Question

To find out the concentration of SO2 in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:
Concentration of SO2 (in ppm)
0.00−0.04
0.04−0.08
0.08−0.12
0.12−0.16
0.16−0.20
0.20−0.24
Frequency
4
9
9
2
4
2
Find the mean of concentration of SO2 in the air.

Answer 1

STEP DEVIATION METHOD:

Step deviation method is used in the cases where the deviation from the assumed mean 'A' are multiples of a common number. If the values of ‘di’ for each class is a multiple of ‘h’ the calculation become simpler by taking ui= di/h = (xi - A )/h

Here, h is the class size of each class interval.

★★ Find the class marks of class interval. These class marks would serve as the representative of whole class and are represented by xi.  

★★ Class marks (xi)  = ( lower limit + upper limit) /2

★★ We may take Assumed mean 'A’ to be that xi which lies in the middle of x1 ,x2 …..xn

MEAN = A + h ×(Σfiui /Σfi) , where ui =  (xi - A )/h

[‘Σ’ Sigma means ‘summation’ ]

FREQUENCY DISTRIBUTION TABLE IS IN THE ATTACHMENT  

From the table : Σfiui = -1 ,  Σfi = 30

Let the assumed mean, A = 0.10,  h = 0.04

MEAN = A + h ×(Σfiui /Σfi)

Mean = 0.10 + 0.04 (-1/30)

= 0.10 - 0.04/30

= 0.10 - 4/3000

= 0.10 - 0.001

= 0.099

Mean = 0.099

Hence, the Mean concentration of SO2 in the air is 0.099 ppm.

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

Answer :

The mean concentration is 0.099 ppm.

Step-by-step explanation :

Step deviation method -

We use step deviation method in case where the deviation are multiples of a common number from the assumed mean.

We can calculate it by taking

$u_{i}=\frac{d_{i}}{h}=\frac{x_{i}-A}{h}$

$Mean=A+h\times\frac{\Sigma f_{i}u_i}{\Sigma u_{i}}$

where h is the class size of each class interval.

Frequency Distribution Table -

Let the assumed mean be 0.10 .

$\begin{tabular}{|c|c|c|c|c|c|}\cline{1-6}Concentration & x_i & f_i & d_i & u_i & \Sigma f_{i}u_i\\ \cline{1-6}0.00-0.04 & 0.02 & 4 & 0.08 & -2 & -8\\ \cline{1-6}0.04-0.08 & 0.06 & 9 & -0.04 & -1 & -9\\ \cline{1-6}0.08-0.12 & 0.10 & 9 & 0 & 0 & 0\\ \cline{1-6}0.12-0.16 & 0.14 & 2 & 0.04 & 1 & 2\\ \cline{1-6}0.16-0.20 & 0.18 & 4 & 0.08 & 2 & 8\\ \cline{1-6}0.20-0.24 & 0.22 & 2 & 0.12 & 3 & 6\\ \cline{1-6} & & \Sigma f_{i}=30 & & & \Sigma f_{i}u_{i}=-1\\ \cline{1-6}\end{tabular}$

Since, mean -

$\implies A+h\times\frac{\Sigma f_{i}u_i}{\Sigma u_{i}}$

$\implies 0.10+0.04\times\frac{-1}{30}$

$\implies 0.10-\frac{0.04}{30}$

$\implies 0.10-\frac{4}{3000}$

$\implies 0.10-0.001$

$\implies 0.099$