# Using Statistical Control Charts to Monitor Building Water Consumption: A Case Study on the Replacement of Toilets

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Water Consumption Monitoring

## 3. Statistical Control Charts

#### 3.1. Shewhart Control Chart

_{2}= 3/d

_{2}is a variable that is dependent on the number of samples m. The standard deviation $\sigma $ can be estimated from either the standard deviation or the range (R) of the observations within each sample. Since R is a random variable, the quantity W = R/σ, is also a random variable. The parameters of the probabilistic distribution of W have been determined for any sample size. The mean of the distribution of W is called d

_{2}, which can be found in the literature [29].

_{0}= 370, meaning that, if the process is in control, a signal will be given every 370 samples, on average, and the probability that a single point falls outside the control limits when the process is in control is $\alpha =0.0027$ [29].

#### 3.2. Exponentially Weighted Moving Average Control Chart

_{i}and the number of samples i (or time), defined in Equation (4) [29].

_{i}is the most recent observed value and λ is the weight constant that controls the amount of influence of the previous observations, $0<\lambda <1$. Values near one (1) put almost all weight on the current observation and, for values near zero (0), a small weight is applied to the past observations. Smaller λ values allow changes of lesser magnitude to be detected. Usually, a value of λ = 0.1 or λ = 0.2 is utilized [29]. The initial value is the process target value with Z

_{0}= µ

_{0}, and µ

_{0}is also used as the reference value (or central line) in the chart. When the target value is unknown, the µ

_{0}parameter can be replaced by the average of a large number of observations that are under statistical control. Supposing that the observations x

_{i}are a random independent variable with variance σ

^{2}, then the variance of z

_{i}is given by Equation (5) [29].

## 4. Materials and Methods

- Total volume of water per day: total volume of water, measured in liters/day, consumed by the toilets in a day.
- Average volume of water per flush per day, measured in liters/flush/day.
- Number of flushes: total amount of toilet flushes in a day.

## 5. Results

## 6. Conclusions

^{2}and Multivariate Exponentially Weighted Moving Average control charts. Besides investigating the application of different types of graphics and techniques, comparing the performance of these different approaches is essential to identify the best solutions, considering the specifics of the process. Control charts, in general, possess a high degree of flexibility, thus they can be modified to display more accurate and reliable information for a given process while considering its specificities.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Total Volume Per Day (L/Day) | Average Volume Per Flush Per Day (L/Flush/Day) | Number of Flushes | ||
---|---|---|---|---|

Phase 1 | Minimum | 1024 | 6.55 | 120 |

Maximum | 2359 | 9.02 | 318 | |

Average | 1606 | 7.51 | 215 | |

Standard Deviation | 288.61 | 0.59 | 39.65 | |

Phase 2 | Minimum | 474.1 | 2.83 | 99 |

Maximum | 2364.50 | 6.72 | 364 | |

Average | 936.8 | 5.24 | 179.20 | |

Standard Deviation | 314.20 | 0.68 | 54.38 |

Total Volume Per Day (L/Day) | Average Volume Per Flush Per Day (L/Flush/Day) | Number of Flushes | ||
---|---|---|---|---|

Phase 1 | Minimum | 584.40 | 6.17 | 72.00 |

Maximum | 1289.10 | 8.37 | 197.00 | |

Average | 961.70 | 7.18 | 134.40 | |

Standard Deviation | 179.22 | 0.47 | 25.27 | |

Phase 2 | Minimum | 260.20 | 3.68 | 56.00 |

Maximum | 1570.70 | 6.57 | 239.00 | |

Average | 614.70 | 5.22 | 116.52 | |

Standard Deviation | 215.78 | 0.69 | 31.08 |

Total Volume Per Day (L/Day) | Average Volume Per Flush Per Day (L/Flush/Day) | Number of Flushes | ||
---|---|---|---|---|

Phase 1 | Minimum | 276.40 | 5.60 | 42.00 |

Maximum | 920.40 | 9.17 | 119.00 | |

Average | 570.30 | 7.54 | 75.00 | |

Standard Deviation | 130.16 | 0.83 | 16.51 | |

Phase 2 | Minimum | 155.90 | 1.67 | 27.00 |

Maximum | 793.80 | 7.03 | 234.00 | |

Average | 322.10 | 5.42 | 62.68 | |

Standard Deviation | 113.40 | 1.00 | 32.46 |

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**MDPI and ACS Style**

Freitas, L.L.G.; Kalbusch, A.; Henning, E.; Walter, O.M.F.C. Using Statistical Control Charts to Monitor Building Water Consumption: A Case Study on the Replacement of Toilets. *Water* **2021**, *13*, 2474.
https://doi.org/10.3390/w13182474

**AMA Style**

Freitas LLG, Kalbusch A, Henning E, Walter OMFC. Using Statistical Control Charts to Monitor Building Water Consumption: A Case Study on the Replacement of Toilets. *Water*. 2021; 13(18):2474.
https://doi.org/10.3390/w13182474

**Chicago/Turabian Style**

Freitas, Lucas Lepinski Golin, Andreza Kalbusch, Elisa Henning, and Olga Maria Formigoni Carvalho Walter. 2021. "Using Statistical Control Charts to Monitor Building Water Consumption: A Case Study on the Replacement of Toilets" *Water* 13, no. 18: 2474.
https://doi.org/10.3390/w13182474