Percentiles for Normal Distributions (Empirical Rule)

(a) Percentiles highlighted in 0.5 increments of standard deviations from the mean
(b) Deciles highligted in terms of standard deviations from the mean
Figure 1: The Normal(0, 1) cdf. Provides the cdf for any Normal distribution if the variable is measured in standard deviations from the mean. It’s the same cdf in both figures just with different values highlighted.
Table 1: Select percentiles for a Normal distribution in terms of standard deviations from the mean
(a) In 0.5 increments of standard deviations from the mean
Percentile Standard deviations from the mean
0.02% -3.5
0.1% -3.0
0.6% -2.5
2.3% -2.0
6.7% -1.5
15.9% -1.0
30.9% -0.5
50% 0.0
69.1% 0.5
84.1% 1.0
93.3% 1.5
97.7% 2.0
99.4% 2.5
99.9% 3.0
99.98% 3.5
(b) Deciles and quantiles as standard deviations from the mean
Percentile Standard deviations from the mean
1% -2.33
5% -1.64
10% -1.28
20% -0.84
25% -0.67
30% -0.52
40% -0.25
50% 0.00
60% 0.25
70% 0.52
75% 0.67
80% 0.84
90% 1.28
95% 1.64
99% 2.33
(a) Each sector represents an interval with 5% probability
(b) Each sector represents an interval with 1% probability
(c) Values labeled in increments of 0.5 standard deviations from the mean
Figure 2: A continuous Normal(0, 1) spinner. The same spinner is displayed in all figures, with different features highlighted. This spinner can be used to simulate values from any Normal distribution by simulating values in terms of standard deviations from the mean.

The empirical rule is often described in terms of “within [blank] standard deviations of the mean” as in the following: