Confidence Tables

A common way to calculate confidence limits is with a table of cumulative values for a "standard" Normal distribution. A standard Normal distribution has a mean of 0 and a standard deviation of 1. Most statistics books or books of numerical values will have a table of standard Normal figures. It is important to work with either a "two-tailed" table or double your answers from a "one-tail" table. The "tail" refers to the curve going in both directions from the mean in the center.

A portion of a two-tailed standard Normal table is given in Table 2-6. Look in this table for the "y" value. This is the displacement from the mean along the horizontal axis. Look at y = 1, one standard deviation from the mean. You will see an entry in the cumulative column of 0.6826. This means that the "area under the curve" from ±1σ is 0.6826 of all the area. The confidence of a value falling around the mean, ±1σ, is 0.6826, commonly truncated to 68.3%.

Table 2-6: Standard Normal Probabilities
"y" Value	Probability
0.1	0.0796
0.2	0.1586
0.3	0.2358
0.4	0.3108
0.5	0.4514
0.6	0.5160
0.7	0.5762
0.8	0.6318
1.0	0.6826
1.1	0.7286
1.2	0.7698
1.3	0.8064
1.4	0.8384
1.5	0.8664
1.6	0.8904
1.7	0.9108
1.8	0.9282
1.9	0.9426
2.0	0.9544
2.1	0.9643
2.2	0.9722
2.3	0.9786
2.4	0.9836
2.5	0.9876
2.6	0.9907
2.7	0.9931
2.8	0.9949
2.9	0.9963
3.0	0.9974
For p(-y < X_i < y) where X_i is a standard normal random variable of mean 0 and standard deviation of 1.
For nonstandard Normal distributions, look up y = a/σ, where "a" is the value from a nonstandard distribution with mean = 0 and σ is the standard deviation of that nonstandard Normal distribution.
If the mean of the nonstandard Normal distribution is not equal to 0, then "a" is adjusted to "a = b - μ," where "b" is the value from the nonstandard Normal distribution with mean μ: y = (b - μ)/σ.

^[23]For a normal curve, the slope changes such that the curvature goes from concave to convex at exactly ±1σ from the mean. This curvature change will show up as an inflection on the cumulative probability curve.