Question

1. an article in the transactions of the institution of chemical engineers reported data from an experiment investigating the effect of several process variables on the vapor phase oxidation of naphthalene. A sample of the percentage mole conversion of naphthalene to maleic anhydryde follows 4.2,4.7,4.7,5.0,3.8,3.6,3.0,5.1,3.1,3.8,4.8,4.0,5.2,4.3,2.8,2.0,2.8,3.3,4.8 and 5.0

a calculate the sample mean

b find the medium

c find the mode

d calculate the standard deviation

Answer 1

a) x₁= 4.2,x₂=4.7,x₃=4.7,x₄=5.0,x₅=3.8,   22.4

x₆=3.6,x₇=3.0,x₈=5.1,x₉=3.1,x₁₀=3.8, 18.6

x₁₁=4.8,x₁₂=4.0,x₁₃=5.2,x₁₄=4.3,x₁₅=2.8, 21.1

x₁₆=2.0,x₁₇=2.8,x₁₈=3.3,x₁₉=4.8 ,x₂₀=5.0   17.9

sample mean = (x_{1 +} x_{2 + .......+} x_n ) / n

n= total number of samples = 20

sample mean( $\mu$ ) =(4.2+4.7+4.7+5.0+3.8+3.6+3.0+5.1+3.1+3.8+4.8+4.0+5.2+4.3+2.8+2.0+2.8+3.3+4.8 + 5.0 ) / 20

                     $\mu$ = 4

b) medium :

first arrange the values from minimum to maximum value

[2, 2.8, 2.8, 3, 3.1 , 3.3, 3.6, 3.8 ,3.8, 4 , 4.2, 4.3, 4.7, 4.7, 4.8, 4.8, 5 , 5 , 5.1, 5.2 ]

If n is an even number of data values the median is the average of the two values present in the middle when numbers are in increasing order.

total 20 values are present

here n=20 even number

find the average of 10th and 11 th term in the order.

10th term = 4   11th term=4.2

medium = (4+ 4.2 ) / 2 = 4.1

medium   = 4.1

c) Mode :

Mode is the number which is present in data and that comes maximum number of times.

[2, 2.8, 2.8, 3, 3.1 , 3.3, 3.6, 3.8 ,3.8, 4 , 4.2, 4.3, 4.7, 4.7, 4.8, 4.8, 5 , 5 , 5.1, 5.2 ]

In our case some numbers came two times.So all these numbers considered as mode

Mode =2.8 , 3.8, 4.7, 4.8 and 5.0

d) Standard deviation :

S=$\sqrt{\sum_{i=1}^{i=20}}(x-\mu )^{^{2}} / \sqrt{n-1}$

	x	(x-μ)	(x-μ)^2
1	4.2	0.2	0.04
2	4.7	0.7	0.49
3	4.7	0.7	0.49
4	5	1	1
5	3.8	-0.2	0.04
6	3.6	-0.4	0.16
7	3	-1	1
8	5.1	1.1	1.21
9	3.1	-0.9	0.81
10	3.8	-0.2	0.04
11	4.8	0.8	0.64
12	4	0	0
13	5.2	1.2	1.44
14	4.3	0.3	0.09
15	2.8	-1.2	1.44
16	2	-2	4
17	2.8	-1.2	1.44
18	3.3	-0.7	0.49
19	4.8	0.8	0.64
20	5	1	1
			16.46

standard deviation =sqrt(16.46 ) / sqrt (20-1)

                          =4.057093 / 4.35889

standard deviation =0.931