The marks of 40 students, grouped as in Worked Example 3 (mean =27), are given below. Find the variance and st

Worked Examples · Example 8

The marks of 40 students, grouped as in Worked Example 3 (mean $=27$ ), are given below. Find the variance and standard deviation using the direct method.

Marks	0–10	10–20	20–30	30–40	40–50
Number of students ( $f$ )	4	6	14	10	6

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Step 1 — Tabulate class marks, $fx^2$ .

Class	$f$	Class mark $x$	$fx^2$
0–10	4	5	100
10–20	6	15	1350
20–30	14	25	8750
30–40	10	35	12250
40–50	6	45	12150
Total	40		34600

(Recall $\bar{x}=27$ from Worked Example 3.)

Step 2 — Apply the computational shortcut.

\sigma^2 = \dfrac{34600}{40} - 27^2 = 865 - 729 = 136

\sigma = \sqrt{136} \approx 11.6619

Step 3 — Independent check by the direct deviation-squared method.

Class	$f$	$x-27$	$(x-27)^2$	$f(x-27)^2$
0–10	4	−22	484	1936
10–20	6	−12	144	864

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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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