Thus, the value of p is 8.
Solution 9:
Mean score = Σfixi/Σfi
16.6 = (1228+24p)/100
1228 + 24 p = 16.6 x 100
24 p = 1660 – 1228
p = 432/24 = 18
Thus, the value of p is 18.
Question :10. Find the missing value of p for the following distribution whose mean is 12.58
Solution 10:
Mean score = Σfixi/Σfi
12.58 = (524+7p)/50
524 + 7 p = 12.58 x 50
7 p = 629 – 524
p = 105/7 = 15
Thus, the value of p is 15.
Question :11. Find the missing frequency (p) for the following distribution whose mean is 7.68
Solution 11:
Mean score = Σfixi/Σfi
7.68 = 303 + 9p/41 + p
303 + 9 p = 314.88 + 7.68p
1.32 p = 314.88 – 303
p = 11.88/1.32 = 9
Thus, the value of p is 9.
Question :12. Find the value of p, if the mean of the following distribution is 20
Solution 12:
Mean score = Σfixi/Σfi
20 = ([(295 + (20 + p) 5p)])/(15 + 5p)
295 + 100 p + 5p2 = 300 + 100p
5p2 = 300 – 295
5p2= 5
p2 = 1
p = 1
thus, the value of P is 1.
Exercise 23.3
Find the median of the following data (1 – 8)
Question :1. 83, 37, 70, 29, 45, 63, 41, 70, 34, 54
Solution 1:
Firstly, arrange the given data into ascending order,
29, 34, 37, 41, 45, 54, 63, 70, 70, 83
Now, the number of observations is 10. So, n = 10 (even)
Thus, median = (n/2)th term + (((n+1))/2)th term
Median = (value of 5th term + value of 6th term)/2
= (45+54)/2 = 49.5
Thus, median for given data is 49.5
Question :2. 133, 73, 89, 108, 94,104, 94, 85, 100, 120
Solution 2:
Firstly, arrange the given data into ascending order,
73, 85, 89, 94, 100, 104, 108, 120, 133
Now, the number of observations, n = 10 (even)
Thus, median = (n/2)th term + (((n+1))/2)th term
Median = ((value of 5th term + value of 6th term))/2
= ((94+100))/2 = 97
So, the median for given data is 97
Question :3. 31, 38, 27, 28, 36, 25, 35, 40
Solution 3:
Firstly, arrange the given data into ascending order
25, 27, 28, 31, 35, 36, 38, 40
And number of observations, n = 8 (even)
Thus, median = (n/2)th term + (((n+1))/2)th term
Median = ((value of 4th term + value of 5th term))/2
= ((31+35))/2 = 33
So, the median for given data is 33
Question :4. 15, 6, 16, 8, 22, 21, 9, 18, 25
Solution 4:
Firstly, arrange the given data into ascending order
6, 8, 9, 15, 16, 18, 21, 22, 25
And number of observations, n = 9 (odd)
Thus, median = (((n+1))/2)th term
Median = value of 5th term
= 16
So, the median for given number is 16.
Question :5. 41, 43,127, 99, 71, 92, 71, 58, 57
Solution 5:
Firstly, arrange the given data into ascending order
41, 43, 57, 58, 71, 71, 92, 99, 127
and number of observations, n = 9 (odd)
Thus, median = (((n+1))/2)th term
Median = value of 5th term
= 71
So, the median for given number is 71.
Question :6. 25, 34, 31, 23, 22, 26, 35, 29, 20, 32
Solution 6:
Firstly, arrange the given data into ascending order,
20, 22, 23, 25, 26, 29, 31, 32, 34, 35
And number of observations, n = 10 (even)
Thus, median = (n/2)th term + (((n+1))/2)th term
Median = ((value of 5th term + value of 6th term))/2
= ((26+29))/2 = 27.5
So, the median for given data is 27.5
Question :7. 12, 17, 3, 14, 5, 8, 7, 15
Solution 7:
Firstly, arrange the given data into ascending order,
3, 5, 7, 8, 12, 14, 15, 17
and number of observations, n = 8 (even)
Thus, median = (n/2)th term + (((n+1))/2)th term
Median = ((value of 4th term + value of 5th term))/2
= ((8+12))/2 = 10
So, the median for given data is 10
Question :8. 92, 35, 67, 85, 72, 81, 56, 51, 42, 69
Solution 8:
Firstly, arrange the given data into ascending order,
35, 42, 51, 56, 67, 69, 72, 81, 85, 92
And number of observations, n = 10 (even)
Thus, median = (n/2)th term + (((n+1))/2)th term
Median = ((value of 5th term + value of 6th term))/2
= ((67+69))/2 = 68
So, the median for given data is 68
Question :9. Numbers 50, 42, 35, 2x +10, 2x – 8, 12, 11, 8, 6 are written in descending order and their median is 25, find x.
Solution 9:
The number of observations n is 9.
The median is the n+12th observation, which is the 5th observation, since n is odd.
We begin with the last number because the numbers are ordered in ascending order.
Median = 5th observation.
=> 25 = 2x – 8
=> 2x = 25 + 8
=> 2x = 33
=> x = 33/2
x = 16.5
Question :10. Find the median of the following observations: 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33. If 92 is replaced by 99 and 41 by 43 in the above data, find the new median?
Solution 10:
Arranging the given data in ascending order:
33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92
The number of observations (n) is 11 in this case (odd).
Since the number of observations is odd, therefore,
Thus, median = (((n+1))/2)th term
Median = value of the 5th term
= 58.
So, the median is 58.
If 92 is replaced by 99 and 41 by 43, then the new observations arranged in ascending order are:
33, 35, 43, 46, 55, 58, 64, 77, 87, 90, 99
New median = Value of the 6th term
= 58.
Question :11. Find the median of the following data: 41, 43, 127, 99, 61, 92, 71, 58, 57, If 58 is replaced by 85, what will be the new median?
Solution 11:
Arranging the given data in ascending order,
41, 43, 57, 58, 61, 71, 92, 99,127
The number of observations (n) is 9 in this case (odd).
Thus, median = (((n+1))/2)th term
Median = value of 5th term
So, the median = 61.
If 58 is replaced by 85, then the new observations arranged in ascending order are:
41, 43, 57, 61, 71, 85, 92, 99, 12
New median = Value of the 5th term = 71.
Question :12. The weights (in kg) of 15 students are: 31, 35, 27, 29, 32, 43, 37, 41, 34, 28, 36, 44, 45, 42, 30. Find the median. If the weight 44 kg is replaced by 46 kg and 27 kg by 25 kg, find the new median.
Solution 12:
Arranging the given data in ascending order, we have:
27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 44, 45
The number of observations (n) is 15 in this case (odd).
The number of observations is odd,
Thus, median = (((n+1))/2)th term
Median = value of 8th term
= 35 kg.
So, the median is 35 kg.
If 44 kg is replaced by 46 kg and 27 kg by 25 kg, then the new observations arranged in ascending order are:
25, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 45, 46
∴ New median = Value of the 8th observation = 35 kg.