Number of Samples to be Collected
Solid waste is very heterogeneous in nature and its composition varies with place and time. Even samples obtained from the same place (sampling point) on the same day, but at different time may show totally different characteristics. Due
to this reason the method by which the sample is collected and the number of samples collected is critical.
In the planning of sample survey, a stage is always reached at which some decision must be made about the size of the sample. This decision is extremely important as unduly large number of samples result in waste of resources, while
less number of samples diminish the accuracy and utility of the results.
A method of determining the number of samples by statistical technique has been suggested by Dennis E. Carruth and Albert J. Klee*.
The data on physical analysis of solid waste is presented in percentage.
Since the percentage of one constituent differs greatly from the other, the data follows a multinomial distribution. So the data is subjected to a normalising transformation by using arcsin function.
Y = 2 arcsin Ö X
Where X is the original percentage value of a component expressed as a decimal; and
Y is transformed value of X.
To determine the number of samples required for composition analysis following formula is used.
n = (ZS/d)2
Where,
n = number of samples
Z = the standard normal deviate for confidence leve l desired
S = estimated standard deviation (transformed basis)
* Analysis of Solid Waste Composition, Statistical Technique to Determine Sample Size, SW-19ts, US Department of Health, Education and Welfare, Bureau of Solid Waste Management, 1969
d = sensitivity (transformed basis) = ½2 arcsin ÖX - 2 arcsin ÖX ± D ½
The value for D is set according to the desired level of precision. In this case the values for acceptable precision are obtained from the range e.g. paper content in Indian Cities ranges between 2.91 – 6.43%. The average percentage of paper content is 4.036. Therefore X = p.04036. (Y = 0.4045). There will be two values i.e. 0.02126 and 0.02394. The choice of sign for X±D is positive if X is less than 0.5. Therefore, corresponding values for X±D are 0.0516 & 0.0643 and
transformed values Y -> 0.4582 & 0.5126.Therefore d1 = ½2 arcsin Ö 0.04036 - 2 arcsin Ö 0.0516 ½
= ½0.4045 - 0.4582½
= 0.0537
and d2 = ½2 arcsin Ö 0.04036 - 2 arcsin Ö 0.0643 ½
= ½0.4045 – 0.5126½
= 0.1081
Substituting the values of d1 in equation n = (ZS/d1)2
We get n = 6
Z = 1.96, Z+0.684
Similarly substituting value of d2 in equation n = (ZS/d2)2
We get n = 2
Therefore samples required for paper is in the range of 2-6.
Similarly number of samples required for other constituents were calculated
and results are given in Table 3.2.
Table 3.2 : Critical Statistics Obtained from Typical Indian Data
X Y V Range
(no. of samples)
Paper 0.04036 0.4045 0.0742 2 – 6
Rubber, Leather & Synthetics 0.00596 0.1545 0.0298 13 – 35
Glass 0.00558 0.1495 0.0285 9 – 10
Metals 0.00506 0.1424 0.0277 13 – 20
Total Compostable Matter 0.4221 1.4144 0.1766 1 – 36
Inert 0.4793 1.4979 0.0731 2 – 3
X -> mean of n observations expressed as decimals
Y -> transformed value of X
V -> standard deviation
Data for C/N ratio is transformed as follows:
C/N ratio X Y d no. of samples
Average value 25.66 0.2566 1.0570 0.1137 1 0.0980
Therefore d1 = ½2 arcsin Ö 0.04036 - 2 arcsin Ö 0.0516 ½ = ½0.4045 - 0.4582½ = 0.0537
and d2 = ½2 arcsin Ö 0.04036 - 2 arcsin Ö 0.0643 ½ = ½0.4045 – 0.5126½ = 0.1081
Substituting the values of d1 in equation n = (ZS/d1)2
We get n = 6
Z = 1.96, Z+0.684
Similarly substituting value of d2 in equation n = (ZS/d2)2
We get n = 2
Therefore samples required for paper is in the range of 2-6.
An advantage of this method is that number of samples can also be determined for any important chemical parameter. For example, carbon to nitrogen (C/N) ratio is important for determining the suitability of the solid waste for composting. The number of samples required can be calculated from values of C/N ratio. Normally the range of C/N ratio in Indian Municipal Solid Waste is 21.13-30.94 and the typical average value of C/N ratio is 25.66. The desired precision can be obtained from upper and lower values of the range and the average.
Number of samples can be calculated separately for nitrogen and carbon. The number of samples required in case of nitrogen is about 380 and that for carbon is one. Similarly number of samples can be calculated for Phosphorus,
Potassium and other chemical parameters.
It is evident from the statistical results obtained from the method mentioned above that the number of samples to be taken does not exceed thirty-five in any case. Though larger number of samples will increase precision, the required
number of samples for increased precision increase at a very large disproportionate rate making it very uneconomic and analysis a hard task. The basic aim should be to obtain a sample size which is a compromise between economy and precision.
