STATISTICS

MEAN

1. Formula to find mean when individual observations x1, x2, x3, ... xn

 (ungrouped data) are given is

   Mean= (sum of observations)/(Total number of observations)

  X= (x1 + x2 + x3 + ….....+ xn)/n

  X= (Σ x1)/n

2. Formula to find mean when observations x1, x2, x3, ... xn along with the

 corresponding frequencies f1, f2, f3, ... fn are given (grouped data) is

 Mean= (sum of product of frequencies along with corresponding observations)/(Sum of frequencies)

  X= (f1x1 + f2x2 + f3x3 + ….....+ fnxn)/(f1+f2+f3+.……..+fn)

  X= (Σ f1x1)/Σf1

  X= (Σ f1x1)/N (Note : finding mean using this formula is called Direct method)

3. Formula to find mean when class interval are given (grouped data with

 C.I)

 X= a+ h/N(Σf1a1)(Note : finding mean using this formula is called Step deviation).

 * a → Assumed mean

 * h → Class size (length or width of the class)

 * ui → xi-a/h, where xi - a = di (deviation)

 * N → Sum of frequencies

4. Formula to find mean by using Assumed mean (or) shortcut method

X = a + 1/N (Σfidi)

 * a → Assumed mean

 * di = xi-a (deviation)

 * N → Sum of frequencies

Median

* Formula to find median when individual observations x₁, x₂, x₃, ..., xₙ

 (ungrouped data) are given is

    Case I : If n = odd => M = ((n+1)/2)th observation

    Case II : If n = even => M= ((n/2)th observation + (n/2+1)th observation )/2

* Formula to find median when observations (xᵢ) along with corresponding

 frequencies (fᵢ) are given is

     M = the observation opposite to the cummulative frequency which is just

 greater than N/2 value

* Formula to find median when class intervals (grouped data) are given

    M = l + ((n/2 - cf)/f) * h; where

     f => Median frequency (opposite to median class)

     cf => Cummulative frequency just preceeding the median frequency

     l => Lower unit of the median class

     N => Sum of all frequencies

     h => Class size (length of the class)

Mode

* Formula to find Mode when individual observations (ungrouped data)

 are given is

     Mode (Z) = 'The observation' which occurs most frequently.

* Formula to find Mode when class intervals (grouped data) are given is

     Mode (Z) = l + ((f₁-f₀)/(2f₁-f₀-f₂)) * h

     f₁ => Modal frequency (highest frequency).

     f₀ => Frequency just preceding modal frequency.

     f₂ => Frequency just succeeding modal frequency.

     l => lower limit of the modal frequency.

     h => class size.

* EMPHERICAL Relation between Mean, Median and Mode is given by

         Mode(Z) = 3Median(M) - 2Mean(x̄)