Question

The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures

Number of students per teacher	The number of states/U.T.
$15 - 20$	$3$
$20 - 25$	$8$
$25 - 30$	$9$
$30 - 35$	$10$
$35 - 40$	$3$
$40 - 45$	$0$
$45 - 50$	$0$
$50 - 55$	$2$

Answer 1

Hint: As for finding the mode of the grouped data we use formula $L + \dfrac{{{f_m} - {f_{m - 1}}}}{{({f_m} - {f_{m - 1}}) + ({f_m} - {f_{m + 1}})}} \times w$where L is the lower class boundary of the modal group ,${f_{m - 1}}$ is the frequency of the group before the modal group ,${f_m}$ is the frequency of the modal group , ${f_{m + 1}}$ is the frequency of the group after the modal group , w is the group width and for the mean of grouped data Mean = $\dfrac{{\sum {f \times } X}}{{\sum f }}$ where $X$ is the midpoint of group and f is frequency of that .

Complete step by step answer:
In the first case we have to find the mode of the given data ,
Mode is the number that appears most frequently in a data set . In the grouped data we use the formula for mode that is ,
Estimated mode = $L + \dfrac{{{f_m} - {f_{m - 1}}}}{{({f_m} - {f_{m - 1}}) + ({f_m} - {f_{m + 1}})}} \times w$
Modal group is the group which have maximum frequency .
where L is the lower class boundary of the modal group
${f_{m - 1}}$ is the frequency of the group before the modal group
${f_m}$ is the frequency of the modal group
${f_{m + 1}}$ is the frequency of the group after the modal group
w is the group width .

For the given question
Modal Group is $30 - 35$
L = $30$ (is the lower class boundary of the modal group)
${f_{m - 1}} = 9$ (is the frequency of the group before the modal group)
${f_m} = 10$ (is the frequency of the modal group)
${f_{m + 1}} = 3$ (is the frequency of the group after the modal group)
w = $5$ (is the group width)
Now putting these values in the given Estimated mode equation ,
Mode = $30 + \dfrac{{10 - 9}}{{(10 - 9) + (10 - 3)}} \times 5$
Mode = $30 + \dfrac{1}{{1 + 7}} \times 5$
Mode = $30 + \dfrac{5}{8}$
When we divide $5$ by $8$ we get $0.625$
Hence the mode = $30 + 0.625 = 30.625$
Now for the mean of the given data ,
first we have to find the midpoint of the group data as ,
Midpoint of $15 - 20$is $17.5$
Midpoint of $20 - 25$ is $22.5$
The midpoint of $25 - 30$ is $27.5$ similarly for other as,
After finding the Midpoint of the group, multiple by these to the frequency of the data or Number of states.

Number of students per teacher	A number of states/U.T.Frequency $f$	Midpoint $X$	Midpoint $ \times $ frequencyf $ \times $ X
$15 - 20$	$3$	$17.5$	$52.5$
$20 - 25$	$8$	$22.5$	$180$
$25 - 30$	$9$	$27.5$	$247.5$
$30 - 35$	$10$	$32.5$	$325$
$35 - 40$	$3$	$37.5$	$112.5$
$40 - 45$	$0$	$42.5$	$0$
$45 - 50$	$0$	$47.5$	$0$
$50 - 55$	$2$	$52.5$	$105$

Hence for the mean of the group data the formula is
Mean = $\dfrac{{\sum {f \times } X}}{{\sum f }}$
therefore $\sum {f \times } X = 52.5 + 180 + 247.5 + 325 + 112.5 + 0 + 0 + 105$ = $1022.5$
$\sum {f = 3 + 8 + 9 + 10 + 3 + 0 + 0 + 2 = 35} $
By putting the value in equation the ,
Mean = $\dfrac{{1022.5}}{{35}}$ on dividing we get
Mean $ = 29.21$

$\therefore $ The mode of the given data is 30.625 and the mean is $29.21$

Note:
We find the midpoint of the grouped data as we have to think like $17.5$ Number of students per teacher have $3$ number of states so from this the grouped data is converted into simple data or in another word we will say that
$17.5$ is three times, $22.5$ is eight times, $27.5$ is nine times, $32.5$ is ten times, $37.5$ is three times, $52.5$ two times now find the mean from simply.
For Median of the group data we use formula Median = $ = L + \dfrac{{\dfrac{n}{2} - B}}{G} \times w$ where:
$L$ is the lower class boundary of the group containing the median, $n$ is the total number of values, $B$ is the cumulative frequency of the groups before the median group, $G$ is the frequency of the median group, $w$ is the group width.