Properties Of Standard DeviationBY: SAHIL JINDAL

Standard Deviation

Standard deviation is a widely used measurement of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the "average" (mean, or expected/budgeted value).

Calculation

To calculate SD, you first need to find out variance: Variance (S2) = average squared deviation of values from mean Standard deviation (S) = square root of the variance

ExampleQ. A hen lays eight eggs. Each egg was weighed and recorded as follows:60 g, 56 g, 61 g, 68 g, 51 g, 53 g, 69 g, 54 g.A. First, calculate the mean (59). Then find the standard deviation.

Weight of eggs, in grams

Using the information from the above table, we can see that

The sum of each observation minus the mean, squared equals 320

Weight (x)

(x - mean) (x - mean)^2

60 1 156 -3 961 2 468 9 8151 -8 6453 -6 3669 10 10054 -5 25472 320

TypesLow Standard Deviation indicates that the data points tend to be very close to the mean.High Standard Deviation indicates that the data is spread out over a large range of values.

Properties

Standard deviation is only used to measure spread or dispersion around the mean of a data set.

Standard deviation is never negative. Standard deviation is sensitive to outliers. A

single outlier can raise the standard deviation and in turn, distort the picture of spread.

Cont..

For data with approximately the same mean, the greater the spread, the greater the standard deviation.

If all values of a data set are the same, the standard deviation is zero (because each value is equal to the mean).

Impact on ‘б’ of Change in Origin & Scale If a constant is added to all the observations then their standard deviation will

remain constant, i.e., Standard Deviation is not affected by addition of a constant to all observations. For ex. If S.D. of 1, 2, 3 is 0.82 and 10 added is to these, then Standard Deviation of 11, 12,

13 will also be 0.82.

If a constant is subtracted to all the observations then their standard deviation will remain constant, i.e., Standard Deviation is not affected by subtraction of a constant to all observations. For ex. If S.D. of 11, 12, 13 is 0.82 and 10 subtracted is to these, then Standard Deviation of

1, 2, 3 will also be 0.82.

Cont.

It is minimum when it is calculated through mean.

If all the observations are multiplied by a constant ‘c’ then their standard deviation will also increase ‘c’ times, i.e., Standard Deviation is affected by multiplication of all observations by a constant. For ex. If S.D. of 1, 2, 3 is 0.82, then Standard Deviation of 2, 4, 6 will be 1.64.

If all the observations are divided by a constant ‘c’ then their standard deviation will also decrease ‘c’ times, i.e., Standard Deviation is affected by division of all observations by a constant. For ex.If S.D. of 2, 4, 6 is 1.64, then Standard Deviation of 1, 2, 3 will be 0.82.