## The S Curve

Recall that the cumulative probability accumulates from 0 to 1 regardless of the actual distribution being summed or integrated. We can easily equate the accumulating value as accumulating from 0 to 100%. For example, if we accumulate all the values in a Normal distribution between ±1s of the mean, we will find 68.3% of the total value of the cumulative total. We can say with 68.3% "confidence" that an outcome from a Normal distribution will fall in the range of ±1s of the mean; the corollary is that with 31.7% confidence, an outcome will lie outside this range, either more pessimistic or more optimistic.

Integrating the Normal curve produces an "S" curve. In general, integrating the BETA and Triangular curves will also produce a curve of roughly an "S" shape. [23] Figure 2-7 shows the "S" curve.

Cumulative standard Normal Curve

Cumulative standard Normal Curve

Normalized random variable value/standard deviation Figure 2-7: Confidence Curve for the Normal Distribution.

Normalized random variable value/standard deviation Figure 2-7: Confidence Curve for the Normal Distribution.