Question

Find the mean median mode from the following frequency distribution:

Number 

  8


  9

10

11

12

13

14

15

16

17

Frequency

  3


  8

12

15

17

12

13

5

  8

 ​

Answer 1

Frequency Distribution Table of the 9 observations is given below:

$\begin{tabular}{|c|c|c|}\cline{1-3}&&\\ \sf{x_i} & \sf{f_i} & \sf{f_ix_i} \\&&\\\cline{1-3} \sf{8} & \sf{3} & \sf{24} \\\cline{1-3} \sf{9} & \sf{8} & \sf{72} \\\cline{1-3} \sf{10} & \sf{12} & \sf{120} \\\cline{1-3} \sf{11} & \sf{15} & \sf{165} \\\cline{1-3} \sf{12} & \sf{17} & \sf{204} \\\cline{1-3} \sf{13} & \sf{12} & \sf{156} \\\cline{1-3} \sf{14} & \sf{13} & \sf{182} \\\cline{1-3} \sf{15} & \sf{5} & \sf{75} \\\cline{1-3} \sf{16} & \sf{8} & \sf{128} \\\cline{1-3}\end{tabular}$

Then, mean of the given data,

$\longrightarrow\sf{\bar x=\dfrac{\displaystyle\sum_{i=1}^9f_ix_i}{\displaystyle\sum_{i=1}^9f_i}}$

$\displaystyle\longrightarrow\sf{\bar x=\dfrac{24+72+120+165+204+156+182+75+128}{3+8+12+15+17+12+13+5+8}}$

$\displaystyle\longrightarrow\sf{\bar x=\dfrac{1126}{93}}$

$\displaystyle\longrightarrow\sf{\underline{\underline{\bar x=12.11}}}$

Now the cumulative frequency distribution table is given below.

$\begin{tabular}{|c|c|}\cline{1-2}&\\ \sf{x_i} & \sf{f_i} &&\\\cline{1-2} \sf{\leq8} & \sf{3} &\cline{1-2} \sf{\leq9} & \sf{11} &\cline{1-2} \sf{\leq10} & \sf{23} &\cline{1-2} \sf{\leq11} & \sf{38} &\cline{1-2} \sf{\leq12} & \sf{55} &\cline{1-2} \sf{\leq13} & \sf{67} &\cline{1-2} \sf{\leq14} & \sf{80} &\cline{1-2} \sf{\leq15} & \sf{85} &\cline{1-2} \sf{\leq16} & \sf{93} &\cline{1-2}\end{tabular}$

Since $\displaystyle\sf{\sum_{i=1}^9f_i=93,}$ the median is $\displaystyle\sf{\left(\dfrac{93+1}{2}\right)^{th}=47^{th}}$ term, which is among $\displaystyle\sf{39^{th}}$ to $\displaystyle\sf{55^{th}}$ observations whose number is 12.

Hence the median is,

$\displaystyle\longrightarrow\sf{\underline{\underline{m=12}}}$

The frequency is the greatest for the number 12, i.e., 17. Hence the mode is,

$\displaystyle\longrightarrow\sf{\underline{\underline{M=12}}}$

Answer 2

The mean, medium, mode from the following frequency distribution:

Xi= 8,9,10,11,12,13,14,15,16,17

Fi= 3,8,12,15,17,12,13,5,8

FiXi=24,72,120,165,204,156,182,75,128

Stepwise explanation is given below:

- MEDIAN :-

To find the Median place the numbers in value order and finds the middle number.

                   10,11,12,13,14,15      

- Since, there is an even number of data set, we compute the median by taking the mean of the two middlemost numbers or n=5th.

- Median of given data =

[(n) th term +( n+1) th term]/2

=(12+13)/2

=25/2

=12.5 or 12

The median of given data is 12

- MODE :-

To find the Mode, or modal value, place the numbers in value order then counts how many of each number. The number appears times, more often than the other values, is the Mode. So,

- The mode of a data set is the element that appears most frequently. Since each element in this data set is unique and only appears one time, there is no mode.

- MEAN :-

By using its formula

=(24+72+120+165+204+156+182+75+128)/(3+8+12+15+17+12+13+5+8)

=1126/93

=12.11

Or

- To find the Mean add up all the numbers, then divides total numbers:

=(8+9+10+11+12+13+14+15+16+17)/10

=125/10

=12.5