Answer to Question #301258 in Statistics and Probability for Tarurendra Kushwah

Question

The following data is given Age (in years): Below 10, below20, below30, below40,below50, below60, below70, below 80 and No. of persons: 2,5,9,12,14,15,15.5,15.6

Find mean , median and mode.

Answer 1

Age (years) No. of persons

Below 10 2

Below 20 5

Below 30 9

Below 40 12

Below 50 14

Below 60 15

Below 70 15.5

Below 8 15.6

Here frequencies are in cumulative frequencies so that,

Age (years) Actual class f c.f class mark(x) fx

Below 10 0-10 2 2 5 10

Below 20 10-20 3 5 15 45

Below 30 20-30 4 9 25 100

Below 40 30-40 3 12 35 105

Below 50 40-50 2 14 45 90

Below 60 50-60 1 15 55 55

Below 70 60-70 0.5 15.5 65 32.5

Below 80 70-80 0.1 15.6 75 7.5

Here, $n=15.6$

a) Mean

The value of the mean is given as,

$\bar x={\sum fx\over \sum f}={445\over15.6}=28.525641$

b)Median

To determine the value of the median, we first determine the position of the median class as follows.

The position of the median class is ${n\over2}^{th}={15.6\over2}^{th}=7.8^{th}$ position. Therefore, the median class is 20-30

The value of the median can be computed using the formula below.

$median=l+{({n\over2}-cf)\times c\over f_m}$ where,

$l$ is the lower class boundary of the median class

$cf$ is the cumulative frequency of the class preceding the median class

$c$ is the width of the median class

$f_m$ is the frequency of the median class.

So,

$Median=20+{(7.8-5)\times 10\over4}=20+7=27$

The value of the median is 27

c) Mode

To determine the modal class, we look for the class with the highest frequency. From the given data, the class with the highest frequency is 20-30. This is the modal class.

The formula for finding the median is given as,

$mode=l+{(f_m-f_1)\times c\over 2f_m-f_1-f_2}$ where,

$l$ is the lower class boundary of the modal class

$f_m$ is the frequency of the modal class

$c$ is the width of the modal class

$f_1$ is the frequency of the class preceding the modal class

$f_2$ is the frequency of the class succeeding the modal class

So,

$mode=20+{(4-3)\times 10\over8-3-3 }=20+5=25$

Therefore, the mode is 25