 # Variance and Standard Deviation

## Standard deviation is the positive square root of the mean of the squared deviation from the arithmetic mean.The square of standard deviation is called variance.Three methods for calculating variance and standard deviation are direct method, short-cut method or assumed mean method and step deviation method.Variance and standard deviation from an ungrouped data can be computed using    direct method    short-cut methodVariance and standard deviation from grouped data can be computed using step deviation method.Variance and standard deviation from grouped data within class can be computed using step deviation method.The measure of variability which is independent of units is called coefficient of variation (denoted as C.V.).C.V. = (S. D./Mean) × 100, where mean is not equal to 0.For two series with equal means, the series with greater standard deviation (or variance) is more variable or dispersed than the other. Also, the series with lesser value of standard deviation (or variance) is said to be more consistent than the other.Combined mean is the collective arithmetic mean of several sets of data combined into a single arithmetic mean.Combined Standard deviation is the collective standard deviation of several sets of data combined into a single standard deviation.

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• Q1

The variance of the first ‘n’ natural numbers is

Marks:1

(n2 – 1)/12.

##### Explanation: • Q2

Megha takes a test of 25 marks. The observations of marks of 5 students are 10, 9, 14, 17 and 20. Now, the standard deviation is

Marks:1

4.15.

##### Explanation: • Q3

The standard deviation of the following series is Marks:1

9.

##### Explanation:

Let us assume an arbitrary mean a = 25
Class interval h = 10
Construct the following table: • Q4

If the standard deviation of a set of observation is 4 and if each observation is divided by 4, the standard deviation of the new set of observations will be

Marks:1

1.

##### Explanation:
We know that, if y = x/h, then Sy = Sx/|h|

Since each observation is divided by 4.

Therefore, the S.D of new set of observation will be 4/4 = 1.

• Q5

The standard deviation for the following frequency distribution is Marks:1
 CI Frequency (fi) Mid point xi fi xi  0 – 4 4 – 8 8 – 12 12 -16 4 8 2 1 2 6 10 14 8 48 20 14 – 4 0 4 8 16 0 16 64 Total 15 90 8 96 