spi(t)=bcwp(t)/bcws(t)=incremental Schedule Performance Index
cpi(t)=bcwp(t)/acwp(t)=incremental Cost Performance Index
The cumulative and incremental definitions are linked by the following:
bcws(t)=BCWS(t)–BCWS(t–1) or BCWS(t)=BCWS(t–1)+bcws(t)
bcwp(t)=BCWP(t)–BCWP(t–1) or BCWP(t)=BCWP(t–1)+bcwp(t)
acwp(t)=ACWP(t)–ACWP(t–1) or ACWP(t)=ACWP(t–1)+acwp(t)
That is, acwp(t) is the actual cost of work performed in the time period t, whereas ACWP(t) is the cumulative actual cost of the work performed from the beginning of the project through time period t. The time period is the reporting period, usually a month.
Due to random fluctuations in project conditions, the dimensionless indices spi(t) and cpi(t) will vary. These statistical variations are typically assumed, for convenience, to be drawn from normal distributions, with some defined means and standard deviations. If the project is in a state of statistical control, these means and variances will be stable. The mean values of spi(t) and cpi(t) should be 1.0 (greater than 1.0 is better; less is worse) and the variances of both should be acceptably small. Zero variances, which would indicate exceptional quality of project planning and control, are unlikely to occur. The objective of the analysis is to determine whether a project has gone out of statistical control, which means that either the mean of spi(t) or cpi(t) is changing or the variance is changing, or both. And if a project is going out of control, this may mean that the project will go over schedule or over budget in the future.
To evaluate whether a change is occurring in the mean (the measure of central tendency) of spi(t), one should first establish the average value and the standard deviation based on historical data from projects that are considered to have been under control and good performers. Then, upper and lower natural process limits (UNPL and LNPL), which are conventionally three standard deviations above and below the mean, are derived. This usage is the source for the well-known six-sigma limits in statistical process control charting (one sigma is the standard deviation). Then, the probability that the measured spi(t) will be below the three-sigma LNPL (based on the normal distribution) owing to statistical fluctuations alone is 0.0013, and the probability that spi(t) will be above the UNPL is also 0.0013. That is, if the measured spi(t) is below the LNPL, this indicates that the process may be going out of control, as the probability that this value would occur with the process in control is only about 1/1,000. More specifically, if management were to follow up on every value of spi(t) below the LNPL to investigate a possible adverse change in the process, management would be wrong only once in a thousand times.
As a simple indicator of dispersion or variability, control charting methods often use the period-to-period range, which is the absolute magnitude of the