## Definition and Concept

[mathjax]

Arithmetic Mean (A.M) is given(overline{X} = frac{sum{X}}{n} = frac{Sum of term}{number of term})

The Mean of the Combined Series If n1 and n2are the sizes of the two series and (overline{X_{1}}) and (overline{X_{2}}) are their respective means, then the mean X of the combined series of size n1+n2 is given by

(overline{X}_{12}=frac{n_{1}overline{X_{1}}+n_{1}overline{X_{2}}}{n_{1}+n_{2}})

Weighted Arithmetic Average is given as (overline{X}_{W}= frac{sum {XW}}{sum{W}})

1. For consecutive integers or for equally spaced numbers (AP)

(overline{X}=frac{first term + last term }{2})

2. Count of consecutive numbers inclusive = last term – first term + 1,

Example 9 to 15, total = 7.

3. Count of consecutive numbers exclusive (terms greater than x but less than y) = last term – first term -1. Example: 9 to 15, total = 5

4. If the average of a few consecutive integers is 0, then either all numbers are zero or there will be an odd number of integers.

5. The average of an odd number of consecutive integers is an integer and the average of an even number of consecutive integers is a non-integer.

MEDIAN

1. The median is the value in the middle when there are an odd number of cases or the average of two middle values when there are an even number of cases when the values are ordered ascending or descending.

2. If the number of observations is odd, the median is equal to the middle number.
Exp: 2, 5, 17, 19, 22
The median is 17.

3. If the number of observations is even, the median is equal to the average of the two middle numbers.
Exp: 2, 5, 17, 19, 22, 25
(17+19)/2 = 18

4. For consecutive integers or equally spaced numbers (AP), the median is (First term + Last term)/2.
Thus, in this scenario, Median = Mean.

Range

It is defined as the difference between the two extreme observations of the distribution. Range = Xmax – Xmin

where Xmax is the greatest observation

and Xmin is the smallest observation of the variable value.

If Range = 0, all the observations are equal.

In equal space or consecutive case range of n integer from anywhere are same

Exp:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …….

Range of 5 number from 2 or from 6 or from anywhere are same

Standard deviation. It is defined as positive square root of the A.M. of the squares of the deviations of the given observations, SD =√((∑(x-x)^2)/n)   . It is a measure of how much each value varies from the mean of all the values.

1.Less SD implies more consistency, less variation, less spread, more compactness AND vice versa.

2.If SD = 0, all the observations are equal.

3.Range is always greater than SD, except when all observations are equal, when both are equal to 0.

4. Items + or – some constant = no change in sd

5. Items × some constant = sd increase

6. Item ÷ some constant = sd decrease

7. N sd below the mean = mean – N×sd

8. N sd above the mean = mean + N×sd

9. Sd is always +ve.