 and SD 8. Ben scored 90 on a test with mean 80 and SD 12. Who did better relative to their class?
- Amira: z = (82 − 70)/8 = 1.5. Ben: z = (90 − 80)/12 ≈ 0.83.
- Amira performed better relative to her class.
Example 3: probability below a value
Heights of men are N(175, 7) in cm. Find the probability that a man is shorter than 168 cm.
- z = (168 − 175)/7 = −1.
- P(Z < −1) ≈ 0.1587, about 16%.
Example 4: probability above a value
Using the same distribution, find P(height > 189 cm).
- z = (189 − 175)/7 = 2.
- P(Z > 2) = 1 − 0.9772 = 0.0228.
Example 5: probability between two values
Find P(170 < height < 180).
- z values: (170 − 175)/7 ≈ −0.71 and (180 − 175)/7 ≈ 0.71.
- P(−0.71 < Z < 0.71) ≈ 0.7611 − 0.2389 = 0.522.
Example 6: inverse normal, finding a value
What height marks the tallest 10% of men?
- Find z with 90% below it: z ≈ 1.28.
- x = μ + zσ = 175 + 1.28 × 7 ≈ 184 cm.
Example 7: finding the standard deviation
A machine fills bags with mean 500 g. 5% of bags weigh less than 490 g. Find σ, assuming a normal distribution.
- The z with 5% below it is about −1.645.
- −1.645 = (490 − 500)/σ, so σ = 10/1.645 ≈ 6.08 g.
Reading a z table
A standard normal table gives the area to the left of a z-score. To find P(Z < 1.23), go down the left column to 1.2 and across to the 0.03 column: about 0.8907. For an area to the right, subtract from 1. For an area between two z-scores, subtract the smaller left-area from the larger. Always sketch the curve and shade the region you want; it catches most errors.
Using a calculator
On a TI-84, normalcdf(lower, upper, μ, σ) gives the area between two values, and invNorm(area, μ, σ) gives the value with that area to its left. Use a very large number such as 1E99 for infinity. On the AP exam, write the distribution, the boundary and the probability in words or symbols, such as “P(X > 189) where X ~ N(175, 7)”; calculator syntax alone does not earn full credit.
Is the data actually normal?
Normal calculations only make sense if the distribution is approximately normal. Check with a histogram or dotplot (roughly symmetric and single-peaked), or a normal probability plot (roughly a straight line). Strong skewness or outliers mean normal calculations may be misleading. In sampling distributions, the Central Limit Theorem often makes the sample mean approximately normal even when the population is not, as long as the sample is large enough.
Common mistakes
- Forgetting that tables give areas to the left; subtract from 1 for areas to the right.
- Using the variance instead of the standard deviation in the z formula.
- Applying normal calculations to strongly skewed data.
- Rounding z too early, which shifts the answer.
Practice questions
Try each one before opening the answer.
Q1. X ~ N(50, 10). Find the z-score of 35.
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Q2. X ~ N(50, 10). Find P(X > 70).
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Q3. X ~ N(50, 10). Find the value with 75% below it.
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Q4. About what percentage of a normal distribution lies more than 3 SD from the mean?
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Q5. Scores are N(60, 12). What percentage scored between 48 and 72?
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Frequently asked questions
Is the normal distribution on the AP Statistics exam?
Yes. It appears in Unit 1 for describing data and throughout the course in sampling distributions, confidence intervals and significance tests.
Is the normal distribution in A level Maths?
Yes, in the second year of A level Maths statistics, including inverse normal problems and using the normal distribution in hypothesis tests.
What is a percentile?
The pth percentile is the value with p% of the data below it. A z-score of 0 is the 50th percentile, and z ≈ 1.28 is about the 90th.
Key takeaways
- A normal distribution is set by its mean and standard deviation.
- z = (x − μ)/σ counts standard deviations from the mean.
- Tables give areas to the left of z.
- Use inverse normal to find a value from a probability.
- Check the data is approximately normal first.