Variance & Standard Deviation
Lecture 13
Variance
The arithmetic mean of the square deviation from the mean is called variance, denoted by
When frequencies are given, the variance can be
obtained as:
A large variance indicates that numbers in the set are
far from the mean and far from each other. A small variance, on the other hand,
indicates the opposite.
Note: A variance cannot be negative. That's because
it's mathematically impossible since you can't have a negative value resulting
from a square.
Short cut formula to obtain variance:
When frequencies are given:
Standard Deviation
Variance is a real measure of dispersion, but due to the square of the units, it cannot be easily interpreted. So, a modified measure of
dispersion is known as the standard deviation.
Definition:
The positive square root of the variance
is called the standard deviation, denoted by 
(population) and by S (sample).
A short-cut method to compute the standard deviation is
given below:
Example 4.7: Find variance and standard deviation for
the following data.
10, 12, 14, 16, 18, 20
Solution:
The formula for variance is given below:
The formula for variance is given below:
Properties of variance
1.
The
variance of a constant is zero.
Var (a)
= 0
Proof:
let X = a, a, a, …., a. where a is a constant
and repeated n times.
The mean
of A is
2. If a is a constant and X is a variable, then
3. If a is a & b are constants and X is a variable, then
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