Question

The median and standard deviation (S.D) of distribution are \[20\] and \[4\] respectively. If each item is increased by \[2\], then
A. Median will increase by \[2\]
B. Median will remain the same
C. Standard deviation will remain the same
D. The standard deviation will be increased by 2.

Answer 1

Hint: We can proceed with the basic definition of standard deviation as a quantity expressing by how much the members of a group differ from the mean value for the group. And median as denoting or relating to a value or quantity lying at the midpoint of a frequency distribution of observed values or quantities.

Complete step by step answer:

The terms used for calculation of standard deviation are a deviation from the mean of all the possible observations.
As each number/observation is increased by \[2\]
So, the mean of the observations and the deviations from the mean of the observations will remain the same.
Hence, the standard deviation will not change.
On the other hand, the median gives the average of the middle two numbers, when the total number of terms is even, and for odd observations, the middle number is the median of the observations.
So, the median depends on the observed values, so if all the values increase by 2, then the median also increases by 2.
Thus, it will increase by \[2\].
Hence, option (a) and option (c) are our correct answers.

Note: We can also solve by using the formula of standard deviation as \[{\text{$\sigma$ = }}\sqrt {\dfrac{{\sum {{{{\text{(}}{{\text{x}}_{\text{i}}}{\text{ -$ \mu$ )}}}^{\text{2}}}} }}{{\text{N}}}} \],
\[{\text{$\sigma$ }}\] = population standard deviation
N = the size of the population
\[{{\text{X}}_{\text{i}}}\] = each value from the population
\[{\text{$\mu$ }}\] = the population mean
And in order to calculate the median, the data must first be ranked (sorted in ascending order). The median is the number in the middle. Median = the middle value of a set of ordered
In simple language, the median is the middle term generally for the given observations. And the standard deviation is the deviation of the observations from the mean value of the observations.