To find out concentration of \(SO_2\) 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. Find the mean concentration of \(SO_2\) in the air.

Question :

Concentration of \(SO_2\) (in ppm)Frequency
0.00 – 0.044
0.04 – 0.089
0.08 – 0.129
0.12 – 0.162
0.16 – 0.204
0.20 – 0.242

Solution :

Let the assumed mean a = 0.10

Concentration of \(SO_2\) (in ppm)Frequency (\(f_i\))Mid – Value (\(x_i\))\(u_i\) = \(x_i – 0.10\over 0.04\)\(f_iu_i\)
0.00 – 0.0440.02-2-8
0.04 – 0.0890.06-1-9
0.08 – 0.1290.1000
0.12 – 0.1620.1412
0.16 – 0.2040.1828
0.20 – 0.2420.2236
\(\sum f_i\) = 30\(\sum f_iu_i\) = -1

By step deviation method :

Mean = a + \(\sum f_iu_i\over \sum f_i\) \(\times\) h = 0.10 + \(-1\over 30\) \(\times\) 0.04

= 0.10 – 0.0013 = 0.0987

= 0.099 ppm

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