If x¯ is the mean of n values of x , then ∑i=1(xi−x?). equal to ________. If a has any value other than x? , then ∑i=1(xi−a)2 is_______than (xi−a)2

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar I f x ¯ i s t h e m e a n o f n o b s e r v a t i o n s o f x , t h e n ∑ i = 1 n x i − x ¯ = 0 a n d i f ' a ' h a s t h e v a l u e o t h e r t h a n x ¯ , t h e n ∑ i = 1 n ( x i − x ¯ ) 2 i s l e s s t h a n t h a n ∑ ( x i − a ) 2 . H e n c e , t h e v a l u e o f t h e f i l l e r a r e 0 a n d l e s s .

The mean and standard deviation of a set of n1 observations are x?1 and s1 , respectively, while the mean and standard deviation of another set of n2 observations are x?2 and s2 , respectively. Show that the standard deviation of the combined set of (n1+n2) observations is given by: S.D.=n1s12+n2s22+n1n2(x?1−x?2)2n1+n2

This is a short answer type question as classified in NCERT Exemplar L e t x i , i = 1 , 2 , 3 , 4 , … n 1 a n d y j , j = 1 , 2 , 3 , 4 , … n 2 ∴ x 1 ¯ = 1 n 1 ∑ i = 1 n x i a n d x 2 ¯ = 1 n 2 ∑ i = 1 n y j ⇒ σ 1 2 = 1 n 1 ∑ i = 1 n 1 ( x i − x 1 ¯ ) 2 a n d σ 2 2 = 1 n 2 ∑ j = 1 n 2 ( y j − x 2 ¯ ) 2 N o w m e a n o f t h e c o m b i n e d s e r i e s i s g i v e n b y x ¯ = 1 n 1 + n 2 [ ∑ i = 1 n x i + ∑ i = 1 n y j ] = n 1 x 1 ¯ + n 2 x 2 ¯ n 1 + n 2 T h e r e f o r e , σ 2 o f t h e c o m b i n e d s e r i e s i s σ 2 = 1 n 1 + n 2 [ ∑ i = 1 n 1 ( x i − x ¯ ) 2 + ∑ j = 1 n 2 ( y j − x ¯ ) 2 ] N o w , ∑ i = 1 n 1 ( x i − x ¯ ) 2 = ∑ i = 1 n 1 ( x i − x j ¯ + x j ¯ − x ¯ ) 2 = ∑ i = 1 n 1 ( x i − x j ¯ ) 2 + n 1 ( x j ¯ − x ¯ ) 2 + 2 ( x j ¯ − x ¯ ) ∑ i = 1 n 1 ( x i − x j ¯ ) 2 B u t ∑ i = 1 n ( x i − x i ¯ ) = 0 [ ? The algebraic sum of the deviation of values of first s e r i e s f r o m t h e i r m e a n i s z e r o . ] A l s o ∑ i = 1 n 1 ( x i − x ¯ ) 2 = n 1 s 1 2 + n 1 ( x 1 ¯ − x ¯ ) 2 = n 1 s 1 2 + n 1 d 1 2 w h e r e d 1 = ( x 1 ¯ − x ¯ ) S i m i l a r l y , w e h a v e ∑ j = 1 n 2 ( y j − x ¯ ) 2 = ∑ j = 1 n 2 ( y j − x i ¯ + x i ¯ − x ¯ ) 2 = n 2 s 2 2 + n 2 d 2 2 w h e r e d 2 = ( x 2 ¯ − x ¯ ) Now combined Standard Deviation(SD) σ = n 1 ( s 1 2 + d 1 2 ) + n 2 ( s 2 2 + d 2 2 ) n 1 + n 2 w h e r e d 1 = x 1 ¯ − x ¯ = x 1 ¯ − ( n 1 x 1 ¯ + n 2 x 2 ¯ n 1 + n 2 ) = n 2 ( x 1 ¯ − x 2 ¯ ) n 1 + n 2 d 2 = x 2 ¯ − x ¯ = x 2 ¯ − ( n 1 x 1 ¯ +

The following information relates to a sample of size 60: x2=18000 , ∑x=960 . The variance is: (a) 6.63 (b) 16 (c) 22 (d) 44

This is an Objective Type Questions as classified in NCERT Exemplar W e k n o w t h a t V a r i a n c e ( σ 2 ) = ∑ x i 2 N − ( ∑ x i N ) 2 = 1 8 0 0 0 6 0 − ( 9 6 0 6 0 ) 2 = 3 0 0 − 2 5 6 = 4 4 H e n c e , t h e c o r r e c t o p t i o n i s ( d ) .

Mean deviation for n observations x1,x2,…,xn from their mean x? is given by (a) 1n∑i=1?xi−x?? (b) 1n∑i=1(xi−x?) (c) 1n∑i=1(xi−x?)2 (d) ∑ n ( x i − x ? ) 2 n

This is an Objective Type Questions as classified in NCERT Exemplar M D = 1 n ∑ i = 1 n | x i − x ¯ | H e n c e , t h e c o r r e c t o p t i o n i s ( b ) .

For the curve defined parametrically as y = 3sinθ.cosθ, x = e^θ.sinθ where θ∈[0,π], the tangent is parallel to x-axis when θ is :

π/4

3π/4

π/2

Let b 1 , b 2 , ................ be a geometric sequence such that b 1 + b 2 = a and Σ b i = 2. Given that b? < 0, then the value of b 1 is :

2+√2

2-√2

1+√2

4+√2

The statement A -> (B -> A) is equivalent to:

A -> (  ( A ↔ B )

All possible values of ∈ [ 0 , 2 π ]  for which sin 2q + tan 2q > 0 lie in:

( 0 , π 4 ) ∪ ( π 2 , 3 π 4 ) ∪ ( π , 5 π 4 ) ∪ ( 3 π 2 , 7 π 4 )

( 0 , π 2 ) ∪ ( π 2 , 3 π 4 ) ∪ ( π , 7 π 6 )

( 0 , π 4 ) ∪ ( π 2 , 3 π 4 ) ∪ ( 3 π 2 , 1 1 π 6 )

( 0 , π 2 ) ∪ ( π , 3 π 2 )

If the function f(x) = { ax+b, -∞<x≤2; x²-5x+6, 2<x<3; px²+qx+1, 3≤x<∞ } is differentiable is (-∞, ∞), then:

a=-1, p=-4/9

b=2, q=5/3

a=1, b=2

The equation of the common tangent touching the circle (x-3) 2 + y 2 = 9 and the parabola y 2 = 4x above the x-axis, is

√3y = x+3

√3y = 3x+1

√3y = -(x+3)

The constant term in the expansion of (log 10 (x^(log 10 x)) - log x 2 100) 12 is:

924

1050

A quadratic function f(x) satisfies f(x) ≥ 0 for all real x. If f(2) = 0 and f(4) = 12, then the value of f(6) is :

12

The graph of function f(x) = x 5 /20 - x 4 /12 + 5 has :

One local maximum, one local minimum, one point of inflection.

No local extremum, one point of inflection.

Two local maximum, one local minimum, two point of inflection

One local maximum, one local minimum, two point of inflection.

The value of the integral ∫ s i n θ s i n θ ( s i n 6 θ + s i n 4 θ + s i n 2 θ ) 2 s i n 4 θ + 3 s i n 2 θ + 6 1 − c o s 2 θ d θ (where c is a constant of integration)

1 1 8 [ 9 − c o s 6 θ − 3 c o s 4 θ − 6 c o s 2 θ ] 3 2 + c

1 1 8 [ 9 − 2 s i n 6 θ − 3 s i n 4 θ − 6 s i n 2 θ ] 3 2 + c

1 1 8 [ 1 1 − 1 8 c o s 2 θ + 9 c o s 4 θ − 2 c o s 6 θ ] 3 2 + c

1 1 8 [ 1 1 − 1 8 s i n 2 θ + 9 s i n 4 θ − 2 s i n 6 θ ] 3 2 + c

Let f , g a: N ® N such that f (n + 1) = f (n) +f (1)  ∀ n ∈ N and g be any arbitrary function. Which of the following statements is NOT true?

If f is onto, then f (n) =  n ∀ n ∈ N

If fog is one – one, then g is one – one

If g is onto, then fog is one – one

A JEE aspirant estimates that he will be successful with an 80% chance if he studies 10 hr/day with 60% chance if he studies 7hr/day and with 40% chance if he studies 4 hr/day. Further, he believes that he will study 10 hr, 7hr and 4 hr/day with probability 0.1, 0.2 and 0.7 respectively. Given that he is successful, the probability that he studies for 4 hr/day equals.

7/12

14/19

Maths NCERT Exemplar Solutions Class 11th Chapter Fifteen: Overview, Questions, Easy Tricks, Rules, Preparation

Q: Number of ways in which three distinct numbers can be selected between 1 and 20 both inclusive, whose sum is even is ____________

All even + 2 odd 1 even ¹? C? + ¹? C? × ¹? C? (10×9×8)/6 + (10×9)/2 × 10 120 + 450 = 570

Q: A circle passes through the points (2, 2) and (9, 9) and touches the x-axis. The absolute value of the difference of possible x-coordinate of the point of contact is ______

Family of circle through (2, 2) and (9, 9), (x-2) (x-9) + (y-2) (y-9) + λ (y-x) = 0 Touches y=0 (x-axis) (x-2) (x-9) + 18 - λx = 0 x² - 11x - λx + 36 = 0 x² - (11+λ)x + 36 = 0 has repeated roots D = 0 ⇒ λ = 1, λ = -23 x = 6 or x = -6 Difference = 12

Q: If a,b are any two perpendicular vectors of equal magnitude and |3a+4b| + |4a-3b| = 20, then |a| equals: ____________

|3a+4b|² = 9|a|² + 16|b|² + 24a·b But a·b = 0, |a|=|b|=k |3a+4b| = 5k |4a-3b| 10k = 20? k = 2 = |a| = |b|

Q: The value of ∫₀^(π/2) |xsin²x - 1/2|dx is equal to aπ/b where a, b are co-prime numbers, then a.b is ____________

∫ (from 0 to π/2) (x/2)|cos (2x)|dx ∫ (from 0 to π/4) (x/2)cos (2x)dx - ∫ (from π/4 to π/2) (x/2)cos (2x)dx = [x sin (2x)/4 - ∫sin (2x)/4 dx] (from 0 to π/4) - [x sin (2x)/4 - ∫sin (2x)/4 dx] (from π/4 to π/2) = [x sin (2x)/4 + cos (2x)/8] (from 0 to π/4) - [x sin (2x)/4 + cos (2x)/8] (from π/4 to π/2) = (π/16 - 1/8) - (-1/8 - π/16) = π/8

Q: lim (x→0) (ae2x - bcosx + c)/(xsinx) = 1, then a + b + c is equal to ____________

lim (x→0) (ae²? - bcosx + c) / (x sinx) = lim (x→0) (ae²? - bcosx + c) / x² * lim (x→0) x/sinx For limit to exist a - b + c = 0 2a + b = 0 (4a-b)/2 = 1 c = 2, b = 2, a = -1 a+b+c = 3

Statistics Short Answers Type Question

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

\begin{array}{l} F i r s t n n a t u r a l n u m b e r s a r e 1, 2, 3, \dots, n . H e r e, n i s o d d . \\ ∴ M e a n \bar{x} = \frac{1 + 2 + 3 + \dots + n}{n} = \frac{\frac{n (n + 1)}{2}}{n} = \frac{n + 1}{2} \\ T h e d e v i a t i o n s o f n u m b e r s f r o m m e a n (\frac{n + 1}{2}) a r e \\ 1 - \frac{n + 1}{2}, 2 - \frac{n + 1}{2}, 3 - \frac{n + 1}{2}, \dots, n - \frac{n + 1}{2} \\ i . e ., - \frac{n - 1}{2}, - \frac{n - 3}{2}, \dots, - 2, - 1, 0, 1, 2, \dots, \frac{n - 1}{2} . \\ T h e a b s o l u t e v a l u e s o f d e v i a t i o n f r o m t h e m e a n i . e ., | x_{i} - \bar{x} | a r e \\ \frac{n - 1}{2}, \frac{n - 3}{2}, \dots, 2, 1, 0, 1, 2, \dots, \frac{n - 1}{2} . \\ T h e s u m o f a b s o l u t e v a l u e s o f d e v i a t i o n f r o m t h e m e a n i . e ., | x_{i} - \bar{x} | a r e \\ = 2 (1 + 2 + 3 + \dots t o \frac{n - 1}{2} t e r m s) \\ = 2 . \frac{\frac{n - 1}{2} (\frac{n - 1}{2} + 1)}{2} = \frac{n - 1}{2} . \frac{n + 1}{2} = \frac{n^{2} - 1}{4} . \\ ∴ M e a n d e v i a t i o n a b o u t t h e m e a n = \frac{\sum_{}^{} | x_{i} - \bar{x} |}{n} = \frac{\frac{n^{2} - 1}{4}}{n} = \frac{n^{2} - 1}{4 n} . \end{array}

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

\begin{array}{l} F i r s t n n a t u r a l n u m b e r s a r e 1, 2, 3, 4, 5, 6, \dots, n . H e r e, n i s e v e n . \\ ∴ M e a n \bar{x} = \frac{1 + 2 + 3 + 4 + \dots + n}{n} = \frac{n (n + 1)}{2 n} = \frac{n + 1}{2} \\ ∴ M D = \frac{1}{n} [\begin{array}{l} | 1 - \frac{n + 1}{2} | + | 2 - \frac{n + 1}{2} | + | 3 - \frac{n + 1}{2} | + \dots + | \frac{n - 2}{2} - \frac{n + 1}{2} | + | \frac{n}{2} - \frac{n + 1}{2} | + \\ | \frac{n + 2}{2} - \frac{n + 1}{2} | + \dots + | n - \frac{n + 1}{2} | \end{array}] \\ = \frac{1}{n} [| \frac{1 - n}{2} | + | \frac{3 - n}{2} | + | \frac{5 - n}{2} | + \dots + | \frac{- 3}{2} | + | - \frac{1}{2} | + | \frac{1}{2} | + \dots + | \frac{n - 1}{2} |] \\ = \frac{1}{n} [\frac{1}{2} + \frac{3}{2} + \dots + \frac{n - 1}{2}] (\frac{n}{2}) t e r m s \\ = \frac{1}{n} {(\frac{n}{2})}^{2} = \frac{1}{n} . \frac{n^{2}}{4} = \frac{n}{4} [∵ S u m o f f i r s t o d d n n a t u r a l n u m b e r s = n^{2}] \\ H e n c e, t h e r e q u i r e d M D = \frac{n}{4} . \end{array}

Commonly asked questions

Q:

The mean and standard deviation of a set of $n_{1}$ observations are ${\overset{?}{x}}_{1}$ and $s_{1}$ , respectively, while the mean and standard deviation of another set of $n_{2}$ observations are ${\overset{?}{x}}_{2}$ and $s_{2}$ , respectively. Show that the standard deviation of the combined set of $(n_{1} + n_{2})$ observations is given by:

$S . D . = \sqrt[]{\frac{n_{1} s_{1}^{2} + n_{2} s_{2}^{2} + n_{1} n_{2} {({\overset{?}{x}}_{1} - {\overset{?}{x}}_{2})}^{2}}{n_{1} + n_{2}}}$

Q:

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

Q:

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

Q:

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

Q:

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 seconds, standard deviation = 3.25 seconds. Further, another set of 15 observations $x_{1}, x_{2}, \dots, x_{15}$ , also in seconds, is now available, and we have: $\sum_{i = 1} x_{i} = 279$ $\sum_{i = 1} x_{i}^{2} = 5524$ Calculate the standard deviation based on all 40 observations.

Q:

Two sets, each of 20 observations, have the same standard deviation 5. The first set has a mean of 17, and the second set has a mean of 22. Determine the standard deviation of the set obtained by combining the two given sets.

Q:

The mean life of a sample of 60 bulbs was 650 hours, and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and a standard deviation of 7 hours. Find the overall standard deviation.

Q:

Mean and standard deviation of 100 items are 50 and 4, respectively. Find the sum of all the items and the sum of the squares of the items.

Q:

If for a distribution: ∑ $(x - 5) = 3$ $\sum {(x - 5)}^{2} = 43$ and the total number of items is 18, find the mean and standard deviation.

Statistics Long Answers Type Question

1. Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.

\begin{array}{l} G i v e n t h a t n = 100, \bar{x} = 40, σ = 10 \\ ∴ \bar{x} = \frac{\sum_{}^{} x_{i}}{n} \Rightarrow 40 = \frac{\sum_{}^{} x_{i}}{100} \Rightarrow \sum_{}^{} x_{i} = 4000 \\ C o r r e c t e d \sum_{}^{} x_{i} = 4000 - 30 - 70 + 3 + 27 = 3930 \\ a n d C o r r e c t e d m e a n = \frac{3930}{100} = 39.3 \\ N o w σ^{2} = \frac{\sum_{}^{} x_{i}^{2}}{n} - {(40)}^{2} \Rightarrow 100 = \frac{\sum_{}^{} x_{i}^{2}}{100} - 1600 \\ \Rightarrow \sum_{}^{} x_{i}^{2} = 1700 \times 100 \Rightarrow \sum_{}^{} x_{i}^{2} = 170000 \\ ∴ C o r r e c t e d \sum_{}^{} x_{i}^{2} = 170000 - {(30)}^{2} - {(70)}^{2} + {(3)}^{2} + {(27)}^{2} \\ = 170000 - 900 - 4900 + 9 + 729 = 164938 \\ ∴ C o r r e c t S D = \sqrt[]{\frac{164938}{100} - {(39.3)}^{2}} = \sqrt[]{1649.38 - 1544.49} = \sqrt[]{104.89} = 10.24 \\ H e n c e, t h e r e q u i r e d S D = 10.24 \end{array}

2. While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the mean and variance as 45 and 16, respectively. Find the correct mean and variance.

\begin{array}{l} G i v e n t h a t n = 10, \bar{x} = 45, σ^{2} = 16 \\ ∴ \bar{x} = \frac{\sum_{}^{} x_{i}}{n} \Rightarrow 45 = \frac{\sum_{}^{} x_{i}}{10} \Rightarrow \sum_{}^{} x_{i} = 450 \\ C o r r e c t e d \sum_{}^{} x_{i} = 450 - 52 + 25 = 423 \\ a n d C o r r e c t e d m e a n = \frac{423}{10} = 42.3 \\ N o w σ^{2} = \frac{\sum_{}^{} x_{i}^{2}}{n} - {(\frac{\sum_{}^{} x_{i}}{n})}^{2} \Rightarrow 16 = \frac{\sum_{}^{} x_{i}^{2}}{10} - {(45)}^{2} \\ \Rightarrow 16 = \frac{\sum_{}^{} x_{i}^{2}}{10} - 2025 \Rightarrow \frac{\sum_{}^{} x_{i}^{2}}{10} = 2041 \\ \Rightarrow \sum_{}^{} x_{i}^{2} = 2041 \times 10 \Rightarrow \sum_{}^{} x_{i}^{2} = 20410 \\ ∴ C o r r e c t e d \sum_{}^{} x_{i}^{2} = 20410 - {(52)}^{2} + {(25)}^{2} \\ = 20410 - 2704 + 625 = 18331 \\ a n d C o r r e c t e d v a r i a n c e \\ σ^{2} = \frac{18331}{10} - {(42.3)}^{2} = 1833.1 - 1789.3 = 43.8 \\ H e n c e, t h e r e q u i r e d m e a n = 42.3 a n d v a r i a n c e = 43.8 \end{array}

Commonly asked questions

Q:

While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the mean and variance as 45 and 16, respectively. Find the correct mean and variance.

Q:

Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.

Statistics Objective Type Question

1. Mean deviation for $n$ observations $x_{1}, x_{2}, \dots, x_{n}$ from their mean $\overset{ˉ}{x}$ is given by

(a) $\frac{1}{n} \sum_{i = 1} ∣ x_{i} - \overset{ˉ}{x} ∣$

(b) $\frac{1}{n} \sum_{i = 1} (x_{i} - \overset{ˉ}{x})$

(c) $\sqrt[]{\frac{1}{n} \sum_{i = 1} {(x_{i} - \overset{ˉ}{x})}^{2}}$

(d) $\frac{\sum^{n} {(x_{i} - \overset{ˉ}{x})}^{2}}{n}$

\begin{array}{l} M D = \frac{1}{n} \sum_{i = 1}^{n} | x_{i} - \bar{x} | \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}

2. Let $x_{1}, x_{2}, \dots, x_{n}$ be $n$ observations and $\overset{ˉ}{x}$ be their arithmetic mean. The formula for the standard deviation is given by:

(a) $\sqrt[]{\frac{1}{n} \sum_{i = 1} {(x_{i} - \overset{ˉ}{x})}^{2}}$

(b) $\frac{1}{n} \sum_{i = 1} {(x_{i} - \overset{ˉ}{x})}^{2}$

(c) $\sqrt[]{\frac{\sum^{n} x_{i}^{2}}{n} - {\overset{ˉ}{x}}^{2}}$

(d) $\sum_{i = 1} x_{i}^{2} + n {\overset{ˉ}{x}}^{2}$

\begin{array}{l} T h e f o r m u l a f o r S . D . = σ = \sqrt[]{\frac{\sum_{}^{} {(x_{i} - \bar{x})}^{2}}{n}} \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}

Commonly asked questions

Q:

The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is:

(a) 50000

(b) 250000

(c) 252500

(d) 255000

Q:

The following information relates to a sample of size 60: $x^{2} = 18000$ , $\sum x = 960$ . The variance is:

(a) 6.63

(b) 16

(c) 22

(d) 44

Q:

Mean deviation for $n$ observations $x_{1}, x_{2}, \dots, x_{n}$ from their mean $\overset{?}{x}$ is given by

(a) $\frac{1}{n} \sum_{i = 1} ? x_{i} - \overset{?}{x} ?$

(b) $\frac{1}{n} \sum_{i = 1} (x_{i} - \overset{?}{x})$

(c) $\sqrt[]{\frac{1}{n} \sum_{i = 1} {(x_{i} - \overset{?}{x})}^{2}}$

(d) $\sqrt[]{\frac{\sum^{n} {(x_{i} - \overset{?}{x})}^{2}}{n}}$

Q:

Let $x_{1}, x_{2}, \dots, x_{n}$ be $n$ observations and $\overset{?}{x}$ be their arithmetic mean. The formula for the standard deviation is given by:

(a) $\frac{1}{n} \sum_{i = 1} {(x_{i} - \overset{?}{x})}^{2}$

(b) $\frac{1}{n} \sum_{i = 1} {(x_{i} - \overset{?}{x})}^{2}$

(c) $\frac{\sum^{n} x_{i}^{2}}{n} - {\overset{?}{x}}^{2}$

(d) $\sum_{i = 1} x_{i}^{2} + n {\overset{?}{x}}^{2}$

Q:

Let $a, b, c, d, e$ be the observations with mean $m$ and standard deviation $s$ . The standard deviation of the observations $a + k, b + k, c + k, d + k, e + k$ is:

(a) $s$

(b) $k \cdot s$

(c) $s + k$

(d) $\frac{S}{K}$

Q:

Let $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ be the observations with mean $m$ and standard deviation $s$ . The standard deviation of the observations $k x_{1}, k x_{2}, k x_{3}, k x_{4}, k x_{5}$ is:

(a) $k + s$

(b) s/k

(c) $k \cdot s$

(d) $s$

Q:

Let $x_{1}, x_{2}, \dots, x_{n}$ be $n$ observations. Let $w_{i} = l x_{i} + k$ for $i = 1, 2, \dots, n$ , where $l$ and $k$ are constants. If the mean of $x_{i}$ â€™s is 48 and their standard deviation is 12, the mean of $w_{i}$ â€™s is 55 and the standard deviation of $w_{i}$ â€™s is 15, the values of $l$ and $k$ should be:

(a) $l = 1.25, k = - 5$

(b) $l = 2.5, k = - 5$

(c) $l = - 1.25, k = 5$

(d) $l = 2.5, k = 5$

Q:

Standard deviations for the first 10 natural numbers is:

(a) 5.5

(b) 3.87

(c) 2.97

(d) 2.87

Q:

Consider the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ . If 1 is added to each number, the variance of the numbers so obtained is:

(a) 6.5

(b) 2.87

(c) 3.87

(d) 8.25

Q:

Consider the first 10 positive integers. If we multiply each number by -1 and then add 1 to each number, the variance of the numbers so obtained is:

(a) 8.25

(b) 6.5

(c) 3.87

(d) 2.87

Q:

Coefficient of variation of two distributions are 50 and 60, and their arithmetic means are 30 and 25 respectively. The difference of their standard deviations is:

(a) 0

(b) 1

(c) 1.5

(d) 2.5

Q:

The standard deviation of some temperature data in $?^{?} C$ is 5. If the data were converted into $?^{?} F$ , the variance would be:

(a) 81

(b) 57

(c) 36

(d) 25

Statistics Fill in the blank Type Question

1. Coefficient of variation = .........../mean × 100

\begin{array}{l} G i v e n t h a t σ_{C} = 5 \\ C V = \frac{S D}{M e a n} \times 100 \\ H e n c e, t h e v a l u e o f t h e f i l l e r i s S D . \end{array}

2. If $\bar{x}$ is the mean of $n$ values of $x$ , then $\sum_{i = 1} (x_{i} - \overset{ˉ}{x}) .$ equal to ________

If $a$ has any value other than $\overset{ˉ}{x}$ , then $\sum_{i = 1} {(x_{i} - a)}^{2}$ is_______than ${(x_{i} - a)}^{2}$

\begin{array}{l} I f \bar{x} i s t h e m e a n o f n o b s e r v a t i o n s o f x, t h e n \sum_{i = 1}^{n} x_{i} - \bar{x} = 0 a n d i f' a' h a s t h e v a l u e o t h e r \\ t h a n \bar{x}, t h e n \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} i s l e s s t h a n t h a n \sum_{}^{} {(x_{i} - a)}^{2} . \\ H e n c e, t h e v a l u e o f t h e f i l l e r a r e 0 a n d l e s s . \end{array}

Commonly asked questions

Q:

Coefficient of variation = ............/mean × 100.

Q:

If $\bar{x}$ is the mean of $n$ values of $x$ , then $\sum_{i = 1} (x_{i} - \overset{?}{x}) .$ equal to ________. If $a$ has any value other than $\overset{?}{x}$ , then $\sum_{i = 1} {(x_{i} - a)}^{2}$ is_______than ${(x_{i} - a)}^{2}$

Q:

If the variance of a data is 121, then the standard deviation of the data is ________

Q:

The standard deviation of a data is ________ of any change in origin but is ________ on the change of scale.

Q:

The sum of the squares of the deviations of the values of the variable is ________when taken about their arithmetic mean.

Q:

The mean deviation of the data is ________ when measured from the median.

Q:

The standard deviation is ________to the mean deviation taken from the arithmetic mean.

JEE Mains 2021

JEE Mains

01 Mock Test 2025

Commonly asked questions

Q:

Number of ways in which three distinct numbers can be selected between 1 and 20 both inclusive, whose sum is even is ____________

Q:

A circle passes through the points (2, 2) and (9, 9) and touches the x-axis. The absolute value of the difference of possible x-coordinate of the point of contact is ______

Q:

If a,b are any two perpendicular vectors of equal magnitude and |3a+4b| + |4a-3b| = 20, then |a| equals: ____________

Q:

The value of ∫₀^(π/2) |xsin²x - 1/2|dx is equal to aπ/b where a, b are co-prime numbers, then a.b is ____________

Q:

lim (x→0) (ae^2x - bcosx + c)/(xsinx) = 1, then a + b + c is equal to ____________

Q:

The value of Σ_i?0⁸ ⁸C_i(⁸C_r+1 - ⁸C_r) equals ____________

Q:

Let A be a square matrix of order 2 such that A² – 4A + 4I = 0, where I is an identity matrix of order 2. If B = A? + 4A? + 6A³ + 4A² + A – 162 I, then det(B) is equal to

Q:

If 1+x⁴-x⁵ = Σ_i=0⁵ a?(1+x)ⁱ, for all x ∈ R, then a² is equal to ____________

Q:

Let g(x) = ||x+2|-3|. If a denotes the number of relative minima, b denotes the number of relative maxima and c denotes the product of the zeros. Then the value of (a+2b-c) is

Q:

Normals of parabola y² = 4x at P and Q meets at R(x₂, 0) and tangents at P and Q meets at T(x₁, 0). If x₂ = 3, then find the area of quadrilateral PTQR.

Maths NCERT Exemplar Solutions Class 11th Chapter Fifteen: Overview, Questions, Preparation

Statistics Short Answers Type Question

Commonly asked questions

Statistics Long Answers Type Question

Commonly asked questions

Statistics Objective Type Question

Commonly asked questions

Statistics Fill in the blank Type Question

Commonly asked questions

JEE Mains 2021

Try these practice questions

JEE Mains

01 Mock Test 2025

Try these practice questions

Commonly asked questions

Student Forum

Other Similar chapters for you

News & Updates

GATE Registration 2026 Last Date Today With Regular Fees Live: Steps, Eligibility, Fees, and Exam Updates

CAT Preparation 2025: Know 10 Tips for Time Management in VARC section

Best Books to Study for SLAT 2026, Important Topics, Previous Year Questions

ICSE, ISC Exam Date Sheet 2026 @cisce.org (Soon); Live Updates on CISCE 10th 12th Timetable

Time Management Tips for CEED and UCEED 2026

JEE Main Marks vs Percentile vs Rank 2025: Calculate Percentile and Rank Using Marks

List of NIT Colleges in India 2025 - NIRF Ranking, Courses, Seats, Cutoff, Fees and Placement

CBSE 10th Maths Question Paper 2025, 2024, 2023, 2022, 2020, 2019 PDF Download

NEET Chapter Wise Weightage 2025 PDF by NTA: Physics, Chemistry & Biology

List of Top IITs in India 2025: NIRF Ranking, Courses, Seats, Fees & Placement