If the median of the given data is 32, determine the values of \( x \) and \( y \) when the total frequency is 100. <style> table{width:100%;margin-bottom:5px; margin-top:5px;} table, th, td { border: 1px solid black; border-collapse: collapse; text-align: center;} </style> <table> <tr><td>Class Interval</td><td>0-10</td><td>10-20</td><td>20-30</td><td>30-40</td><td>40-50</td><td>50-60</td></tr> <tr><td>Frequency</td><td>10</td><td>x</td><td>25</td><td>30</td><td>y</td><td>10</td></tr> </table>

Q.If the median of the given data is 32, determine the values of \( x \) and \( y \) when the total frequency is 100.

Class Interval	0-10	10-20	20-30	30-40	40-50	50-60
Frequency	10	x	25	30	y	10

The frequency distribution table of the given class:

Class Interval	Frequency	Cumulative Frequency (Less than type)
0-10	10	10
10-20	\(x\)	10+\(x\)
20-30	25	35+\(x\)
30-40	30	65+\(x\)
40-50	\(y\)	65+\(x+y\)
50-60	10	75+\(x+y=n\)

Here, \( n = 100 \) (Given).
According to the condition, \( 75 + x + y = 100 \)
Or, \( x + y = 25 \) ---- (i)

Again, since the median \( = 32 \), The median class is (30-40).
∴ Median formula: \[ = l + \left[\cfrac{\cfrac{n}{2} - cf}{f}\right] \times h \] [Here, \( l = 30 \), \( n = 100 \), \( cf = 35 + x \), \( f = 30 \), \( h = 10 \)]
\[ = 30 + \left[\cfrac{50 - (35 + x)}{30}\right] \times 10 \] \[ = 30 + \cfrac{15 - x}{30} \times 10 \] \[ = 30 + \cfrac{15 - x}{3} \]
According to the condition, \[ 30 + \cfrac{15 - x}{3} = 32 \] Or, \[ \cfrac{15 - x}{3} = 2 \] Or, \[ 15 - x = 6 \] Or, \[ -x = -9 \] Or, \[ x = 9 \]
Substituting \( x \) in equation (i), \[ 9 + y = 25 \] Or, \[ y = 16 \]
∴ The required values are \( x = 9 \), \( y = 16 \).

30–40	40–50	50–60
15	y	5

Class Interval	0-20	20-40	40-60	60-80	80-100
Frequency	7	11	K	9	13

Marks \((x_i)\)	0–4	5–9	10–14	15–19	20–24	25–29	30–34	35–39
Frequency \((f_i)\)	4	5	7	8	7	5	6	3

Class Interval	0–20	20–40	40–60	60–80	80–100
Frequency	7	11	k	9	13

Class Interval	0–10	10–20	20–30	30–40	40–50	50–60
Frequency	7	10	23	50	6	4

Class Interval (Marks)	0-10	10-20	20-30	30-40	40-50
Number of Students	15	20	35	p	10

Class Boundaries	Less than 10	Less than 20	Less than 30	Less than 40	Less than 50	Less than 60	Less than 70	Less than 80
Frequency	4	16	40	76	96	112	120	125

Class Interval	0-5	5-10	10-15	15-20	20-25	25-30	30-35
Frequency	5	12	18	28	17	12	8

Weight (kg)	Less than 38	Less than 40	Less than 42
Number of Students	0	4	6

Less than 44	Less than 46	Less than 48	Less than 50	Less than 52
9	12	28	32	35

Marks	0–5	5–10	10–15	15–20	20–25	25–30	30–35	35–40
Number of Students	2	6	10	16	22	11	8	5

Class Interval	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60
Frequency	7	5	6	12	8	2

Marks Obtained	0–10	10–30	30–60	60–70	70–90
Number of Students	15	25	30	4	10

\( x_i \)	10	15	\( a \)	25	35
\( f_i \)	3	10	25	7	5

Variable (\(x_i\))	10	15	\(a\)	25	35
Frequency (\(f_i\))	3	10	25	7	5

Variable	5	10	15	20	25
Frequency	6	p	6	10	5

Size (\(x\))	2	3	4	5	6	7	8	9
Frequency (\(f_i\))	3	4	5	3	5	4	3	2