If the median of the following data is 32, find 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 following data is 32, find 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

Frequency distribution table:

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 + \(x + y = n\)

Given: \(n = 100\) From the condition: \(75 + x + y = 100\) ⇒ \(x + y = 25\) ----(i) Again, since the **median = 32**, the median class is (30–40) To find median: \[ \text{Median} = l + \left[\frac{n}{2} - cf}{f} \right] \times h \] Where, \(l = 30\), \(n = 100\), \(cf = 35 + x\), \(f = 30\), \(h = 10\) So, \[ \text{Median} = 30 + \left[\frac{50 - (35 + x)}{30} \right] \times 10 = 30 + \frac{15 - x}{3} \] Given median = 32 So, \[ 30 + \frac{15 - x}{3} = 32 \Rightarrow \frac{15 - x}{3} = 2 \Rightarrow 15 - x = 6 \Rightarrow x = 9 \] Substitute into (i): \(9 + y = 25\) ⇒ \(y = 16\) ∴ Required values are \(x = 9\), \(y = 16\) (Answer)

30-40	40-50	50-60
15	y	5

30-40	40-50	50-60
15	y	5

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

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

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

Marks	Less than 10	Less than 20	Less than 30	Less than 40	Less than 50	Less than 60	Less than 70	Less than 80
Number of students	4	16	40	76	96	112	120	125

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	1–5	6–10	11–15	16–20	21–25	26–30	31–35
Frequency	2	3	6	7	5	4	3

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

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

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	0–5	5–10	10–15	15–20	20–25	25–30	30–35
Frequency	5	12	18	28	17	12	8

Class	45–54	55–64	65–74	75–84	85–94	95–105
Frequency	8	13	19	32	12	6

Class Interval	5-14	15-24	25-34	34-44	45-54	55-64
Frequency	3	6	18	20	10	3

Variable	5	10	15	20	25
Frequency	6	\(k\)	6	10	5

Class Interval	4–8	8–12	12–16
Frequency	9	10	18

16–20	20–24	24–28	28–32
14	10	6	3

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

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