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

Hint: 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.
\[{\text{m = }}\dfrac{{{\text{Sum of the terms}}}}{{{\text{Number of terms}}}}\].