Statistical Concepts > Confidence intervals

A random variable assumes various values with various probabilities, according to its probability (density) distribution. According to this distribution, there is a certain value y of the random variable that is exceeded only in p % of cases. Also, there is a certain value x of that random variable such that only p % of the values are smaller than x. This means that 2p % of the values are outside the range from x to y, whereas the rest, (100 - 2p) %, is somewhere between x and y. Stated differently, there is a (100 - 2p) % chance that a value of the random variable is between x and y. The range from x to y is called the 2p % confidence interval of the random variable.

In the cncPAVE program, decision variables such as % shattered concrete, or life cost, are obtained from input random variables by means of Monte Carlo simulation. For this reason the decision variables are also random variables. Consider a random variable such as life cost, LC . Consider the following example. The average life cost, as obtained from 100 000 simulations, is, say, R300/m2. Further say that the probability distribution of this variable indicates that 5 % of the simulated life-cost figures are smaller than R250/m2, and that 5 % of the figures are higher than R400/m2. In this case, the range from R250/m2 to R400/m2 is the 90 % confidence interval of the life-cost random variable.

The best estimate of the life cost is the average, i.e. R300/m2. However, due to uncertainty, this estimate may be wrong - the actual value can be smaller or higher than the average. If we want to make an estimate with a 90 % probability of being correct, then the estimate cannot be a single number, it has to be a range - the 90 % confidence interval, which, in our case, is the interval from R250/m2 to R400/m2.

It is easy to determine confidence intervals for decision variables. This is best done by using graphs that appear on the Distributions Page after a successful simulation run. To learn the technique please try the following:

Create a Scenario of a Jointed Concrete Type. Select this Scenario. Run the calculation process. After the simulation has been completed, notice the average value displayed by the %SH meter - on average about 2.9 % of the slabs are likely to be broken at the time of the first rehabilitation, which is after 20 years.

The question is: What is the 90 % confidence interval for the percentage of broken slabs in this case?

To answer:

Click the Distributions tab. Four graphs will appear. Look at the red curve in the Shattered Concrete graph. The descending red curve crosses the 95 % probability level at about 0.7 % (measured on the horizontal axis), and the 5 % probability level at about 7.2 %. (Red lines are drawn for your convenience at these two levels). One can see that 5 % of the simulated values were smaller than 0.7 %, and 5 % of the simulated values of the shattered concrete percentage were larger than 7.2 %. The remaining 90 % of the values were thus between 0.7 % and 7.2 %.

The range from 0.7 % to 7.2 % is the 90 % confidence interval for the percentage of shattered concrete (broken slabs).

Should you wish to get the confidence interval for another probability rating, say, 95 %, just look where the solid red curve crosses the 97,5 % and 2,5 % probability levels (at 0.55 % and 9.16 %, respectively). The range from 0.55 % to 9.16 % is thus the 95 % confidence interval. Compare this confidence interval with the previous one and notice the trade-off: When describing uncertainty, one can either indicate a relatively narrow interval with relatively small confidence, or a relatively wide interval with a relatively great confidence.

The above techniques are applicable to all the other graphs of decision variables.