Calculation of Arithmetic Mean

It is defined as sum of observations divided by the number of observations.

(a) x̄ =

∑x/n

(Formula for raw data)

∑ x = Sum of all observations.

N = n = Number of observations .

(b) x̄ =

∑fx/∑ f

(Formula for ungrouped data)
∑ f = n = N = Number of observations .

(c) x̄ =

∑fx/n

(Formula for grouped data) . x = Mid-point

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Short Cut Method

This method is used to calculate A.M , when the values of x_i is large . This method reduces the calculation .

Let A be the assumed mean, we can write

d_i = x_i - A
or
∑ f_id_i = ∑ f_i(x_i - A) = ∑ f_ix_i - A ∑ f_i

Dividing both sides by N = ∑f_i, we get

∑ f_id_i /N

=

∑ f_ix_i /N

- A

x̄ = A +

∑ f_id_i /N

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Step - Deviation or Coding Method : -

In a frequency distribution , If x_i values has equal distance or (has a common multiplier), then this method will further reduce the calculation .

We can define u_i =

x_i - A/h

,

Where A is assumed mean , h is the difference of any two successive observations.

∑ f_iu_i = ∑ f_i(

x_i - A/h

) =

1/h

∑ f_i (x_i - A)

h ∑ f_iu_i = ∑ f_ix_i - A∑ f_i

Dividing both sides by N , we get
x̄ = A + h

∑ f_iu_i /N

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Weighted Arithmetic Mean

If the case of simple arithmetic mean, it is assumed that all the items or observations of the distribution are of equal importance or having equal weight. However, if all the observations have no equal importance (or weight) then weighted Arithmetic Mean is applicable.

Let w₁ , w₂ , w₃ ---------- w_n be the respective weights of x₁ , x₂ , x₃ ---------- x_n , then

Weighted Arithmetic Mean , (x̄ w) =

∑ w_ix_i/∑w_i

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Properties of Arithmetic Mean :

(1) The sum of deviations of a set of observations is zero when these deviations are taken from their arithmetic mean .
∑ (x_i - x̄) = 0

(2) The sum of square of deviations of a set of observations is minimum , when the deviations are taken from their arithmetic mean .

∑ f_i(x_i - x̄)² ≤ ∑ f_i(x_i - A)²

Where A is any arbitrary constant .

(3) Arithmetic Mean is capable of being treated algebraically . (It means if two values are known, third value can be calculated).


(4) If N₁ and x̄₁ are the number of observations and mean of series (denoted as series 1) and N₂ and x̄₂ are the respective values of series 2 . Then on combining two series the mean will be

x̄₁₂ =

N₁ x̄₁ + N₂ x̄₂/ N₁ + N₂

(for two groups or series)

x̄₁₂₃ =

N₁ x̄₁ + N₂ x̄₂ + N₃ x̄₃ /N₁ + N₂ + N₃

(for three groups or series) .

It can also be calculated for more than 3 groups or series .

(5) It has impact of both change of origin and change of scale .

→ If all the observations of the series are added or subtracted by a constant K then the mean also gets added or subtracted by a constant K . (change of origin) .

→ If all the observations of the series are divided or multiplied by a constant K , then the mean also gets divided or multiplied .

(6) If all the observations assumed by a variable are constants,say k , then the Arithmetic Mean is also k .

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Merits of Arithmetic Mean :-

(1) A.M is rigidly defined average , it is easy to understand and to calculate .

(2) A.M. is based on all observations .

(3) It is widely used in business world .

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Demerits of Arithmetic Mean :-

(1) A.M. cannot be computed for qualitative data .

(2) A.M. is unduly affected by extreme observations .

(3) It cannot be computed for an open ends distribution .

(4) A.M. cannot be calculated in case missing cbservation data .