The Use of Confidence Limits and Variance Analysis for Water Audits

The use of 95% confidence limits to validate the degree of uncertainty in individual components of the water balance is nowadays best practice among qualified water loss management professionals.

In order to understand the concept of 95% confidence limits, it is first necessary to understand normal distributions which are an important class of statistical distributions. All normal distributions are symmetric and have bell-shaped density curves with a single peak. To speak specifically of any normal distribution, two quantities have to be specified: the mean ^ where the peak of the density occurs, and the standard deviation s, which indicates the spread or girth of the bell curve. Different values of ^ and s yield different normal density curves and hence different normal distributions.

The normal density can be actually specified by means of an equation. The height of the density at any value x is given by

Although there are many normal curves, they all share an important property which is often referred to as the empirical rule:

• 68% of the observations fall within one standard deviation of the mean, that is, between ^ - sand ^ + s.

• 95% of the observations fall within two standard deviations of the mean, that is, between ^ - 2 s and ^ + 2 s.

• 99.7% of the observations fall within three standard deviations of the mean, that is, between ^ - 3 s and ^ + 3 s.

Thus, for a normal distribution, almost all values lie within three standard deviations of the mean as can be seen in Fig. 7.6.

Using 95% confidence intervals allows generating a lower and upper limit for the water balance component. The interval estimate or lower and upper limit gives an indication of how much uncertainty there is in the volume used for each water balance component. The narrower the interval, the more precise is the value used.

The 95% confidence limits also allow for the calculation of the variance related to each water balance component. Variance is a measure of dispersion around the mean. Components with a large variance will have the biggest impact on 95% confidence limit related to the final result of the water balance. The final derived result of the water balance is the volume of real losses. This component will have a 95% confidence limit that is an accumulated value based on the variance related to each component of the water balance. The variance analysis is based on standard statistical principles of normal distribution and uses the root-mean-square (RMS) method for accumulation of error on derived values (see Table 7.5).

Statistical Variance
m - a m m + a
Figure 7.6 Normal distribution curve. (Source: WSO).

Component

Annual Volume (Million gal)

95% Confidence Limits

Variance (gal2 x 1012)

Source #1

7,512.80

2.6%

9,553.7

Source #2

10,519.84

2.6%

18,732.1

Source#3

6,580.71

2.6%%

7,330.2

Source#4

4,411.61

2.60%

3,294.3

Source#4

7.60

2.6%%

0.0

Total system input volume (a)

29,032.56

1.3%

38,910.3

Billed metered authorized consumption

24,778.64

1.10%

20,237.7

Billed un-metered authorized consumption

0.0

NA

NA

Total billed authorized consumption (b)

24,778.64

1.1%

20,237.7

Nonrevenue water [(a) - (b) ]

4,253.92

11.2%

Table 7.5 Calculation of Confidence Limits for Nonrevenue Water

Source: SFPUC

Table 7.5 Calculation of Confidence Limits for Nonrevenue Water

The standard approach to calculate the variance related to a certain volume of the water balance, based on its 95% confidence limit, is as follows:

Variance = (volume in million gal x 95% confidence limit/1.96)2

The aggregated confidence limit related to a calculated volume of the water balance is based on accumulation of error on derived values. Following these principles the 95% confidence limit related to the calculated volume of nonrevenue water is calculated as follows:

95% confidence limit for nonrevenue water

= 1.96 -yj(var iance a + var iance b) / (annual volume nonrevenue water)

The above equation explains the standard approach for calculating an aggregated confidence limit based on accumulation of error on derived values. Table 7.5 provides an example of the calculation of the 95% confidence limits and variances relevant for the calculation of 95% confidence limits for nonrevenue water.

Since the real losses have a confidence limit that is an accumulated value based on the variance related to each component of the water balance it is very important to accurately assign 95% confidence limits to all components of the water balance in order to see which of the components has the biggest impact (which components have the highest variance) on the confidence related to the calculated real loss volume. Once this information is available, it is best practice to take actions (e.g., improving the accuracy of metering devices or installing new metering devices where no meter was in place) in order to improve the confidence related to the real loss volume by improving the confidence related to those components that showed the highest variance.

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