Sorting out the rubbish

Introduction

This case study is based on a real court case between a Council in the state of Victoria, Australia, and a garbage collection company. To maintain the anonymity of the parties involved, some false names are used here: the garbage collection company is called Dumpers, for example. One aspect of the dispute was about the weighbridge that records the weight of garbage trucks entering and leaving the Council tip. Garbage companies are charged a fee on the basis of the weight of garbage dumped at the tip. Dumpers alleged that there was a problem with the weighbridge; they asked an adviser, Dr. Johns, to examine the weighbridge records to see if there were any “irregularities” in the weighbridge operation. Dr. Johns provided a report suggesting that there were problems with the weighbridge. The Council decided to call Ian Gordon, a statistical consultant, to review Dr Johns’ report and conclusion.

The key questions that the Council wanted to answer were:

Was Dr. Johns’ conclusion justified?
Was there anything wrong with the weighbridge?

Hear from Ian Gordon

Timeline

January to July 1997: Data collection
Early 1998: Dumpers asks Dr Johns to review the weighbridge operation.
Study design and the problem described.
August 1998: Dr Johns reports.
August 1998: Council asks Ian Gordon to review Dr Johns’ report.
Data analysis.
September 1998: Ian Gordon reports to Council.

Graph showing distribution of net weight of garbage dumped. — This figure shows the frequency distribution of the net weight in tonnes of garbage dumped by trucks between January and June, 1997.

Available data

Weighbridge data from January to June 1997 were available for analysis. During this time, trucks from 40 different companies were weighed. Some companies dumped rubbish on very few occasions over the 6 month period; others dumped many hundreds of loads.

Dr Johns’ conclusion

“The pattern of tonnages is unlikely to have occurred by chance.”

In particular:

There are unusual counts of net garbage weights for a number of companies; these counts would not be expected by chance.
Company 2 shows an extremely unusual set of observations; all 29 net weights are 0.40 tonnes.
There are a small number of data entry errors, as some weights do not round to 20kg. (Note that weights were measured to the nearest 20kg.)

Dr Johns’ analysis

Part 1: Organising the data.

Split the dataset into subsets according to company. Each company should be analysed separately as they have different-sized trucks and carry different kinds of rubbish.
Divide the net weight data into one-tonne ranges (e.g. 0 to 1 tonne, 1 to 2 tonnes, etc.).
Within any one tonne range, there are 50 possible weight values, as weights are measured to the nearest 20kg.
Tabulate the distribution of weights in a one-tonne range.

Part 2: Analysing the weights in any given one-tonne range, for a particular company.

Let the number of weights observed = n
Assume that the distribution of weights should be uniform.
Model the count, X, at any specific weight in the range, as a binomial random variable: X ~ Bi(n, 1/50).
Calculate the expected frequency of each count according to the binomial model, across the 50 weights.
Make a judgment about how unusual the observed count is compared with that predicted by the binomial model.

Example of Dr Johns’ analysis

Company 11 had 253 net weights in the range 7 to 8 tonnes. There are two weights that do not round to 20kg. As the reasons for these errors are unknown, these weights were removed from the analysis.

Part 1
Distribution of net weights between 7.00 and 7.98 tonnes for Company 11 (n = 251)

Weight	Count	Weight	Count	Weight	Count	Weight	Count	Weight	Count
7	5	7.2	5	7.4	4	7.6	3	7.8	5
7.02	7	7.22	4	7.42	2	7.62	4	7.82	6
7.04	4	7.24	4	7.44	0	7.64	4	7.84	2
7.06	3	7.26	7	7.46	4	7.66	7	7.86	8
7.08	3	7.28	5	7.48	6	7.68	5	7.88	8
7.1	6	7.3	6	7.5	6	7.7	10	7.9	5
7.12	2	7.32	5	7.52	2	7.72	4	7.92	4
7.14	3	7.34	4	7.54	5	7.74	8	7.94	11
7.16	3	7.36	5	7.56	5	7.76	6	7.96	7
7.18	8	7.38	8	7.58	3	7.78	5	7.98	5

Part 2

Assuming the distribution of weights is uniform within this one tonne range, the expected count (for any weight) is 251 x 1/50 = 5.02.
Note that, for example, a count of 5 occurs 12 times, as is shown in bold in the table above. The distribution of net weights between 7.00 and 7.98 tonnes for Company 11 (n = 251) is also shown in the figure below. Counts of 5 are highlighted in green.
Assuming the counts at any single weight, X, are distributed as Bi(251, 0.02), Pr(X = 5) = 0.1772.
The binomial model predicts 50 x 0.1772 = 8.86, or about 9 values of 5 across 50 weight categories.
Twelve counts of 5 were observed, compared to 8.86 expected – Dr. Johns concluded that this was unusually high.

Study design

Variables

Garbage company
Net weight of garbage

Statistician’s description of the design

The data set is simply a set of consecutive observations over time on a number of garbage trucks owned by 40 companies.

Statistician’s description of the problem

The problem is to determine if there are any characteristics of the observations that suggest that there is anything wrong with the weighbridge or the weighing procedures.

Protocol at the weighbridge

Inclusion criteria	Every commercial garbage truck must be weighed at the tip weighbridge prior to entry to the tip and following dumping of their load at the tip.
Data to be recorded before entry to the tip	The weighbridge supervisor records the following information for each vehicle: Date Company name Total weight on weighbridge (in tonnes, to nearest 20kg) TARE (empty weight of truck, as listed on the side of the truck, in tonnes to nearest 20kg)
Data to be calculated	The weighbridge supervisor enters the weight before entry and the TARE into a computer program. The weighbridge supervisor’s computer calculates: Net weight of garbage = Total weight before – TARE The computer automatically updates the daily records data file.
Data analysis	Daily records are forwarded by the supervisor to the council for later analysis.

Analysis

Summary

Ian Gordon disagreed with Dr Johns’ analysis. Ian proposed alternative methods of assessing the weighbridge data; he suggested that the assumption that net weights were uniform over one tonne ranges was questionable, and commented on the selectivity of the analyses presented.