Find the mean and standard deviation of the following continuous frequency distribution by the step-deviation

Worked Examples · Example 7

Find the mean and standard deviation of the following continuous frequency distribution by the step-deviation method:

Class	0–10	10–20	20–30	30–40	40–50
$f$	5	8	15	8	4

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Take class mid-points $x$ , assumed mean $A = 25$ , class width $h = 10$ , and $d = \dfrac{x - A}{h}$ .

Class	$x$	$f$	$d = \tfrac{x-25}{10}$	$fd$	$fd^2$
0–10	5	5	$-2$	$-10$	20
10–20	15	8	$-1$	$-8$	8
20–30	25	15	0	0	0
30–40	35	8	1	8	8
40–50	45	4	2	8	16
Total		40		$-2$	52

Here $N = 40$ , $\sum fd = -2$ , $\sum fd^2 = 52$ .

Mean:

\bar{x} = A + h\left(\frac{\sum fd}{N}\right) = 25 + 10\left(\frac{-2}{40}\right) = 25 - 0.5 = 24.5.

Standard deviation: …

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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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