# 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

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 f‘d‘ f‘d‘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 Σf‘d‘ = 96 Σf‘d‘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.