The z-scores worth knowing by heart
Most people carry the 68, 95, 99.7 rule around: roughly 68% of a bell-shaped group sits within one standard deviation of the mean, 95% within two, 99.7% within three. The 95 is a rounding. Exactly 95% falls inside ±1.96, and ±2 holds 95.45%. The odd-looking rows below, 1.645 and 2.58 among them, are the ones confidence intervals and hypothesis tests lean on.
| z | Share below | Share above | Rarity, this far out | Where you meet it |
|---|---|---|---|---|
| −3.00 | 0.13% | 99.87% | 1 in 741 | Lower control limit on a Shewhart chart |
| −2.00 | 2.28% | 97.72% | 1 in 44 | WHO cutoff for low height-for-age in children |
| −1.00 | 15.87% | 84.13% | 1 in 6 | One SD under the mean, an IQ of 85 |
| 0.00 | 50.00% | 50.00% | 1 in 2 | Exactly average |
| 1.00 | 84.13% | 15.87% | 1 in 6 | An IQ of 115 |
| 1.28 | 89.97% | 10.03% | 1 in 10 | Top 10% cutoff |
| 1.645 | 95.00% | 5.00% | 1 in 20 | One-tailed test at alpha 0.05, 90% interval |
| 1.96 | 97.50% | 2.50% | 1 in 40 | 95% confidence interval |
| 2.00 | 97.72% | 2.28% | 1 in 44 | The usual “unusual” line, an IQ of 130 |
| 2.33 | 99.01% | 0.99% | 1 in 101 | Top 1% cutoff |
| 2.58 | 99.51% | 0.49% | 1 in 202 | 99% confidence interval |
| 3.00 | 99.87% | 0.13% | 1 in 741 | Upper control limit, common outlier rule |
Negative rows mirror the positive ones, so −1.96 has 2.5% below it. Rarity counts one side only, from the value outward. For “this far from the mean in either direction,” halve the rarity: z = 2 becomes 1 in 22.
Worked case: one light bottle off a 500 ml line
A quality tech pulls a bottle off the line and it weighs in at 489 ml. The filler targets 500 ml, and a month of spot checks puts the standard deviation at 4 ml. Should the line stop?
- Subtract the mean.
489 − 500 = −11ml. The bottle is 11 ml short. - Divide by the standard deviation.
−11 / 4 = −2.75. The shortfall equals two and three quarter standard deviations. - Read the area. The curve puts 0.30% of bottles at 489 ml or lighter, about 1 in 336.
- Scale to the shift. A filler running 20,000 bottles a shift turns out around 60 bottles this light by chance alone, with nothing broken.
- Make the call. One bottle at −2.75 is no reason to stop the line. Five of them inside an hour is, since a stable process would almost never cluster them. The z-score tells you how rare one value is. The pattern over time tells you whether the process moved.
Load the bottle example above to see the same numbers on the curve, then switch the shade to Beyond ±z to get the two-sided figure an auditor would quote.
Where z-scores turn up under other names
- Control charts
- Shewhart limits sit at the process mean ±3 standard deviations. A stable process puts a point outside them about 0.27% of the time, so one breach triggers a look and two in a row trigger a stop.
- Child growth charts
- WHO growth standards report height-for-age and weight-for-height as z-scores against a reference population. Below −2 counts as stunting or wasting, below −3 as severe.
- IQ and T-scores
- Test publishers rescale z to friendlier numbers. IQ is
100 + 15z. Psychology T-scores are50 + 10z. Bone density T-scores compare you to healthy young adults, with −2.5 or lower read as osteoporosis. - Z-tests
- The test statistic is a z-score of a sample mean, so it divides by the standard error
σ / √ninstead of σ. Plugging in σ when you meant the standard error is the most common homework mistake with this formula. - Altman Z-score
- A bankruptcy-risk formula built from five balance-sheet ratios. Same letter, unrelated math. This calculator does not compute it.
When the percentile on this page is wrong
The z-score itself is plain arithmetic and always correct for the inputs you give. The percentile next to it is a claim about shape, and the claim fails more often than textbooks admit.
- Skewed data breaks the percentile. Household income, support ticket wait times, and file sizes have long right tails. A z of +2 on that kind of data often has far more than 2.3% of values above it. List mode prints a rank percentile beside the normal one, so you see the gap on your own numbers.
- Short lists hide their own outliers. With a sample standard deviation, no value in a list of n numbers scores beyond
(n − 1) / √n. For 10 values the ceiling is 2.85, so a “flag anything past 3” rule never fires, no matter how bad the reading. The fill-weight example shows a 486.0 ml bottle, 14 ml under target, scoring only about −2.8. Below 11 values, compare against a known mean and SD instead. - The bad value drags the SD with it. One extreme reading inflates the standard deviation used to judge it. If a value looks suspicious, recompute without it, or use the median and the median absolute deviation, which resist this.
- Sample or population changes the answer slightly. On 10 values the n − 1 version of SD runs about 5% larger, which shrinks every z by the same 5%. Past 30 values the difference stops mattering in practice.
- A guessed SD gives a guessed z. If σ came from a rule of thumb or a different year’s data, the third decimal of the result means nothing.