What percentile is 1, 1.5, 2 or 3 standard deviations from the mean?
In a normal distribution, a value 1 standard deviation above the mean is at the 84.13th percentile, 1.5 SD above is at the 93.32th, 2 SD above at the 97.72th and 3 SD above at the 99.87th. Below the mean the percentiles mirror these: −1 SD is the 15.87th, −3 SD the 0.13th.
━ Standard normal curve
Standard deviations to percentiles: the table
The number of standard deviations from the mean is the z-score. The percentile is the share of a normal distribution that lies below it, Φ(z):
| SDs from the mean (z) | Percentile | % above |
|---|---|---|
| −3 | 0.13 | 99.87% |
| −2.5 | 0.62 | 99.38% |
| −2 | 2.28 | 97.72% |
| −1.5 | 6.68 | 93.32% |
| −1 | 15.87 | 84.13% |
| −0.5 | 30.85 | 69.15% |
| 0 | 50.00 | 50.00% |
| +0.5 | 69.15 | 30.85% |
| +1 | 84.13 | 15.87% |
| +1.5 | 93.32 | 6.68% |
| +2 | 97.72 | 2.28% |
| +2.5 | 99.38 | 0.62% |
| +3 | 99.87 | 0.13% |
The table is symmetric: the percentile at −z is 100 minus the percentile at +z. For other values, such as 1.28 or 2.33, use the z-score calculator, which gives the exact percentile for any z.
What percentile is 1.5 standard deviations above the mean?
About the 93rd (93.32%). Suppose exam scores have a mean of 70 and an SD of 8. A score of 70 + 1.5 × 8 = 82 is 1.5 SD above the mean, so it beat about 93% of the class, if the scores are roughly normal. A score of 58, 1.5 SD below, is at the 6.68th percentile.
What percentile is 1 standard deviation below the mean?
About the 16th (15.87%). That is where the “68” of the 68–95–99.7 rule comes from: the range from −1 SD to +1 SD runs from the 15.87th to the 84.13th percentile, which is 68.26% of the values.
What percentile is z = −3, or 3 SDs below?
The 0.13th percentile. Only about 1 value in 741 lies that far below the mean, and the same share lies more than 3 SD above it. That rarity is why ±3 SD is a common outlier cut-off and the edge of a control chart.
How the empirical rule fits in
The empirical rule gives the share between two points rather than below one: about 68% within 1 SD, 95% within 2 SD and 99.7% within 3 SD. Halve what is left for each tail and you get the percentiles above: (100 − 95.45) ÷ 2 = 2.28% below −2 SD, so +2 SD is the 97.72nd percentile.
When the percentiles do not apply
Every number on this page assumes a normal distribution. Incomes, waiting times and reaction times are skewed, and for them a value 2 SD above the mean may sit well below or above the 98th percentile. If you have the data, rank them directly with the percentile calculator. To get z first, compute (x − x̄) ÷ s, where the standard deviation calculator gives s.
Related calculators
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Z-score calculator
z and its percentile from a value, the mean and the SD.
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Percentile calculator
Percentiles of a data set, or from a score, mean and SD.
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Empirical rule calculator
The 68–95–99.7 ranges for your mean and SD.
Common questions
What percentile is 1.5 standard deviations above the mean?
The 93.32th percentile, about the 93rd. In a normal distribution 93.32% of values lie below z = 1.5 and 6.68% lie above it.
What percentile is z = −3?
The 0.13th percentile: only 0.13% of a normal distribution lies more than three standard deviations below the mean, about 1 value in 741.
What percentile is 1 standard deviation below the mean?
The 15.87th percentile, about the 16th. One SD above the mean is the 84.13th, so 68.26% of values fall within one standard deviation either side.
What percent is 1.5 standard deviations?
86.64% of a normal distribution lies within 1.5 standard deviations of the mean (between z = −1.5 and z = 1.5), with 6.68% in each tail.
Does this work if my data are not normal?
Not exactly. These percentiles come from the normal curve. For skewed data the real percentile can be far off; use the percentile rank calculator on the data themselves. Chebyshev's theorem gives a guarantee for any distribution, but only a loose one: at least 75% within 2 SD.
