What are different methods of measuring dispersion. Sample of polythene bags from two manufactures, A, B, are tested by a prospective buyer for bursting pressure and the results are as follows.

Bursting Pressure 5-10 10-15 15-20 20-25 25-30 30-35
Number of bags manufactured by A 2 9 29 54 11 5
B 9 11 18 32 27 13

Which set of bags has more uniform pressure? If price are the same, Which manufacture’s bags would be preferred by buyer? Use appropriate statistical tool

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Range, interquartile range, standard deviation. and variance are the three commonly used measures of dispersion.

Solution:

Let X be the mid value so that \(d^{”} = \frac{17.45}{5}\). Let f and f be the number of bags of manufactures A and B respectively.

Class x \(d^{‘} = \frac{x – 17.45}{5}\) f f fd fd‘2 fd fd‘2
5-9.9 7.45 -2 2 9 -4 8 -18 36
10-14.9 12.45 -2 9 11 -9 9 -11 11
15-19.9 17.45 0 29 18 0 0 0 0
20-24.9 22.45 1 54 32 54 54 32 32
25-29.9 27.45 2 11 27 22 22 54 108
30-34.9 32.45 3 5 13 15 15 39 117
Σd = 3 N=10 N = 110 Σfd=78 Σfd‘2=160 Σfd = 96 Σfd‘2=304

For manufacture A:

\(Mean(\bar{X}_A) = A + \frac{\sum fd^{‘}}{N} \times h\)

= \(17.45 + \frac{78}{100} \times 5\)

= 21

Standard Deviation(σA) = \(h\sqrt{\frac{\sum fd^{‘2}}{N} – \left ( \frac{\sum fd^{‘}}{N} \right )^2}\)

= \(5 \times \sqrt{\frac{160}{110} – \left ( \frac{78}{110} \right )^2}\)

= 5 x 0.975

= 4.875

Covariance(A) = \(\frac{σ_{A}}{\bar{X}_{A}} \times 100%\)

= \(\frac{4.875}{21} \times 100%\)

= 23.2%

For manufacture B:

\(Mean(\bar{X}_B) = A + \frac{\sum fd^{‘}}{N} \times h\)

= \(17.45 + \frac{96}{100} \times 5\)

= 21.8

Standard Deviation(σA) = \(h\sqrt{\frac{\sum fd^{‘2}}{N} – \left ( \frac{\sum fd^{‘}}{N} \right )^2}\)

= \(5 \times \sqrt{\frac{204}{110} – \left ( \frac{96}{110} \right )^2}\)

= 5 x 1.41

= 7.05

Covariance(A) = \(\frac{σ_{A}}{\bar{X}_{A}} \times 100%\)

= \(\frac{7.05}{21.8} \times 100%\)

= 32.34%

Since, the cofficient of variation for A is less than that for B, the set of bags manufactured by A has more uniform pressure.Hence, the buyer would perfer to buy bags manufactured by A.

Another Method:

Brusting Pressure M A B AM BM
5-10 7.5 2 9 15 67.5
10-15 12.5 9 11 112.5 137.5
15-20 17.5 29 18 507.5 315
20-25 22.5 54 32 1215 720
ΣA = 94 ΣB = 70 ΣAM = 1850 ΣBM = 1240

Brusting pressure of Polythene A = \(\frac{\sum AM}{\sum A}\)

= \(\frac{1850}{94}\)

= 19.68

Brusting pressure of Polythene B = \(\frac{\sum BM}{\sum B}\)

= \(\frac{1240}{70}\)

= 17.68

Since, the brusting pressure of polythene A is greater than of polythene B. So, Buyers perfers the polythene A.

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