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, shortcut method or assumed mean method and step deviation method.
Variance and standard deviation from an ungrouped data can be computed using
direct method
shortcut method
Variance 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:1Answer:
(n^{2} – 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:1Answer:
4.15.
Explanation:

Q3
The standard deviation of the following series is
Marks:1Answer:
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:1Answer:
1.
Explanation:
We know that, if y = x/h, then S_{y} = S_{x}/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:1Answer:
2.53.
Explanation:
CI
Frequency (f_{i})
Mid point x_{i}
f_{i} x_{i}
0 – 4
4 – 8
8 – 12
12 164
8
2
1
2
6
10
14
8
48
20
14
– 4
0
4
8
16
0
16
64
Total
15
90
8
96