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Measure of Central tendency is a single value which is used to represent an entire set of data. This is also called average.
There are basically three types of average –
1. Mathematical Average
- Arithmetic Mean (AM)
- Geometric Mean (GM)
- Harmonic Mean (HM)
2. Positional Average
- Median
- Mode.
3. Commercial Average
- Moving Average.
- Progressive average.
Table of Contents
What is Arithmetic Mean?
Arithmetic Mean is define as the sum of the values of all observation divided by the number of observation – Arithmetic Mean can be calculate as –
- Individual Series- X’= ∑x/n
- Continuous Series- ∑ƒx/N
- Short cut method.- X’= A+ ∑ƒd’/N×i
Merits and Demerits of Arithmetic Mean
Merits –
- Arithmetic Mean is based on all item.
- It is very easy to understand and calculate.
- It is rigidly define.
- It is capable of further mathematical treatment.
Demerits –
- Mean can’t be computed graphically.
- It is affected by stream value (Upper and lower value).
- It is not possible to calculate mean in case of open class .
- Mean can’t be computed or calculate in case of qualitative data such as beauty, honesty.
Explain the Properties of Arithmetic Mean?
The properties of Arithmetic mean are explain below –
(i) The sum of the deviation of a given set of individual observation from the Arithmetic mean is always zero, i.e –
∑f(x-x’)=0
For example –
| X | (x-x’) |
| 2 | 2-3= -1 |
| 3 | 3-3= 0 |
| 4 | 4-3= 1 |
| 1 | 1-3= -2 |
| 5 | 5-3= 2 |
| ∑ƒ=15 | 0 |
X’= 15/5
= 3
(ii) The sum of squares of deviations of a set of observation is the minimum when deviations are taken AM, i.e-
∑(x-x’)²= is always minimum.
As for example –
| X | (x-x’) | (x-x’)² |
| 2 | -1 | 1 |
| 3 | 0 | 0 |
| 4 | 1 | 1 |
| 1 | -2 | 4 |
| 5 | 2 | 4 |
| ∑ƒ= 15 | 0 | ∑(x-x’)= 10 |
X’= 15/5
= 3
(iii) If each value of a variable ‘X’. X is increased or decreased or multiply or divided by a same constant ‘K’, the AM also increase or decrease or multiply or divided the same constant ‘k’.
As for example –
| X | (X+2) | (X-2) |
| 2 | 4 | 0 |
| 3 | 5 | 1 |
| 4 | 6 | 2 |
| 1 | 3 | -1 |
| 5 | 7 | 3 |
| ∑x= 15 | ∑x+2= 25 | ∑x-2= 5 |
(iv) If all the observation of a says are constant k, then mean also be k.
As for example –
| X |
| 4 |
| 4 |
| 4 |
| 4 |
| 4 |
| ∑x= 20 |
So, X’= 4
(v) If each item in the series replace by the mean then the sum of substitution will be equal to the sum of individual item.
As for example –
| X | X’ |
| 3 | 5 |
| 8 | 5 |
| 4 | 5 |
| ∑x= 15 | ∑ƒ= 15 |
X’= 15/3
= 5
(vi) If we are given the AM and number of item of two groups, we can computed the combined average of this groups by applying the following formula.
Formula of Arithmetic Mean
Individual Series- X’= ∑x/n
Continuous Series- ∑ƒx/N
Short cut method.- X’= A+ ∑ƒd’/N×i
