Question

To find out the concentration of $S{O_2}$ in the air (in parts per million, which is ppm), the data was collected for $30$ localities in a certain city and is presented below:

Concentration of $S{O_2}$(in ppm)	Frequency
$0.00 - 0.04$	$4$
$0.04 - 0.08$	$9$
$0.08 - 0.12$	$9$
$0.12 - 0.16$	$2$
$0.16 - 0.20$	$4$
$0.20 - 0.24$	$2$

Find the mean concentration of $S{O_2}$ in the air.

Answer 1

The mean (or average) of observations, is the sum of the values of all the observations divided by the total number of observations.
If ${x_1},{x_2},{x_3},......,{x_n}$ are observations with respective frequencies ${f_1},{f_2},{f_3},........,{f_n}$ then this means observation ${x_1}$ occurs ${f_1}$ times, ${x_2}$ occurs ${f_2}$ times, and so on.
Now, the sum of the values of all the observations =${f_1}{x_1} + {f_2}{x_2} + ...... + {f_n}{x_n}$,and sum of the number of observations = ${f_1} + {f_2} + {f_3} + ........ + {f_n}$

Formula used: So, the mean $x$ of the data is given by
$x = \dfrac{{{f_1}{x_1} + {f_2}{x_2} + ...... + {f_n}{x_n}}}{{{f_1} + {f_2} + {f_3} + ........ + {f_n}}}$
Or
$x = \dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}$

Complete step-by-step answer:
It is given that, the data was collected for $30$ localities in a certain city and the concentration of $S{O_2}$ in the air is presented below:

Concentration of $S{O_2}$(in ppm)	Frequency
$0.00 - 0.04$	$4$
$0.04 - 0.08$	$9$
$0.08 - 0.12$	$9$
$0.12 - 0.16$	$2$
$0.16 - 0.20$	$4$
$0.20 - 0.24$	$2$

We need to find out the mean concentration of $S{O_2}$ in the air.
The observation ${x_i}$ is given by $\dfrac{{\left( {{\text{upper class limit + lower class limit}}} \right)}}{2}$

Concentration of $S{O_2}$(in ppm)	Frequency $({f_i})$	Observation ${x_i}$	${f_i}{x_i}$
$0.00 - 0.04$	$4$	$\dfrac{{0.00 + 0.04}}{2} = 0.02$	$4 \times 0.02 = 0.08$
$0.04 - 0.08$	$9$	$\dfrac{{0.04 + 0.08}}{2} = \dfrac{{0.12}}{2} = 0.06$	$9 \times 0.06 = 0.54$
$0.08 - 0.12$	$9$	$\dfrac{{0.08 + 0.12}}{2} = \dfrac{{0.20}}{2} = 0.10$	$9 \times 0.10 = 0.90$
$0.12 - 0.16$	$2$	$\dfrac{{0.12 + 0.16}}{2} = \dfrac{{0.28}}{2} = 0.14$	$2 \times 0.14 = 0.28$
$0.16 - 0.20$	$4$	$\dfrac{{0.16 + 0.20}}{2} = \dfrac{{0.36}}{2} = 0.18$	$4 \times 0.18 = 0.72$
$0.20 - 0.24$	$2$	$\dfrac{{0.20 + 0.24}}{2} = \dfrac{{0.44}}{2} = 0.22$	$2 \times 0.22 = 0.44$

$\sum\limits_{i = 1}^n {{f_i}} = 4 + 9 + 9 + 2 + 4 + 2 = 30$
$\sum\limits_{i = 1}^n {{f_i}{x_i}} = 0.08 + 0.54 + 0.90 + 0.28 + 0.72 + 0.44 = 2.96$
Mean Concentration of $S{O_2}$(in ppm) = $\dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }} = \dfrac{{2.96}}{{30}} = 0.09866 = 0.099$ppm.

Note: Mean
There are several kinds of means in mathematics, especially in statistics. For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.
${\text{m = }}\dfrac{{{\text{Sum of the terms}}}}{{{\text{Number of terms}}}}$.