Problems – Variance and Standard Deviation
1)
The
mean of 100 observations is 50 and their standard deviations is 5. Find the sum
of all square of the observations.
Solution.
Given, ͞x = 50, n = 100 and σ = 5
We have, σ²
= 
- ( ͞x)²
ð
= σ²
= ( ͞x)²
ð 
= n[σ²
+ ( ͞x)²]
= 100[5²
+ (50)²]
= 100[25 + 2500]
= 100 x 2525
= 252500
Hence, the sum of all squares of all the
observations is 252500.
2)
The
marks obtained by seven students are 8, 9, 11, 13, 14, 15, and 21. Find the
variance and standard deviation of these marks.
Solution.
Mean, ͞x = 
= 13
|
Marks |
|
( |
|
8 9 11 13 14 15 21 |
-5 -4 -2 0 1 2 8 |
25 16 4 0 1 4 64 |
|
Total |
|
114 |
Here, N =
7, Σ
͞x)²
Therefore,
σ² = 
Σ ͞x)²
= 
= 16.29
Σ =
√16.29 = 4.04
Therefore,
Variance = 16.29 and standard deviation = 4.04
3)
Find
the variance and standard deviation of the data 6, 5, 9, 13, 12, 8 and 10.
Solution.
Mean, ͞x = 
= 9
|
Marks |
|
( |
|
6 5 9 13 12 8 10 |
-3 -4 0 4 3 -1 1 |
9 16 0 16 9 1 1 |
|
Total |
|
52 |
Here, N =
7, Σ ͞x)²
Therefore,
σ² = Σ ͞x)²
= 
= 7.42
Σ = √7.42
= 2.72
Therefore,
Variance = 7.42 and standard deviation = 2.72
4)
The
following related to sample of size σ₀ Σx2 = 18000 and Σx = 960, find the varience.
Solution.
σ² = 
= 
- (
)²
= 300 – (16)²
= 300 – 256
= 44
Therefore,
Variance = 44