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For the following distribution, calculate mean using all suitable methods:

 Size of items:  1 - 4  4 - 9  9 - 16  16 - 27
 Frequency:  6  12  26  20

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By direct method

 Class interval  Mid value (xi)  Frequency (fi)  fixi
 1 - 4  2.5  6  15
 4 - 9  6.5  12  78
 9 - 16  12.5  26  325
 16 - 27  21.5  20 430
 N = 64 \(\sum\)fiui = 848

Mean \(=\frac{\sum f_ix_i}{N}+A\) 

\(=\frac{848}{64} = 13.25\)

By assumed mean method

Let, 

the assumed mean (A) = 6.5

 Class interval  Mid value (xi)  ui = xi - A = xi - 6.5  Frequency (fi)  fiui
 1 - 4  2.5 -4 6 -24
 4 - 9  6.5 0 12 0
 9 - 16  12.5  6 26 156
 16 - 27  21.5  15  20  300
 N = 64 \(\sum\)fiui = 432


Mean = A + \(\frac{\sum f_iu_i}{N}\)

\(=6.5+\frac{432}{64}\)

\(=6.5+6.75\)

\(=13.25\) 

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