There are 60 students in a class. The following is the frequency distribution of the marks obtained by the students in a test: Marks 0 1 2 3 4 5 Frequency x - 2 x x 2 x + 1 2 2 x x + 1 where, x is positive integer. Determine the mean and standard deviation of the marks.

Given that sum of frequencies is 60 . ∴ x - 2 + x + x 2 + ( x + 1 ) 2 + 2 x + ( x + 1 ) = 60 ⇒ 4 x - 2 + x 2 + x 2 + 2 x + 1 + x + 1 = 60 ⇒ 2 x 2 + 7 x - 60 = 0 ⇒ 2 x 2 + 15 x - 8 x - 60 = 0 ⇒ x ( 2 x + 15 ) - 4 ( 2 x + 15 ) = 0 ⇒ ( 2 x + 15 ) ( x - 4 ) = 0 ⇒ x - 4 = 0 and 2 x + 15 = 0 ⇒ x = 4 a n d x = - 15 2 [Rejected] ∵ x ∈ I + So, x = 4 Now put x = 4 in the frequency distribution table x i f i d i = x i - 3 f i d i f i d i 2 0 2 - 3 - 6 18 1 4 - 2 - 8 16 2 16 - 1 - 16 16 3 25 0 0 0 4 8 1 8 8 5 5 2 10 20 N = 60 ∑ f i d i = - 12 ∑ f j d i 2 = 78 Let assumed mean A = 3 Mean = A + ∑ f i d i N = 3 + - 12 60 = 3 - 1 5 = 14 5 = 2 . 8 and S . D . ( σ ) = ∑ f i d i 2 N - ∑ f i d i N 2 = 78 60 - - 12 60 2 = 1 . 3 - 0 . 04 = 1 . 26 = 1 . 12

HARD

11th CBSE

IMPORTANT

Earn 100

Calculate the mean deviation about the mean of the set of first $n$ natural numbers when $n$ is an odd number.

Important Questions on Statistics

MEDIUM

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 15 - Statistics>EXERCISE>Q 4

Calculate the mean deviation about the mean of the set of first

n

natural numbers when

n

is an even number.

MEDIUM

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 15 - Statistics>EXERCISE>Q 5

Find the standard deviation of the first $n$ natural numbers.

HARD

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 15 - Statistics>EXERCISE>Q 6

The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results: Number of observations

= 25

, mean

= 18.2

= \frac{n^{2} - 1}{4}

, standard deviation

= 3.25

s

. Further, another set of

15

observations

= \frac{\sum |x_{i} - \bar{x}|}{n}

, also in

seconds

, is now available and we have

\sum_{i = 1}^{15} x_{i} = 279

and

\sum_{i = 1}^{15} {x_{i}}^{2} = 5524

. Calculate the standard deviation (correct up to

2

decimal places) based on all

40

observations.

HARD

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 15 - Statistics>EXERCISE>Q 7

The mean and standard deviation of a set of

n_{1}

observations are

x_{1}

and

s_{1}

, respectively while the mean and standard deviation of another set of

n_{2}

observations are

x_{2}

and

s_{2}

, respectively. Show that the standard deviation of the combined set of

(n_{1} + n_{2})

observations is given by

S . D = \sqrt{\frac{n_{1} {(s_{1})}^{2} + n_{2} {(s_{2})}^{2}}{n_{1} + n_{2}} + \frac{n_{1} n_{2} {({\bar{x}}_{1} - {\bar{x}}_{2})}^{2}}{{(n_{1} + n_{2})}^{2}}}

MEDIUM

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 15 - Statistics>EXERCISE>Q 8

Two sets each of

20

observations, have the same standard deviation

5

. The first set has a mean

17

and the second mean

22

. Determine the standard deviation (correct up to two decimal places) of the

x

sets obtained by combining the given two sets.

MEDIUM

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 15 - Statistics>EXERCISE>Q 9

The frequency distribution:

$x$	$A$	$2 A$	$3 A$	$4 A$	$5 A$	$6 A$
$f$	$2$	$1$	$1$	$1$	$1$	$1$

where $A$ is a positive integer, has a variance of $160$ . Determine the value of $A$ .

MEDIUM

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 15 - Statistics>EXERCISE>Q 10

For the frequency distribution:

$\begin{matrix} x & 2 & 3 & 4 & 5 & 6 & 7 \\ f & 4 & 9 & 16 & 14 & 11 & 6 \end{matrix}$

Find the standard deviation (round off to two decimal places).

HARD

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 15 - Statistics>EXERCISE>Q 11

There are $60$ students in a class. The following is the frequency distribution of the marks obtained by the students in a test:

Marks	$0$	$1$	$2$	$3$	$4$	$5$
Frequency	$x - 2$	$x$	$x^{2}$	${(x + 1)}^{2}$	$2 x$	$x + 1$

where, $x$ is positive integer. Determine the mean and standard deviation of the marks.

Calculate the mean deviation about the mean of the set of first n natural numbers when n is an odd number.

Important Questions on Statistics

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Calculate the mean deviation about the mean of the set of first $n$ natural numbers when $n$ is an odd number.