Question

If variance of ten observations \[10,20,30,40,50,...100\] is \[A\] and the other ten observations \[22,42,62,82,102,...,202\] is \[B\], then, \[\dfrac{B}{A}\] is
A. \[0\]
B. \[1\]
C. \[2\]
D. \[4\]

Answer 1

Hint: We know that the variance is,
\[{\sigma ^2} = \dfrac{1}{n}\sum\limits_{i = 1}^n {({x_i} - \overline x } {)^2}\]
Here, \[\overline x \] is the mean.
At first, we will find the mean and variance of the ten observations separately.
Using these values, we will find the required answer.

Complete step-by-step solution:
It is given that; the variance of ten observations \[10,20,30,40,50,...100\] is \[A\] and the other ten observations \[22,42,62,82,102,...,202\] is \[B\].
We know that the variance is,
\[{\sigma ^2} = \dfrac{1}{n}\sum\limits_{i = 1}^n {({x_i} - \overline x } {)^2}\]
Here, \[\overline x \] is the mean.
Now, mean of first ten observation \[10,20,30,40,50,...100\] is \[\dfrac{{10 + 20 + 30 + 40 + 50 + ... + 100}}{{10}} = 55\]
Substitute the values in the above formula we get,
\[{\sigma ^2} = \dfrac{1}{{10}}\sum\limits_{i = 1}^{10} {({x_i} - 55} {)^2}\]
Expanding the series, we get,
\[{\sigma ^2} = \dfrac{2}{{10}} \times {5^2}({1^2} + {3^2} + {5^2} + {7^2} + {9^2})\]
Simplifying we get,
\[{\sigma ^2} = 825\]
So, \[A = 825\]
Now, mean of first ten observation \[22,42,62,82,102,...,202\] is \[\dfrac{{22 + 42 + 62 + 82 + ... + 202}}{{10}} = 112\]
Substitute the values in the above formula we get,
\[{\sigma ^2} = \dfrac{1}{{10}}\sum\limits_{i = 1}^{10} {({x_i} - 112} {)^2}\]
Expanding the series, we get,
\[{\sigma ^2} = \dfrac{2}{{10}} \times {10^2}({1^2} + {3^2} + {5^2} + {7^2} + {9^2})\]
Simplifying we get,
\[{\sigma ^2} = 3300\]
So, \[B = 3300\]
So, \[\dfrac{B}{A} = \dfrac{{3300}}{{825}} = 4\]

Hence, the correct option is D.

Note: Variance is a measure of how data points differ from the mean. According to Layman, a variance is a measure of how far a set of data (numbers) are spread out from their mean (average) value.
Variance means to find the expected difference of deviation from actual value. Therefore, variance depends on the standard deviation of the given data set.
The more the value of variance, the data is more scattered from its mean and if the value of variance is low or minimum, then it is less scattered from mean. Therefore, it is called a measure of spread of data from mean.
Variance is symbolically represented by σ2, s2, or Var(X).
For the purpose of solving questions, the formula for variance is given by:
\[{\sigma ^2} = \dfrac{1}{n}\sum\limits_{i = 1}^n {({x_i} - \overline x } {)^2}\]