Answer:
The value of missing frequency x = [tex]3[/tex] and missing frequency y = [tex]32[/tex].
Step-by-step explanation:
Given,
Marks: [tex]0-10[/tex] [tex]10-20[/tex] [tex]20-30[/tex] [tex]30-40[/tex] [tex]40-50[/tex] [tex]50-60[/tex]
N ( [tex]f_{i}[/tex] ): [tex]10[/tex] x [tex]25[/tex] [tex]20[/tex] y [tex]10[/tex]
N = The number of students
The median of the given data = [tex]36[/tex]
The sum of all the given frequencies, [tex]\sum f_{i}[/tex] = [tex]100[/tex]
To find:
- The value of the missing frequencies x and y
Now, if the median is [tex]36[/tex], then the median class is [tex]30-40[/tex].
Now, we have to find out the cumulative frequency:
Marks Number of students ( [tex]f_{i}[/tex] ) Cumulative frequency
[tex]0-10[/tex] [tex]10[/tex] [tex]10[/tex]
[tex]10-20[/tex] x [tex]10+x[/tex]
[tex]20-30[/tex] [tex]25[/tex] [tex]35+x[/tex]
[tex]30-40[/tex] [tex]20[/tex] [tex]55+x[/tex]
[tex]40-50[/tex] y [tex]55+x+y[/tex]
[tex]50-60[/tex] [tex]10[/tex] [tex]65+x+y[/tex]
Total [tex]100[/tex]
Now, the total number of students [tex]\sum f_{i}[/tex] = [tex]100[/tex]
Therefore,
- [tex]10+x+25+20+y+10[/tex] = [tex]100[/tex]
- [tex]65+x+y = 100[/tex]
- [tex]x+y =35[/tex] -------equation (1)
Now,
- Median = [tex]l+\frac{[\frac{N}{2}-c.f] }{f} \times h[/tex]
Here,
- l = The median class's lower limit = [tex]30[/tex]
- N = The total frequency of the given data = [tex]100[/tex]
- c.f = The cumulative frequency of the preceding class = [tex]35+x[/tex]
- f = The frequency of the median class = [tex]20[/tex]
- h = The class size/height = [tex]10[/tex]
Thus,
- Median = [tex]l+\frac{[\frac{N}{2}-c.f] }{f} \times h[/tex]
- [tex]36[/tex] = [tex]30+\frac{[\frac{100}{2}-(35+x)] }{20} \times 10[/tex]
- [tex]36[/tex] = [tex]30+\frac{[50-35-x)] }{2}[/tex]
- [tex]36[/tex] = [tex]30+\frac{[15-x)] }{2}[/tex]
- [tex]36[/tex] = [tex]\frac{60+15-x}{2}[/tex]
- [tex]75-x = 72[/tex]
- x = [tex]3[/tex]
Now, after putting the value of x in equation (1), we get:
- [tex]3+y =35[/tex]
- y = [tex]32[/tex]
Hence, the value of missing frequency x = [tex]3[/tex] and missing frequency y = [tex]32[/tex].