The number of defective items produced per hour by a machine, recorded over 20 hours, is given below. Find the

Worked Examples · Example 7

The number of defective items produced per hour by a machine, recorded over 20 hours, is given below. Find the variance and standard deviation using the direct method.

Defective items ( $x$ )	1	2	3	4	5
Number of hours ( $f$ )	3	5	8	3	1

West Bengal WbchseTextbookSubjectiveImportance★★★★★est

79% · 11/14 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Step 1 — Find the mean.

$x$	$f$	$fx$	$fx^2$
1	3	3	3
2	5	10	20
3	8	24	72
4	3	12	48
5	1	5	25
Total	20	54	168

$\bar{x} = 54/20 = 2.7$ .

Step 2 — Apply the computational shortcut.

\sigma^2 = \dfrac{\sum fx^2}{N} - \bar{x}^2 = \dfrac{168}{20} - (2.7)^2 = 8.4 - 7.29 = 1.11

\sigma = \sqrt{1.11} \approx 1.0536

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

$x$	$f$	$x-2.7$	$(x-2.7)^2$	$f(x-2.7)^2$
1	3	−1.7	2.89	8.67
2	5	−0.7	0.49	2.45

Start your 14-day free trial to unlock the full solution →

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 $= 144$ :

$\sigma = \sqrt{144} = 12$ …

Start your 14-day free trial to see the full solution →
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: $\text{Variance} = \sigma^2 = (\text{Standard Deviation})^2$ .

…

Start your 14-day free trial to see the full solution →
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 $=\sqrt{\text{variance}}$ .

…

Start your 14-day free trial to see the full solution →
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 $= 0.012$ : …

Start your 14-day free trial to see the full solution →
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 $=1.6$ …

Start your 14-day free trial to see the full solution →
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.

…

Start your 14-day free trial to see the full solution →

←Previous questionExample 6 — The marks of 50 students in an examination are grouped as follows. Fin…→Next questionExample 8 — The marks of 40 students, grouped as in Worked Example 3 (mean =27), a…

Concept understanding — Variance and Standard Deviation of Data

Variance and Standard Deviation of Data

🎓Unlock everything free for 14 days

Unlock everything free for 14 days