Measures of Central Tendency

To make all the study of data useful, we need to use measures of central tendencies. Some of the tendencies are

1. Mean

The mean is the average of the number of observations. It is calculated by dividing the sum of the values of the observations by the total number of observations.

It is represented by x bar or  \bar x.

The mean  \bar x of n values  x_1,x_2,x_3,......x_n is given by

\bar x=\dfrac{x_1,x_2,x_3....x_n}{n}

Mean of Grouped Data (Without Class Interval)

If the data is organized in such a way that the frequency is given but there is no class interval then we can calculate the mean by

\bar{\text{x}}=\dfrac{f_1x_1+f_2x_2+.....+f_nx_n}{f_1+f_2+.....+f_n}

=\dfrac{\sum\limits_{i=1}^{n} f_{1} x_{1}}{\sum\limits_{i=1}^{n} \text{f}_{1}}

where, x_1,x_2,x_3....x_n  are the observations

f_1,f_2,f_3.....f_n are the respective frequencies of the given observations.

Example

Here, x_1,x_2,x_3,x_4 , and x5 are 20, 40, 60, 80,100 respectively.

and  f_1,f_2,f_3,f_4,f_5 are 40, 60, 30, 50, 20 respectively.

\bar{x}=\dfrac{f_{1} x_{1}+f_{2} x_{2}+\cdots+f_{n} x_{n}}{f_{1}+f_{2}+\cdots+f_{n}}


\bar{x}=\dfrac{11000}{200}=55

2. Median

The median is the middle value of the given number of the observation which divides into exactly two parts.

For median of ungrouped data, we arrange it in ascending order and then calculated as follows

  • If the number of the observations is odd then the median will be \left(\dfrac{n+1}{2}\right)^{th}

As in the above figure the no. of observations is 7 i.e. odd, so the median will be \left(\dfrac{n+1}{2}\right)^{th} term.

  • If the number of observations is even then the median is the average of  \dfrac{n}{2} and \left(\dfrac{n}{2}\right)+1 term.

3. Mode

The mode is the value of the observation which shows the number that occurs frequently in data i.e. the number of observations which has the maximum frequency is known as the Mode.

Example

Find the Mode of the following data:

15, 20, 22, 25, 30, 20,15, 20,12, 20

Solution

Here the number 20 appears the maximum number of times so

Mode = 20.

The empirical relation between the three measures of central tendency is

3 Median = Mode + 2 Mean

Question 1: The number of family members in 10 flats of society are

2, 4, 3, 3, 1, 0, 2, 4, 1, 5.

Find the mean number of family members per flat.

Solution:

Number of family members in 10 flats – 2, 4, 3, 3, 1, 0, 2, 4, 1, 5.

So, we get,

\mathrm{Mean=\dfrac{sum\ of\ observation}{total\ no.\ of\ observations}}

\mathrm{Mean=\dfrac{(2 + 4+ 3 + 3 + 1 + 0 + 2 + 4 + 1 + 5)}{10}}

\mathrm{Mean=\dfrac{25}{10}=2.5}

Question 2  .The following is the list of number of coupons issued in a school canteen during a week:

105, 216, 322, 167, 273, 405 and 346.

Find the average no. of coupons issued per day.

Solution:

Number of coupons issued in a week: 105, 216, 322, 167, 273, 405 and 346.

So, we get,

\mathrm{Mean=\dfrac{sum\ of\ observation}{total\ no.\ of\ observations}}

\mathrm{Mean=\dfrac{(106+ 215+ 323+166+273+405+346)}{7}=\dfrac{1834}{7}}

\mathrm{Mean=262}

Question 3.If the mean of six observations y, y + 1, y + 4, y + 6, y + 8, y + 5 is 13, find the value of y.

Solution:

\mathrm{Mean=\dfrac{sum\ of\ observation}{total\ no.\ of\ observations}}

 13=\dfrac{(y + y + 1+ y + 4+ y + 6+ y + 8+ y + 5)}{6}

13=\dfrac{(6y + 24)}{6}

(13 \times 6) = 6y +24

 (13 \times 6) - 24 = 6y

(13 \times 6) - 6 \times 4 = 6y

6(13 - 4) = 6y

y = 9

Question 4. The mean weight of a class of 34 students is 46.5 kg. If the weight of the new boy is included, the mean is rises by 500 g. Find the weight of the new boy.

Solution:

The mean weight of 34 students = 46.5

Sum of the weight of 34 students =(46.5\times34)=1581

Change or increase in the mean weight when the weight of a new boy is added  =0.5

So, the new mean = (46.5 +0.5) = 47

So, let the weight of the new boy be y.

So, \dfrac{\text{(sum of weight of 34 students + weight of new boy)}}{35}=47

\dfrac{(1581+ y)}{35}=47

1581 + y = 1645

y = 1645 - 1581 = 64

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