Normal distributions and z-scores (Statistics) · lesson

Finding a z-score

💡 A z-score counts how many standard deviations a value is above or below the mean.

z = (value − mean) ÷ standard deviation.

A positive z is above the mean. A negative z is below it. z = 0 is exactly the mean.

🧠 Let's work it out

Scores have a mean of 60 and a standard deviation of 8. Find the z-score of 72.

Subtract the mean: 72 − 60 = 12.
Divide by the standard deviation: 12 ÷ 8 = 1.5.

What do you think comes next?

z-score: how many standard deviations a value is from the mean
normal distribution: a bell-shaped spread of data

🔗 Khan Academy: Modeling data distributions · Khan Academy (free) ↗

From a z-score back to a value

💡 value = mean + z × standard deviation.

Undo the z-score formula. Multiply z by the standard deviation to get the distance from the mean. Then add it to the mean.

A negative z gives a value below the mean.

🧠 Let's work it out

Weights have a mean of 200 and a standard deviation of 15. What value has z = −2?

Go 2 standard deviations below: −2 × 15 = −30.
Add that to the mean: 200 − 30 = 170.

What do you think comes next?

standard deviation: the typical distance of values from the mean

The 68-95-99.7 rule

💡 In a normal distribution, about 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of the mean.

These three numbers are worth knowing by heart. They work for any normal distribution.

The curve is symmetric, so what's left outside is split evenly between the two tails. Outside 1 standard deviation is 100 − 68 = 32%, so 16% is in each tail.

🧠 Let's work it out

About what percent of a normal data set is more than 1 standard deviation above the mean?

Within 1 SD is 68%.
Outside is 100 − 68 = 32%.
Half is above, so 16%.

What do you think comes next?

empirical rule: the 68-95-99.7 rule for normal distributions
tail: the far end of a distribution, on either side

Adding up pieces of the curve

💡 Turn the cutoffs into z-scores, then build the area from halves of 68, 95, and 99.7.

From the mean to z = 1 is half of 68%, or 34%. From the mean to z = 2 is 47.5%. From the mean to z = 3 is 49.85%. The same is true below the mean.

Half of all data is below the mean. For a band between z = 1 and z = 2, subtract: 47.5 − 34 = 13.5%.

🧠 Let's work it out

Lengths have a mean of 30 and a standard deviation of 5. About what percent are between 35 and 40?

35 is z = 1 and 40 is z = 2.
Mean to z = 2 is 47.5%.
Mean to z = 1 is 34%.
Subtract: 47.5 − 34 = 13.5.

What do you think comes next?

symmetric: the same on both sides of the center

Comparing with z-scores

💡 To compare values from different groups, compare their z-scores, not their raw numbers.

A score of 90 on a hard test may beat a 95 on an easy one. z-scores put both on the same scale.

The bigger the z (ignoring the sign), the more unusual the value. A small spread makes the same gap from the mean more unusual.

🧠 Let's work it out

Maya scored 78 where the mean was 70 and the SD was 4. Leo scored 88 where the mean was 80 and the SD was 5. Who did better compared with their group?

Maya's z = 8 ÷ 4 = 2.
Leo's z = 8 ÷ 5 = 1.6.
2 is more than 1.6, so Maya did better compared with her group.

What do you think comes next?

raw score: the actual value, before comparing to the mean