# Measures of Central Tendency - Class 9 : Notes

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

**(i) Arithmetic mean:**

**Find the mean of 994, 996, 998, 1002 and 1000.**

*For Example:***Given,**No. of values n = 5

**We know, mean**

**() =**

**So, mean**

**=**

=

= 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=

**Example:**Find the Geometric mean of the values 10, 5, 15, 8, 12

**Solution:**Here , , , , and

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:
**

**Find the median of this data: 83,37,70,29,45,63,41,70,34,54**

*For Example:***Solution:**Arrange the data in ascending order, we get-

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:
**

**Find out the mode of the following marks obtained by 15 students in class:**

*For Example:*Marks: 4,6,5,7,9,8,10,4,7,6,5,9,8,7,7.

**Solution:**Arrange the data in the form of a frequency table-Since. the value of 7 occurs maximum number of times i.e 4.

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,
**

**Find the mean of 994, 996, 998,1002 and 1000.**

*For Example:***Solution:**No. of values n = 5

**We know, mean**

**() =**

**So, mean**

**=**

=

= 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
**

**Calculate the mean for the following distribution**

*For Example:***Solution:**Calculation

**of the Arithmetic mean-Now, mean = = = 7.025**

Hence, value of mean is 7.025

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

**(i) The algebraic sum of the deviations about is 0, i.e.**

**Duration of sunshine (in hours) in Amritsar for first 10 days of August 1997 as reported by the Meteorological Department are given below:**

*For Example:*9.6,5.2,3.5,1.5,1.6,2.4,2.6,8.4,10.3,10.9

Verify that

**Solution:**We have to 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
**

**Mean of**

*For Example:*But mean (given)

……..(1)

The observation are

Mean

Thus, the given statement is true.

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

**Proof:**We have,

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.
**

**Find the coordinates of the points which divide the line segment A(-2, 2) & B(2, 8) into four equal parts.**

*For Example:***Solution:**From the figure, it can be observed that points P, Q, R are dividing the line segment in a ratio 1:3, 1:1, 3:1 respectively.

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

**Solution:Arrange the data in ascending order, we get-**

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

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

Now, medians is-

49.5

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

**Solution:**Arrange the data in the ascending order, we get-

**6, 8, 9, 15, 16, 18, 21, 22, 25**

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

Now, median =

=

=

=

= 16

Hence, value of median is 16.

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