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 2020Subjective· 7mImportance★★★★★

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Compute $N=\sum f_i$ , $\sum f_ix_i$ and $\sum f_ix_i^2$ , get the mean, then variance $=\dfrac{\sum f_ix_i^2}{N}-(\text{mean})^2$ and SD $=\sqrt{\text{variance}}$ .

$x_i$	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
$f_ix_i^2$	48	320	1089	1445	1600	1728	1024

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

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

$\sum f_ix_i^2 = 48+320+1089+1445+1600+1728+1024 = 7254$

Mean: …

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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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