RD Sharma 2020 solution class 9 chapter 24 Measures of Central Tendency FBQS

FBQS

Page-24.22

Question 1:

Fill In the Blanks 

If  X¯ represents the mean of n observations x1,x2,...,xn, then the value of  i=1n xi - X¯ is _____________.

Answer 1:


It is given that, the mean of n observations x1, x2,..., xn is X¯.

X¯=x1+x2+...+xnn   
   
x1+x2+...+xn=nX¯        .....(1)

Now,

i=1nxi-X

=x1-X+x2-X+...+xn-X

=x1+x2+...+xn-X+X+...+Xn times

=nX-nX          [From (1)]

=0

Thus, the value of i=1nxi-X is 0.

If X¯ represents the mean of n observations x1,x2,...,xn, then the value of  i=1n xi - X¯ is ____0____.

Question 2:

Fill In the Blanks 

If each obsevation of the data is increased by 5, then their mean is _____________ by 5.

Answer 2:


Let the mean of n observations x1, x2,..., xn be X¯.

X¯=x1+x2+...+xnn              Mean=Sum of observationsNumber of observations
   
x1+x2+...+xn=nX¯        .....(1)

If each observation is increased by 5, then the new observations are x1 + 5, x2 + 5,..., xn + 5.

∴ Mean of new observations

=x1+5+x2+5+...+xn+5n

=x1+x2+...+xn+5+5+...+5n timesn

=nX+5nn

=nX+5n

=X+5

Thus, the mean of new observations is increased by 5.

If each observation of the data is increased by 5, then their mean is ___increased___ by 5.

Question 3:

Fill In the Blanks 

If the mean of the data x1,x2,...,xn is X¯, then the mean of ax1 + b,ax2+b,...,axn + b is ____________ .

Answer 3:


It is given that, the mean of x1, x2,..., xn  is X¯.

X¯=x1+x2+...+xnn   
   
x1+x2+...+xn=nX¯        .....(1)

Let X' be the mean of data ax1 + b, ax2 + b,..., axn + b.

X'¯=ax1+b+ax2+b+...+axn+bn

X'¯=ax1+ax2+...+axn+b+b+...+bn timesn

X'¯=ax1+x2+...+xn+nbn

X'¯=anX+nbn          [Using (1)]

X'¯=naX+bn

X'¯=aX+b

Thus, the mean of data ax1 + b, ax2 + b,..., axn + b is aX+b.

If the mean of the data x1, x2,..., xn is X¯, then the mean of ax1 + b, ax2 + b,..., axn + b is      aX+b      .

Question 4:

Fill In the Blanks 

If mode and median of the certain data are 3 and 3 respectively, then mean is ___________.

Answer 4:


Mode of the data = 3

Median of the data = 3

Now,

Mode = 3Median − 2Mean

⇒ 3 = 3 × 3 − 2Mean

⇒ 2Mean = 9 − 3 = 6

⇒ Mean = 62 = 3

Thus, the mean of the data is 3.

If mode and median of the certain data are 3 and 3 respectively, then mean is ___3___.

Question 5:

Fill In the Blanks 

If Mode - Mean = 36, then Median - Mean = ___________.

Answer 5:

It is given that,

Mode − Mean = 36

We know

Mode = 3Median − 2Mean

⇒ Mode − Mean = 3Median − 3Mean

⇒ Mode − Mean = 3(Median − Mean)

⇒ 3(Median − Mean) = 36           (Given)

⇒ Median − Mean = 363 = 12

Thus, the value of Median − Mean is 12.

If Mode − Mean = 36, then Median − Mean = ____12____.

Question 6:

Fill In the Blanks 

The mean and mode of a data are 24 and 12 respectively, then median of the data is ___________.

Answer 6:


Mean of the data = 24

Mode of the data = 12

We know

Mode = 3Median − 2Mean

⇒ 12 = 3Median − 2 × 24

⇒ 3Median = 48 + 12 = 60

⇒ Median = 603 = 20

Thus, the median of data is 20.

The mean and mode of a data are 24 and 12 respectively, then median of the data is ___20___.

Question 7:

Fill In the Blanks 

If the mean of 12 observations is 15. If two observations 20 and 25 are removed, then the mean of remaining observations is _________.

Answer 7:


We know

Mean = Sum of observationsNumber of observations

⇒ Sum of observations = Mean × Number of observations

Mean of 12 observations = 15      (Given)

∴ Sum of 12 observations = 15 × 12 = 180

If two observations 20 and 25 are removed, then

Sum of remaining 10 observations = Sum of 12 observations − 20 − 25

⇒ Sum of remaining 10 observations = 180 − 20 − 25 = 135

∴ Mean of remaining 10 observations = Sum of remaining 10 observation10=13510=13.5

Thus, the mean of remaining observations is 13.5.

If the mean of 12 observations is 15. If two observations 20 and 25 are removed, then the mean of remaining observations is ___13.5___.

Question 8:

Fill In the Blanks 

If Mean : Median = 2 : 3, then Mode : Mean = ________ 

Answer 8:


Mean : Median = 2 : 3          (Given)

Let mean = 2x and median = 3x, where x is constant           .....(1)

We know

Mode = 3Median − 2Mean

⇒ Mode = 3 × 3x − 2 × 2x            [From (1)]

⇒ Mode = 9x − 4x = 5x

∴ Mode : Mean = 5x : 2x = 5 : 2

Thus, the ratio of mode to mean is 5 : 2.

If Mean : Median = 2 : 3, then Mode : Mean = ___5 : 2___.

Question 9:

Fill In the Blanks 

The mean of a set of 12 observations is 10 and another set of 8 observations is 12. The mean of all 20 observations is _______.

Answer 9:

We know

Mean = Sum of observationsNumber of observations

⇒ Sum of observations = Mean × Number of observations

Mean of a set of 12 observations = 10                 (Given)

∴ Sum of a set of 12 observations = 10 × 12 = 120                   .....(1)

Mean of another set of 8 observations = 12         (Given)

∴ Sum of another set of 8 observations = 12 × 8 = 96              .....(2)

Now,

Sum of all 20 observations 

= Sum of a set of 12 observations + Sum of another set of 8 observations

= 120 + 96                    [From (1) and (2)]

= 216

∴ Mean of all 20 observations = Sum of all 20 observations20=21620 = 10.8

Thus, the mean of all 20 observations is 10.8.

The mean of a set of 12 observations is 10 and another set of 8 observations is 12. The mean of all 20 observations is ___10.8___.

Question 10:

Fill In the Blanks 

The mean of the following distribution 

x :    3    5    7    4
y :    2    a    5    b

is 5. Then the value of b is ____________.

Answer 10:


We know

Mean = fixii=1       n       fii=1  n  

The mean of given distribution is 5.

5=3×2+5×a+7×5+4×b2+a+5+b

5=6+5a+35+4b7+a+b

57+a+b=41+5a+4b

35+5a+5b=41+5a+4b

5b-4b=41-35

b=6

Thus, the value of b is 6.


The mean of the following distribution 

x :    3    5    7    4
y :    2    a    5    b

is 5. Then the value of b is ____6____.

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