Count how many standard deviations from the mean, and the empirical rule hands you the percentage — three numbers cover nearly every exam question on the bell curve.
The problem
Test scores are normally distributed with mean 80 and standard deviation 5. About what percent of scores lie between 75 and 85?
75 and 85 sit exactly one standard deviation either side of 80 — the ±1σ band holds about 68%. C is the ±2σ band (70 to 90), and D is only HALF the band, mean to +1σ.
Heights, test scores, measurement errors — many real quantities pile up near their average and thin out symmetrically. The mean sits at the centre; the standard deviation σ sets how spread the pile is. Half of everything lies below the mean, half above — that 50% is free.
| Band | Holds about | Here (mean 80, σ 5) |
|---|---|---|
| mean ± 1σ | 68% | 75 – 85 |
| mean ± 2σ | 95% | 70 – 90 |
| mean ± 3σ | 99.7% | 65 – 95 |
What percent score above 90 — that is, above +2σ?
outside ±2σ: 100 − 95 = 5%
both tails together
one tail: 5 ÷ 2 = 2.5%
symmetry halves it
mean to +1σ: 68 ÷ 2 = 34%
half-bands work the same way
“Above 90” is one side only. 100 − 95 = 5% is the share below 70 and above 90 together, so stopping there doubles the answer. Halve it: 2.5% score above 90.
z = value − meanσ
"how many σ away am I?"
score 90: z = 90 − 805 = +2
then the bands take over
A question about the 68–95–99.7 bands chooses numbers that land on whole σ steps. If your z comes out as 1.37 in one of those, reread the problem — a number was probably misassigned.
20 is −3σ, 60 is +1σ
convert to σ-counts
−3σ to the mean: 99.7 ÷ 2 = 49.85%
half the 99.7% band
the mean to +1σ: 68 ÷ 2 = 34%
half the 68% band
total ≈ 83.85%
add the halves
There is no rule for −3σ to +1σ in one piece. Split at the mean and take half of each band.
When the question asks “how many”, finish by multiplying the percent by the group size: if 400 people were measured, about 83.85% of 400 ≈ 335 of them fall in this range.
| Scored higher than about | Position |
|---|---|
| 2.5% | mean − 2σ |
| 16% | mean − 1σ |
| 50% | the mean |
| 84% | mean + 1σ |
| 97.5% | mean + 2σ |
16% below means 1σ below the mean
50 − 34 = 16
score = 60 − 8 = 52
Beating only 16% is below average, so the σ-step is subtracted. 68 would be the score that beats about 84%.
Two different tests. Ana scored 80 on a test with mean 70 and σ = 5. Ben scored 85 on a test with mean 75 and σ = 10. Who did better relative to their group?
Ana: z = (80 − 70) ÷ 5 = 2
2σ above her mean
Ben: z = (85 − 75) ÷ 10 = 1
1σ above his mean
Ana, although her raw score is lower
compare z, not points
What is different: both are 10 points above their means, but Ben’s test is twice as spread out. Ana beat about 97.5% of her group; Ben about 84% of his.
A band far out on one side. Values are normal with mean 40 and σ = 4. About what percent lie between 28 and 32?
28 is −3σ, 32 is −2σ
both below the mean
(99.7 − 95) ÷ 2 = 4.7 ÷ 2
the strip between the bands, one side
= 2.35%
What is different: the region does not touch the mean. Take the difference of two bands, then halve it for one side.
“Above” a score below the mean. Same data: about what percent lie above 36?
36 is −1σ
(36 − 40) ÷ 4 = −1
−1σ to the mean: 34%; above the mean: 50%
split at the mean
34 + 50 = 84%
more than half, as it must be
What is different: “above” a below-average score crosses the mean, so the free 50% joins in.
16% is the share below 36. A score under the mean has most of the group above it.
μ + σ = 70
one SD above
μ − 2σ = 40
two SDs below
3σ = 30, σ = 10
subtract the equations
μ = 70 − 10 = 60
(70 + 40) ÷ 2 = 55 is the mean only when the two scores are the same number of σ from it.
Expect a percent between two scores (often uneven, such as −1σ to +2σ), a count out of N people, a score from a percentile, or μ and σ from two statements. Convert every score to σ-steps, split at the mean, and halve the bands with symmetry. Ten-second check: the pieces you add must not overlap, and the total cannot pass 100%.
The problem
IQ scores are normal with mean 100 and standard deviation 15. About what percent score above 130?
130 is 130 − 10015 = +2σ. Outside ±2σ lies 5%, and the upper tail alone is 2.5%. B forgets to halve; A is the +1σ tail (that would be a score of 115); D is the +3σ tail, score 145.
