Calculate the variance and standard deviation for – Class 12 Mathematics

Question of 165

Calculate the variance and standard deviation for the following discrete frequency distribution :

$x_i$	4	8	11	17	20	24	32
$f_i$	3	5	9	5	4	3	1

Andhra Pradesh BieapBIEAP Intermediate Board 2019Subjective· 7mImportance★★★★★

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Using $\text{Var}=\dfrac{\sum f_ix_i^2}{N}-\left(\dfrac{\sum f_ix_i}{N}\right)^2$ with $N=30$ gives variance $45.8$ and SD $\approx6.77$ .

Step 1 — Find $N$ .

$N=\sum f_i = 3+5+9+5+4+3+1=30$ .

Step 2 — Compute $\sum f_ix_i$ and the mean.

$x_i$	4	8	11	17	20	24	32
$f_i$	3	5	9	5	4	3	1
$f_ix_i$	12	40	99	85	80	72	32

$\sum f_ix_i = 12+40+99+85+80+72+32=420$ .

Mean $\bar x=\dfrac{420}{30}=14$ .

Step 3 — Compute $\sum f_ix_i^2$ .

$f_ix_i^2$ : $4^2(3)=48$ , $8^2(5)=320$ , $11^2(9)=1089$ , $17^2(5)=1445$ , $20^2(4)=1600$ , $24^2(3)=1728$ , $32^2(1)=1024$ .

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL1 markMCQ
$Q. If the variance of the data is 144 then the standard deviation of the data is$
(a) 144
(b) -12
(c) 12
(d) 14
›Reveal solutionSolution

Standard deviation is defined as the positive square root of variance.

By definition, standard deviation $\sigma = \sqrt{\text{variance}}$ . Given variance $= 144$ :

$\sigma = \sqrt{144} = 12$ …

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CBSE 2026Set ANNUAL1 markMCQ
$Q. The square of standard deviation is —$
(a) Mean-deviation
(b) Range
(c) Variance
(d) Mean
›Reveal solutionSolution

The square of the standard deviation is called the variance, option (c).

Standard deviation ( $\sigma$ ) measures the spread of data about the mean. By definition, variance ( $\sigma^2$ ) is exactly the square of the standard deviation: $\text{Variance} = \sigma^2 = (\text{Standard Deviation})^2$ .

…

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CBSE 2026Set ANNUAL1 mark
$Q. Fill in the blank: If variance of given data is 225, then the standard deviation of the same data will be ______.$
›Reveal solutionSolution

Standard deviation $=\sqrt{\text{Variance}}$ ; here $\sqrt{225}=15$ .

By definition, variance is the square of the standard deviation, so standard deviation $=\sqrt{\text{variance}}$ .

…

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CBSE 2024Set ANNUAL1 markMCQ
$Q. If the standard deviation of a data is 0.012, find the variance.$
(a) 0.144
(b) 0.00144
(c) 0.000144
(d) None of these
›Reveal solutionSolution

Variance is defined as the square of the standard deviation.

By definition, variance $= (\text{standard deviation})^2$ .

Given standard deviation $= 0.012$ : …

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CBSE 2024Set ANNUAL1 mark
$Q. If the standard deviation of obtained marks of a class students is 1.6, then find the variance.$
›Reveal solutionSolution

Given standard deviation $=1.6$ , the variance is $2.56$ .

By definition, variance $=(\text{standard deviation})^2$ . Here standard deviation $=1.6$ …

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CBSE 2024Set ANNUAL1 mark
$Q. Fill in the blank: In a series, the difference between the maximum value and the minimum value is called ______.$
›Reveal solutionSolution

This difference is called the range of the series, the simplest measure of dispersion.

Step 1. Range is defined as: Range $=$ Maximum value $-$ Minimum value.

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Concept understanding — Variance and Standard Deviation of Data

Variance and Standard Deviation of Data

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