Confidence Intervals and Limits for Projects


Confidence Intervals and Limits for Projects

The whole point of studying statistics in the context of projects is to make it easier to forecast outcomes and put plans in place to affect those outcomes if they are not acceptable or reinforce outcomes if they present a good opportunity for the project. It often comes down to "confidence" rather than a specific number. Confidence in a statistical sense means "with what probability will the outcome be within a range of values?" Estimating confidence stretches the project-forecasting problem from estimating the probability of a specific value for an outcome to the problem of forecasting an outcome within certain limits of value.

Mathematically, we shift our focus from the PDF to the cumulative probability function. Summing up or integrating the probability distribution over a range of values produces the cumulative probability function. The cumulative probability equals the sum (or integral) of the probability distribution over all possible outcomes.

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 1σ 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 1σ 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.

click to expand
Figure 2-7: Confidence Curve for the Normal Distribution.

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 < Xi < y) where Xi 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.