Page 1
1. Write the median class of the following distribution:
Class 0 10 10 20 20 30 30 40 40 50 50 60 60 70
Frequency 4 4 8 10 12 8 4
Sol:
To find median let us put the data in the table given below:
Class Frequency Cumulative frequency (cf)
0-10 4 4
10-20 4 8
20-30 8 16
30-40 10 26
40-50 12 38
50-60 8 46
60-70 4 50
Total N=
Now,
The cumulative frequency just greater than 25 is 26, and the corresponding class is 30-40.
Thus, the median class is 30-40.
2. What is the lower limit of the modal class of the following frequency distribution?
Age
(in years)
0 - 10 10- 20 20 -30 30 40 40 50 50 60
Number of
patients
16 13 6 11 27 18
Sol:
Here the maximum class frequency is 27, and the class corresponding to this frequency is
40-50 So the modal class is 40 -50.
Now,
Modal class = 40-50, lower limit (/) of modal class = 40.
Thus, lower limit (/) of modal class is 40
3. The monthly pocket money of 50 students of a class are given in the following distribution:
Monthly pocket
0 - 50 50 100 100 150 150 -200 200 250 250 - 300
Number of
Students
2 7 8 30 12 1
Find the modal class and give class mark of the modal class.
Sol:
Here the maximum class frequency is 30, and the class corresponding to the frequency is
150-200. So, the modal class is 150-200.
Page 2
1. Write the median class of the following distribution:
Class 0 10 10 20 20 30 30 40 40 50 50 60 60 70
Frequency 4 4 8 10 12 8 4
Sol:
To find median let us put the data in the table given below:
Class Frequency Cumulative frequency (cf)
0-10 4 4
10-20 4 8
20-30 8 16
30-40 10 26
40-50 12 38
50-60 8 46
60-70 4 50
Total N=
Now,
The cumulative frequency just greater than 25 is 26, and the corresponding class is 30-40.
Thus, the median class is 30-40.
2. What is the lower limit of the modal class of the following frequency distribution?
Age
(in years)
0 - 10 10- 20 20 -30 30 40 40 50 50 60
Number of
patients
16 13 6 11 27 18
Sol:
Here the maximum class frequency is 27, and the class corresponding to this frequency is
40-50 So the modal class is 40 -50.
Now,
Modal class = 40-50, lower limit (/) of modal class = 40.
Thus, lower limit (/) of modal class is 40
3. The monthly pocket money of 50 students of a class are given in the following distribution:
Monthly pocket
0 - 50 50 100 100 150 150 -200 200 250 250 - 300
Number of
Students
2 7 8 30 12 1
Find the modal class and give class mark of the modal class.
Sol:
Here the maximum class frequency is 30, and the class corresponding to the frequency is
150-200. So, the modal class is 150-200.
Also, class mark of the modal class is
4. A data has 25 observations arranged in a descending order. Which observation represents the
median?
Sol:
If the number of observations is odd, then the median is observation.
Thus, observation represents the median.
5. For a certain distribution, mode and median were found to be 1000 and 1250 respectively.
Find mean for this distribution using an empirical relation.
Sol:
There is an empirical relationship between the three measures of central tendency:
3median = mode + 2Mean
=
=
Thus, the mean is 1375.
6. In a class test, 50 students obtained marks as follows:
Marks
obtained
0 20 20 40 40 60 60 80 80 100
Number of
Students
4 6 25 10 5
Find the modal class and the median class.
Sol:
Here the maximum class frequency is 25, and the class corresponding to this frequency is
40-60.
So, the modal class is 40-60.
Now, to find the median class let us put the data in the table given below:
Marks Obtained Number of students Cumulative frequency (cf)
0-20 4 4
20-40 6 10
40-60 25 35
60-80 10 45
80-100 5 50
Total N=
Now,
The cumulative frequency just greater than 25 is 35, and the corresponding class is 40-60.
Thus, the median class is 40-60.
Page 3
1. Write the median class of the following distribution:
Class 0 10 10 20 20 30 30 40 40 50 50 60 60 70
Frequency 4 4 8 10 12 8 4
Sol:
To find median let us put the data in the table given below:
Class Frequency Cumulative frequency (cf)
0-10 4 4
10-20 4 8
20-30 8 16
30-40 10 26
40-50 12 38
50-60 8 46
60-70 4 50
Total N=
Now,
The cumulative frequency just greater than 25 is 26, and the corresponding class is 30-40.
Thus, the median class is 30-40.
2. What is the lower limit of the modal class of the following frequency distribution?
Age
(in years)
0 - 10 10- 20 20 -30 30 40 40 50 50 60
Number of
patients
16 13 6 11 27 18
Sol:
Here the maximum class frequency is 27, and the class corresponding to this frequency is
40-50 So the modal class is 40 -50.
Now,
Modal class = 40-50, lower limit (/) of modal class = 40.
Thus, lower limit (/) of modal class is 40
3. The monthly pocket money of 50 students of a class are given in the following distribution:
Monthly pocket
0 - 50 50 100 100 150 150 -200 200 250 250 - 300
Number of
Students
2 7 8 30 12 1
Find the modal class and give class mark of the modal class.
Sol:
Here the maximum class frequency is 30, and the class corresponding to the frequency is
150-200. So, the modal class is 150-200.
Also, class mark of the modal class is
4. A data has 25 observations arranged in a descending order. Which observation represents the
median?
Sol:
If the number of observations is odd, then the median is observation.
Thus, observation represents the median.
5. For a certain distribution, mode and median were found to be 1000 and 1250 respectively.
Find mean for this distribution using an empirical relation.
Sol:
There is an empirical relationship between the three measures of central tendency:
3median = mode + 2Mean
=
=
Thus, the mean is 1375.
6. In a class test, 50 students obtained marks as follows:
Marks
obtained
0 20 20 40 40 60 60 80 80 100
Number of
Students
4 6 25 10 5
Find the modal class and the median class.
Sol:
Here the maximum class frequency is 25, and the class corresponding to this frequency is
40-60.
So, the modal class is 40-60.
Now, to find the median class let us put the data in the table given below:
Marks Obtained Number of students Cumulative frequency (cf)
0-20 4 4
20-40 6 10
40-60 25 35
60-80 10 45
80-100 5 50
Total N=
Now,
The cumulative frequency just greater than 25 is 35, and the corresponding class is 40-60.
Thus, the median class is 40-60.
7. Find the class marks of classes 10 -25 and 35 55.
Sol:
Class mark =
class mark of 10-25 =
= 17.5
And class mark of 35-55=
= 45
8. While calculating the mean of a given data by the assumed-mean method, the following
values were obtained.
A=25, 110
i i
f d , 50
i
f
Find the mean.
Sol:
According to assumed-mean method,
Thus, mean is 27.2.
9. The distribution X and Y with total number of observations 36 and 64, and mean 4 and 3
respectively are combined. What is the mean of the resulting distribution X + Y?
Sol:
According to the question,
And total number of observations = 36 + 64 = 100
Thus, mean = = 3.36.
10. In a frequency distribution table with 12 classes, the class-width is 2.5 and the lowest class
boundary is 8.1, then what is the upper class boundary of the highest class?
Sol:
Upper class boundary = Lowest class boundary + width × number of classes
=
=
=
Thus, upper class boundary of the highest class is 38.1.
Page 4
1. Write the median class of the following distribution:
Class 0 10 10 20 20 30 30 40 40 50 50 60 60 70
Frequency 4 4 8 10 12 8 4
Sol:
To find median let us put the data in the table given below:
Class Frequency Cumulative frequency (cf)
0-10 4 4
10-20 4 8
20-30 8 16
30-40 10 26
40-50 12 38
50-60 8 46
60-70 4 50
Total N=
Now,
The cumulative frequency just greater than 25 is 26, and the corresponding class is 30-40.
Thus, the median class is 30-40.
2. What is the lower limit of the modal class of the following frequency distribution?
Age
(in years)
0 - 10 10- 20 20 -30 30 40 40 50 50 60
Number of
patients
16 13 6 11 27 18
Sol:
Here the maximum class frequency is 27, and the class corresponding to this frequency is
40-50 So the modal class is 40 -50.
Now,
Modal class = 40-50, lower limit (/) of modal class = 40.
Thus, lower limit (/) of modal class is 40
3. The monthly pocket money of 50 students of a class are given in the following distribution:
Monthly pocket
0 - 50 50 100 100 150 150 -200 200 250 250 - 300
Number of
Students
2 7 8 30 12 1
Find the modal class and give class mark of the modal class.
Sol:
Here the maximum class frequency is 30, and the class corresponding to the frequency is
150-200. So, the modal class is 150-200.
Also, class mark of the modal class is
4. A data has 25 observations arranged in a descending order. Which observation represents the
median?
Sol:
If the number of observations is odd, then the median is observation.
Thus, observation represents the median.
5. For a certain distribution, mode and median were found to be 1000 and 1250 respectively.
Find mean for this distribution using an empirical relation.
Sol:
There is an empirical relationship between the three measures of central tendency:
3median = mode + 2Mean
=
=
Thus, the mean is 1375.
6. In a class test, 50 students obtained marks as follows:
Marks
obtained
0 20 20 40 40 60 60 80 80 100
Number of
Students
4 6 25 10 5
Find the modal class and the median class.
Sol:
Here the maximum class frequency is 25, and the class corresponding to this frequency is
40-60.
So, the modal class is 40-60.
Now, to find the median class let us put the data in the table given below:
Marks Obtained Number of students Cumulative frequency (cf)
0-20 4 4
20-40 6 10
40-60 25 35
60-80 10 45
80-100 5 50
Total N=
Now,
The cumulative frequency just greater than 25 is 35, and the corresponding class is 40-60.
Thus, the median class is 40-60.
7. Find the class marks of classes 10 -25 and 35 55.
Sol:
Class mark =
class mark of 10-25 =
= 17.5
And class mark of 35-55=
= 45
8. While calculating the mean of a given data by the assumed-mean method, the following
values were obtained.
A=25, 110
i i
f d , 50
i
f
Find the mean.
Sol:
According to assumed-mean method,
Thus, mean is 27.2.
9. The distribution X and Y with total number of observations 36 and 64, and mean 4 and 3
respectively are combined. What is the mean of the resulting distribution X + Y?
Sol:
According to the question,
And total number of observations = 36 + 64 = 100
Thus, mean = = 3.36.
10. In a frequency distribution table with 12 classes, the class-width is 2.5 and the lowest class
boundary is 8.1, then what is the upper class boundary of the highest class?
Sol:
Upper class boundary = Lowest class boundary + width × number of classes
=
=
=
Thus, upper class boundary of the highest class is 38.1.
11. The observation 29, 32, 48, 50, x, x+2, 72, 78, 84, 95 are arranged in ascending order. What
is the value of x if the median of the data is 63?
Sol:
If number of observations is even, then the median will be the average of th and the
th observations.
In the given case,
Thus,
Thus, the value of x is 62.
12. The median of 19 observations is 30. Two more observation are made and the values of these
are 8 and 32. Find the median of the 21 observations taken together.
Hint Since 8 is less than 30 and 32 is more than 30, so the value of median (middle value)
remains unchanged.
Sol:
Since, 8 is less than 30 and 32 is more than 30, so the middle value remains unchanged
Thus, the median of 21 observations taken together is 30.
13. If the median of and , where x > 0, is 8, find the value of x.
Hint Arranging the observations in ascending order, we have Median= .
Sol:
Arranging the observations in ascending order, we have
Thus, the value of x is 24.
14. What is the cumulative frequency of the modal class of the following distribution?
Class 3 6 6 9 9 12 12 15 15 18 18 21 21 24
Frequency 7 13 10 23 54 21 16
Sol:
Here the maximum class frequency is 23, and the class corresponding to this frequency is
12-15.
Page 5
1. Write the median class of the following distribution:
Class 0 10 10 20 20 30 30 40 40 50 50 60 60 70
Frequency 4 4 8 10 12 8 4
Sol:
To find median let us put the data in the table given below:
Class Frequency Cumulative frequency (cf)
0-10 4 4
10-20 4 8
20-30 8 16
30-40 10 26
40-50 12 38
50-60 8 46
60-70 4 50
Total N=
Now,
The cumulative frequency just greater than 25 is 26, and the corresponding class is 30-40.
Thus, the median class is 30-40.
2. What is the lower limit of the modal class of the following frequency distribution?
Age
(in years)
0 - 10 10- 20 20 -30 30 40 40 50 50 60
Number of
patients
16 13 6 11 27 18
Sol:
Here the maximum class frequency is 27, and the class corresponding to this frequency is
40-50 So the modal class is 40 -50.
Now,
Modal class = 40-50, lower limit (/) of modal class = 40.
Thus, lower limit (/) of modal class is 40
3. The monthly pocket money of 50 students of a class are given in the following distribution:
Monthly pocket
0 - 50 50 100 100 150 150 -200 200 250 250 - 300
Number of
Students
2 7 8 30 12 1
Find the modal class and give class mark of the modal class.
Sol:
Here the maximum class frequency is 30, and the class corresponding to the frequency is
150-200. So, the modal class is 150-200.
Also, class mark of the modal class is
4. A data has 25 observations arranged in a descending order. Which observation represents the
median?
Sol:
If the number of observations is odd, then the median is observation.
Thus, observation represents the median.
5. For a certain distribution, mode and median were found to be 1000 and 1250 respectively.
Find mean for this distribution using an empirical relation.
Sol:
There is an empirical relationship between the three measures of central tendency:
3median = mode + 2Mean
=
=
Thus, the mean is 1375.
6. In a class test, 50 students obtained marks as follows:
Marks
obtained
0 20 20 40 40 60 60 80 80 100
Number of
Students
4 6 25 10 5
Find the modal class and the median class.
Sol:
Here the maximum class frequency is 25, and the class corresponding to this frequency is
40-60.
So, the modal class is 40-60.
Now, to find the median class let us put the data in the table given below:
Marks Obtained Number of students Cumulative frequency (cf)
0-20 4 4
20-40 6 10
40-60 25 35
60-80 10 45
80-100 5 50
Total N=
Now,
The cumulative frequency just greater than 25 is 35, and the corresponding class is 40-60.
Thus, the median class is 40-60.
7. Find the class marks of classes 10 -25 and 35 55.
Sol:
Class mark =
class mark of 10-25 =
= 17.5
And class mark of 35-55=
= 45
8. While calculating the mean of a given data by the assumed-mean method, the following
values were obtained.
A=25, 110
i i
f d , 50
i
f
Find the mean.
Sol:
According to assumed-mean method,
Thus, mean is 27.2.
9. The distribution X and Y with total number of observations 36 and 64, and mean 4 and 3
respectively are combined. What is the mean of the resulting distribution X + Y?
Sol:
According to the question,
And total number of observations = 36 + 64 = 100
Thus, mean = = 3.36.
10. In a frequency distribution table with 12 classes, the class-width is 2.5 and the lowest class
boundary is 8.1, then what is the upper class boundary of the highest class?
Sol:
Upper class boundary = Lowest class boundary + width × number of classes
=
=
=
Thus, upper class boundary of the highest class is 38.1.
11. The observation 29, 32, 48, 50, x, x+2, 72, 78, 84, 95 are arranged in ascending order. What
is the value of x if the median of the data is 63?
Sol:
If number of observations is even, then the median will be the average of th and the
th observations.
In the given case,
Thus,
Thus, the value of x is 62.
12. The median of 19 observations is 30. Two more observation are made and the values of these
are 8 and 32. Find the median of the 21 observations taken together.
Hint Since 8 is less than 30 and 32 is more than 30, so the value of median (middle value)
remains unchanged.
Sol:
Since, 8 is less than 30 and 32 is more than 30, so the middle value remains unchanged
Thus, the median of 21 observations taken together is 30.
13. If the median of and , where x > 0, is 8, find the value of x.
Hint Arranging the observations in ascending order, we have Median= .
Sol:
Arranging the observations in ascending order, we have
Thus, the value of x is 24.
14. What is the cumulative frequency of the modal class of the following distribution?
Class 3 6 6 9 9 12 12 15 15 18 18 21 21 24
Frequency 7 13 10 23 54 21 16
Sol:
Here the maximum class frequency is 23, and the class corresponding to this frequency is
12-15.
So, the modal class is 12.15.
Now, to find the cumulative frequency let us put the data in the table given below:
Class Frequency Cumulative frequency
3-6 7 7
6-9 13 20
9-12 10 30
12-15 23 53
15-18 4 57
18-21 21 78
21-24 16 94
Total
Thus, the cumulative frequency of the modal class is 53.
15. Find the mode of the given data:
Class Interval 0 20 20 40 40 60 60 80
Frequency 15 6 18 10
Sol:
Here the maximum class frequency is 18, and the class corresponding to this frequency is
40-60.
So, the modal class is 40-60.
Now,
Modal class = 40-60, lower limit (/) of modal class 40, class size (h)=20,
Frequency of the modal class =18,
Frequency of class preceding the modal class =6,
Frequency of class succeeding the modal class = 10.
Now, let us substitute these values in the formula:
=
=
=
= 52
Hence, the mode is 52.
16. The following are the ages of 300 patients getting medical treatment in a hospital on a
particular day:
Age
(in years)
10 20 20 30 30 40 40 50 50 60 60 -70
Number of
patients
6 42 55 70 53 20
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