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**(1) Arithmetic mean (AM), Geometric mean(GM), Harmonic mean(HM), Median and Mode are various measures of central tendency.
**

= 998

Hence, mean of the given numbers is 998

**(ii) Geometric mean:
**If we have a series of n positive values such as are repeated times respectively then geometric mean will become:

G.M of X= = (For Grouped Data)

Where n=

G.M of X= =

= = 9.36

**(iii) Harmonic mean: **Harmonic mean is quotient of “number of the given values” and “sum of the reciprocals of the given values”.

Harmonic mean in mathematical terms is defined as follows:**Example: **Calculate the harmonic mean of the numbers: 13.5, 14.5, 14.8, 15.2 and 16.1

**Solution: **The harmonic mean is calculated as below:H.M of X = =

H.M of X = = = 14.63

**(iv) Median:
**

29, 34, 37, 41, 45, 54, 63, 70, 70, 83

Here, the number of observation n = 10 (even)

Now, medians is-

49.5

Hence the value of median is 49.5.

**(v) Mode:
**

Marks: 4,6,5,7,9,8,10,4,7,6,5,9,8,7,7.

Hence, the mode value is 7

**(2) (i) If are n values of a variable X, then the arithmetic mean of these values is given by **** ****or,
**

= 998

Hence, mean of the given numbers is 998

**(ii) If a variate X take values with corresponding frequencies respectively, then the arithmetic mean of these values is given by ****or, , where
**

Hence, value of mean is 7.025

**(3) If is the mean of n observations , then
**

9.6,5.2,3.5,1.5,1.6,2.4,2.6,8.4,10.3,10.9

Verify that

Taking LHS,

(9.6+ 5.2+3.5+1.5+1.6+2.4+2.6+8.4+10.3+10.9) - 10 x

56 - 10 x 5.6

56 - 56

0 = RHS Hence Proved

**(ii) Prove that the mean of the observations is
**Given, …..(i)

The observation are .

Mean

From Equation (1); we get

Mean

So that given statement is true.

**(iii) Prove that the mean of the observations is
**

But mean (given)

……..(1)

The observation are

Mean

Thus, the given statement is true.

**(iv) Prove that the mean of the observations is
**

Let be the mean of . Then,

**(4) Media of a distribution is the value of the variable which divides the distribution into two equal parts.
**

Coordinates of P

Coordinates of Q

Coordinates of R

**(5) lf are n values of a variable arranged in ascending or descending order, then**

** Median = value of observation, if n is odd
Median = (value of observation + value of observation)/2, if n is even**

** For Example:**(i) Find the median of this data : 83,37,70,29,45,63,41,70,34,54

29, 34, 37, 41, 45, 54, 63, 70, 70, 83

Here, the number of observation n = 10 (even)

Now, medians is-

Hence the value of median is 49.5.

(ii) Find the median of this data : 15,6,16,8,22,21,9,18,,25

Here, the number of observation n=9 (odd)

Now, median =

=

=

=

= 16

Hence, value of median is 16.

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Message *really cool n I have learned many things.