A class XII has 20 students whose marks (out of 30) are 14, 17, 25, 14, 21, 17, 17, 19, 18, 26, 18, 17, 17, 26, 19, 21, 21, 25, 14 and 19. If random variable X denotes the marks of a selected student given that the probability of each student to be selected is equally likely. Find mean, variance and standard deviation of X.

Mean = 19.25, Variance = 13.89, Standard Deviation = 3.73

The correct answer is Option (1) → Mean = 19.25, Variance = 13.89, Standard Deviation = 3.73

Let us prepare the following frequency table:

Total number of students = 20.

Given that X = marks of a selected student.

So $P(X=14)=\frac{3}{20};P(X=17)=\frac{5}{20}=\frac{1}{4};P(X=18)=\frac{2}{20}=\frac{1}{10};$

$P(X=19)=\frac{3}{20};P(X=21)=\frac{3}{20};P(X=25)=\frac{2}{20}=\frac{1}{10};$

$P(X=26)=\frac{2}{20}=\frac{1}{10};$

Hence, the required probability distribution is

To calculate mean, variance and standard deviation, we construct the following table:

∴ Mean $μ = Σp_ix_i=\frac{385}{20}=\frac{77}{4}=19.25$

Variance $σ^2 = Σp_i{x_i}^2-μ^2=\frac{7689}{20}-(\frac{77}{4})^2=\frac{7689}{20}-\frac{5929}{16}$

$=\frac{30756-29645}{80}=\frac{1111}{80}=13.89$

Standard deviation $σ = \sqrt{\text{Variance}} = \sqrt{13.89} = 3.73$