Detailed Notes: Statistics | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Statistics 
1. Measures of central tendency: 
An average value or central value of a distribution is the value of variable which is 
representative of the entire distribution, this represented value are called the measures 
of central tendency are of following type. 
(A) Mathematical average 
(B) Positional average 
(i) Arithmetic mean or mean 
(i) Median 
(ii) Geometrical mean 
(ii) Mode 
(iii) Harmonic mean 
2. Mean (Arithmetic mean) 
If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? are ?? values of variate ?? ?? then their A.M. ?? ? is defined as 
?? ? =
?? 1
+ ?? 2
+ ?? 3
+ ? … + ?? ?? ?? =
?
?? =1
?? ??? ?? ?? 
If ?? 1
, ?? 2
, ?? 3
, … . ?? ?? are values of veriate with frequencies ?? 1
, ?? 2
, ?? 3
, … … . . ?? ?? then their A.M. is 
given by 
?? ? =
?? 1
?? 1
×?? 2
?? 2
+?? 3
?? 3
+?…?? ?? ?? ?? ?? 1
+?? 2
+?? 3
+?..+?? ?? =
?
?? =1
?? ??? ?? ?? ?? ?? , where ?? = ?
?? =1
?? ??? ?? 
(i) Properties of arithmetic mean: 
(a) Sum of deviation of variate from their A.M. is always zero that is ?? ( ?? ?? - ?? ?)= 0. 
(b) Sum of square of deviation of variate from their A.M. is minimum that is 
?? ( ?? ?? - ?? )
2
 is minimum 
(c) If ?? ? is mean of variate ?? then 
A.M. of ( ?? ?? + ?? )= ?? ? + ?? 
A.M. of ?? 1
· ?? ?? = ?? . ?? ? 
A.M. of ( ???? + ?? )= ???? ? + ?? 
(ii) Merits of arithmetic mean: 
(a) It is rigidly defined. 
(b) It is based on all the observation taken. 
Page 2


Statistics 
1. Measures of central tendency: 
An average value or central value of a distribution is the value of variable which is 
representative of the entire distribution, this represented value are called the measures 
of central tendency are of following type. 
(A) Mathematical average 
(B) Positional average 
(i) Arithmetic mean or mean 
(i) Median 
(ii) Geometrical mean 
(ii) Mode 
(iii) Harmonic mean 
2. Mean (Arithmetic mean) 
If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? are ?? values of variate ?? ?? then their A.M. ?? ? is defined as 
?? ? =
?? 1
+ ?? 2
+ ?? 3
+ ? … + ?? ?? ?? =
?
?? =1
?? ??? ?? ?? 
If ?? 1
, ?? 2
, ?? 3
, … . ?? ?? are values of veriate with frequencies ?? 1
, ?? 2
, ?? 3
, … … . . ?? ?? then their A.M. is 
given by 
?? ? =
?? 1
?? 1
×?? 2
?? 2
+?? 3
?? 3
+?…?? ?? ?? ?? ?? 1
+?? 2
+?? 3
+?..+?? ?? =
?
?? =1
?? ??? ?? ?? ?? ?? , where ?? = ?
?? =1
?? ??? ?? 
(i) Properties of arithmetic mean: 
(a) Sum of deviation of variate from their A.M. is always zero that is ?? ( ?? ?? - ?? ?)= 0. 
(b) Sum of square of deviation of variate from their A.M. is minimum that is 
?? ( ?? ?? - ?? )
2
 is minimum 
(c) If ?? ? is mean of variate ?? then 
A.M. of ( ?? ?? + ?? )= ?? ? + ?? 
A.M. of ?? 1
· ?? ?? = ?? . ?? ? 
A.M. of ( ???? + ?? )= ???? ? + ?? 
(ii) Merits of arithmetic mean: 
(a) It is rigidly defined. 
(b) It is based on all the observation taken. 
(c) It is calculated with reasonable ease. 
(d) It is least affected by fluctuations in sampling. 
(e) It is based on each observation and so it is a better representative of the data. 
(f) It is relatively reliable 
(g) Mathematical analysis of mean is possible. 
(iii) Demerits of Arithmetic Mean: 
(a) It is severely affected by the extreme values. 
(b) It cannot be represented in the actual data since the mean does not coincide with any 
of the observed value. 
(c) It cannot be computed unless all the items are known. 
Problem 1: Find mean of data 2, 4, 5, 6, 8, 17. 
Solution:   Mean =
2+4+5+6+8+17
6
= 7 
Problem 2: Find the mean of the following distribution: 
?? : 4 6 9 10 15 ?? : 5 10 10 7 8  
Solution: Calculation of Arithmetic Mean 
 ?? ?? ?? ?? ?? ?? ??  4 5 20  6 10 60  9 10 90   10 7 70   15 8 120  ?? = ? ?
?
?
 ?? ?? = 40 ? ?
?
?
 ?? ?? ?? ?? = 360   
?  ???????? = ?? ? =
? ?
?
?
 ?? ?? ?? ?? ? ?
?
?
 ?? ?? =
360
40
= 9  
Problem 3: Find the mean wage from the following data: 
 
 ???? . ???? ?????????????? :  7 14 19 25 20 10 5  
Solution: Let the assumed mean be ?? = 900 and h = 20. 
Calculation of Mean 
Wage (in Rs) ?? ?? No. of workers ?? ?? ?? ?? = ?? ?? - ?? = ?? ?? - 900 
?? ?? =
?? ?? - 900
20
 
?? ?? ?? ?? 
800 7 -100 -5 -35 
820 14 -80 -4 -56 
960 19 -40 -2 -38 
920 25 0 1 0 
 20 100 5 20 
Page 3


Statistics 
1. Measures of central tendency: 
An average value or central value of a distribution is the value of variable which is 
representative of the entire distribution, this represented value are called the measures 
of central tendency are of following type. 
(A) Mathematical average 
(B) Positional average 
(i) Arithmetic mean or mean 
(i) Median 
(ii) Geometrical mean 
(ii) Mode 
(iii) Harmonic mean 
2. Mean (Arithmetic mean) 
If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? are ?? values of variate ?? ?? then their A.M. ?? ? is defined as 
?? ? =
?? 1
+ ?? 2
+ ?? 3
+ ? … + ?? ?? ?? =
?
?? =1
?? ??? ?? ?? 
If ?? 1
, ?? 2
, ?? 3
, … . ?? ?? are values of veriate with frequencies ?? 1
, ?? 2
, ?? 3
, … … . . ?? ?? then their A.M. is 
given by 
?? ? =
?? 1
?? 1
×?? 2
?? 2
+?? 3
?? 3
+?…?? ?? ?? ?? ?? 1
+?? 2
+?? 3
+?..+?? ?? =
?
?? =1
?? ??? ?? ?? ?? ?? , where ?? = ?
?? =1
?? ??? ?? 
(i) Properties of arithmetic mean: 
(a) Sum of deviation of variate from their A.M. is always zero that is ?? ( ?? ?? - ?? ?)= 0. 
(b) Sum of square of deviation of variate from their A.M. is minimum that is 
?? ( ?? ?? - ?? )
2
 is minimum 
(c) If ?? ? is mean of variate ?? then 
A.M. of ( ?? ?? + ?? )= ?? ? + ?? 
A.M. of ?? 1
· ?? ?? = ?? . ?? ? 
A.M. of ( ???? + ?? )= ???? ? + ?? 
(ii) Merits of arithmetic mean: 
(a) It is rigidly defined. 
(b) It is based on all the observation taken. 
(c) It is calculated with reasonable ease. 
(d) It is least affected by fluctuations in sampling. 
(e) It is based on each observation and so it is a better representative of the data. 
(f) It is relatively reliable 
(g) Mathematical analysis of mean is possible. 
(iii) Demerits of Arithmetic Mean: 
(a) It is severely affected by the extreme values. 
(b) It cannot be represented in the actual data since the mean does not coincide with any 
of the observed value. 
(c) It cannot be computed unless all the items are known. 
Problem 1: Find mean of data 2, 4, 5, 6, 8, 17. 
Solution:   Mean =
2+4+5+6+8+17
6
= 7 
Problem 2: Find the mean of the following distribution: 
?? : 4 6 9 10 15 ?? : 5 10 10 7 8  
Solution: Calculation of Arithmetic Mean 
 ?? ?? ?? ?? ?? ?? ??  4 5 20  6 10 60  9 10 90   10 7 70   15 8 120  ?? = ? ?
?
?
 ?? ?? = 40 ? ?
?
?
 ?? ?? ?? ?? = 360   
?  ???????? = ?? ? =
? ?
?
?
 ?? ?? ?? ?? ? ?
?
?
 ?? ?? =
360
40
= 9  
Problem 3: Find the mean wage from the following data: 
 
 ???? . ???? ?????????????? :  7 14 19 25 20 10 5  
Solution: Let the assumed mean be ?? = 900 and h = 20. 
Calculation of Mean 
Wage (in Rs) ?? ?? No. of workers ?? ?? ?? ?? = ?? ?? - ?? = ?? ?? - 900 
?? ?? =
?? ?? - 900
20
 
?? ?? ?? ?? 
800 7 -100 -5 -35 
820 14 -80 -4 -56 
960 19 -40 -2 -38 
920 25 0 1 0 
 20 100 5 20 
 
We have, 
 ?? = 100, ? ?
?
?
 ?? ?? ?? ?? = -44, ?? = 900 ?????? h = 20 ?   ???????? = ?? ? = ?? + h (
1
?? ? ?
?
?
 ?? ?? ?? ?? )
? ?? ? = 900 + 20 ×
-44
100
= 900 - 8.8 = 891.2  
Hence, mean wage = Rs. 891.2 
3. Geometric mean: 
If ?? 1
, ?? 2
, ?? 3
, … . . ?? 3
 are ?? positive values of variate then their geometric mean ?? is given by 
?? = ( ?? 1
?? 2
?? 3
… . . ?? ?? )
1/?? 
? ?? = ?????????????? [
1
?? ?
?? =1
?? ??????? ?? ?? ] 
4. Median: 
The median of a series is values of middle term of series when the values are written is 
ascending order or descending order. Therefore median, divide on arranged series in two 
equal parts 
(i) For ungrouped distribution: 
If ?? be number of variates in a series then 
 ???????????? = { (
?? + 1
2
)
?? h 
 ???????? , ( ?? h???? ?? ???? ?????? )   ???????? ???? (
?? 2
)
?? h 
 ?????? (
?? 2
+ 2)
?? h 
 ???????? ( ?? h???? ?? ???? ???????? )   
(ii) For ungrouped frequency distribution: 
First we calculate cumulative frequency (sum of all frequencies). Let it be ?? then 
Median { (
?? +1
2
)
?? h 
 ???????? ( ?? h???? ?? ?????? )   ???????? ???? (
?? 2
)& (
?? +2
2
) ( ?? h???? ?? ???? ???????? )   
 
5. Merits and demerits of median: 
The following are some merits and demerits of median: 
(i) Merits: 
(a) It is easy to compute and understand. 
Page 4


Statistics 
1. Measures of central tendency: 
An average value or central value of a distribution is the value of variable which is 
representative of the entire distribution, this represented value are called the measures 
of central tendency are of following type. 
(A) Mathematical average 
(B) Positional average 
(i) Arithmetic mean or mean 
(i) Median 
(ii) Geometrical mean 
(ii) Mode 
(iii) Harmonic mean 
2. Mean (Arithmetic mean) 
If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? are ?? values of variate ?? ?? then their A.M. ?? ? is defined as 
?? ? =
?? 1
+ ?? 2
+ ?? 3
+ ? … + ?? ?? ?? =
?
?? =1
?? ??? ?? ?? 
If ?? 1
, ?? 2
, ?? 3
, … . ?? ?? are values of veriate with frequencies ?? 1
, ?? 2
, ?? 3
, … … . . ?? ?? then their A.M. is 
given by 
?? ? =
?? 1
?? 1
×?? 2
?? 2
+?? 3
?? 3
+?…?? ?? ?? ?? ?? 1
+?? 2
+?? 3
+?..+?? ?? =
?
?? =1
?? ??? ?? ?? ?? ?? , where ?? = ?
?? =1
?? ??? ?? 
(i) Properties of arithmetic mean: 
(a) Sum of deviation of variate from their A.M. is always zero that is ?? ( ?? ?? - ?? ?)= 0. 
(b) Sum of square of deviation of variate from their A.M. is minimum that is 
?? ( ?? ?? - ?? )
2
 is minimum 
(c) If ?? ? is mean of variate ?? then 
A.M. of ( ?? ?? + ?? )= ?? ? + ?? 
A.M. of ?? 1
· ?? ?? = ?? . ?? ? 
A.M. of ( ???? + ?? )= ???? ? + ?? 
(ii) Merits of arithmetic mean: 
(a) It is rigidly defined. 
(b) It is based on all the observation taken. 
(c) It is calculated with reasonable ease. 
(d) It is least affected by fluctuations in sampling. 
(e) It is based on each observation and so it is a better representative of the data. 
(f) It is relatively reliable 
(g) Mathematical analysis of mean is possible. 
(iii) Demerits of Arithmetic Mean: 
(a) It is severely affected by the extreme values. 
(b) It cannot be represented in the actual data since the mean does not coincide with any 
of the observed value. 
(c) It cannot be computed unless all the items are known. 
Problem 1: Find mean of data 2, 4, 5, 6, 8, 17. 
Solution:   Mean =
2+4+5+6+8+17
6
= 7 
Problem 2: Find the mean of the following distribution: 
?? : 4 6 9 10 15 ?? : 5 10 10 7 8  
Solution: Calculation of Arithmetic Mean 
 ?? ?? ?? ?? ?? ?? ??  4 5 20  6 10 60  9 10 90   10 7 70   15 8 120  ?? = ? ?
?
?
 ?? ?? = 40 ? ?
?
?
 ?? ?? ?? ?? = 360   
?  ???????? = ?? ? =
? ?
?
?
 ?? ?? ?? ?? ? ?
?
?
 ?? ?? =
360
40
= 9  
Problem 3: Find the mean wage from the following data: 
 
 ???? . ???? ?????????????? :  7 14 19 25 20 10 5  
Solution: Let the assumed mean be ?? = 900 and h = 20. 
Calculation of Mean 
Wage (in Rs) ?? ?? No. of workers ?? ?? ?? ?? = ?? ?? - ?? = ?? ?? - 900 
?? ?? =
?? ?? - 900
20
 
?? ?? ?? ?? 
800 7 -100 -5 -35 
820 14 -80 -4 -56 
960 19 -40 -2 -38 
920 25 0 1 0 
 20 100 5 20 
 
We have, 
 ?? = 100, ? ?
?
?
 ?? ?? ?? ?? = -44, ?? = 900 ?????? h = 20 ?   ???????? = ?? ? = ?? + h (
1
?? ? ?
?
?
 ?? ?? ?? ?? )
? ?? ? = 900 + 20 ×
-44
100
= 900 - 8.8 = 891.2  
Hence, mean wage = Rs. 891.2 
3. Geometric mean: 
If ?? 1
, ?? 2
, ?? 3
, … . . ?? 3
 are ?? positive values of variate then their geometric mean ?? is given by 
?? = ( ?? 1
?? 2
?? 3
… . . ?? ?? )
1/?? 
? ?? = ?????????????? [
1
?? ?
?? =1
?? ??????? ?? ?? ] 
4. Median: 
The median of a series is values of middle term of series when the values are written is 
ascending order or descending order. Therefore median, divide on arranged series in two 
equal parts 
(i) For ungrouped distribution: 
If ?? be number of variates in a series then 
 ???????????? = { (
?? + 1
2
)
?? h 
 ???????? , ( ?? h???? ?? ???? ?????? )   ???????? ???? (
?? 2
)
?? h 
 ?????? (
?? 2
+ 2)
?? h 
 ???????? ( ?? h???? ?? ???? ???????? )   
(ii) For ungrouped frequency distribution: 
First we calculate cumulative frequency (sum of all frequencies). Let it be ?? then 
Median { (
?? +1
2
)
?? h 
 ???????? ( ?? h???? ?? ?????? )   ???????? ???? (
?? 2
)& (
?? +2
2
) ( ?? h???? ?? ???? ???????? )   
 
5. Merits and demerits of median: 
The following are some merits and demerits of median: 
(i) Merits: 
(a) It is easy to compute and understand. 
(b) It is well defined an ideal average should be 
(c) It can also be computed in case of frequency distribution with open ended classes. 
(d) It is not affected by extreme values. 
(e) It can be determined graphically. 
(f) It is proper average for qualitative data where items are not measured but are scored. 
(ii) Demerits: 
(a) For computing median data needs to be arranged in ascending or descending order. 
(b) It is not based on all the observations of the data. 
(c) It cannot be given further algebraic treatment. 
(d) It is affected by fluctuations of sampling. 
(e) It is not accurate when the data is not large. 
(f) In some cases median is determined approximately as the mid-point of two 
observations whereas for mean this does not happen. 
Problem 4: Find the median of observations 4, 6, 9, 4, 2, 8, 10 
Solution:   Values in ascending order are 2,4,4,6,8,9,10 
here ?? = 7 so 
?? +1
2
= 4 
so median = 4
?? h 
 observaiton = 6 
Problem 5: Obtain the median for the following frequency distribution: 
?? 11 13 15 18 21 23 30 40 50 
?? 8 10 11 16 20 25 15 9 6 
 
 
 
Solution: 
?? ?? ???? 
11 8 8 
13 10 18 
15 11 29 
18 16 45 
21 20 65 
Page 5


Statistics 
1. Measures of central tendency: 
An average value or central value of a distribution is the value of variable which is 
representative of the entire distribution, this represented value are called the measures 
of central tendency are of following type. 
(A) Mathematical average 
(B) Positional average 
(i) Arithmetic mean or mean 
(i) Median 
(ii) Geometrical mean 
(ii) Mode 
(iii) Harmonic mean 
2. Mean (Arithmetic mean) 
If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? are ?? values of variate ?? ?? then their A.M. ?? ? is defined as 
?? ? =
?? 1
+ ?? 2
+ ?? 3
+ ? … + ?? ?? ?? =
?
?? =1
?? ??? ?? ?? 
If ?? 1
, ?? 2
, ?? 3
, … . ?? ?? are values of veriate with frequencies ?? 1
, ?? 2
, ?? 3
, … … . . ?? ?? then their A.M. is 
given by 
?? ? =
?? 1
?? 1
×?? 2
?? 2
+?? 3
?? 3
+?…?? ?? ?? ?? ?? 1
+?? 2
+?? 3
+?..+?? ?? =
?
?? =1
?? ??? ?? ?? ?? ?? , where ?? = ?
?? =1
?? ??? ?? 
(i) Properties of arithmetic mean: 
(a) Sum of deviation of variate from their A.M. is always zero that is ?? ( ?? ?? - ?? ?)= 0. 
(b) Sum of square of deviation of variate from their A.M. is minimum that is 
?? ( ?? ?? - ?? )
2
 is minimum 
(c) If ?? ? is mean of variate ?? then 
A.M. of ( ?? ?? + ?? )= ?? ? + ?? 
A.M. of ?? 1
· ?? ?? = ?? . ?? ? 
A.M. of ( ???? + ?? )= ???? ? + ?? 
(ii) Merits of arithmetic mean: 
(a) It is rigidly defined. 
(b) It is based on all the observation taken. 
(c) It is calculated with reasonable ease. 
(d) It is least affected by fluctuations in sampling. 
(e) It is based on each observation and so it is a better representative of the data. 
(f) It is relatively reliable 
(g) Mathematical analysis of mean is possible. 
(iii) Demerits of Arithmetic Mean: 
(a) It is severely affected by the extreme values. 
(b) It cannot be represented in the actual data since the mean does not coincide with any 
of the observed value. 
(c) It cannot be computed unless all the items are known. 
Problem 1: Find mean of data 2, 4, 5, 6, 8, 17. 
Solution:   Mean =
2+4+5+6+8+17
6
= 7 
Problem 2: Find the mean of the following distribution: 
?? : 4 6 9 10 15 ?? : 5 10 10 7 8  
Solution: Calculation of Arithmetic Mean 
 ?? ?? ?? ?? ?? ?? ??  4 5 20  6 10 60  9 10 90   10 7 70   15 8 120  ?? = ? ?
?
?
 ?? ?? = 40 ? ?
?
?
 ?? ?? ?? ?? = 360   
?  ???????? = ?? ? =
? ?
?
?
 ?? ?? ?? ?? ? ?
?
?
 ?? ?? =
360
40
= 9  
Problem 3: Find the mean wage from the following data: 
 
 ???? . ???? ?????????????? :  7 14 19 25 20 10 5  
Solution: Let the assumed mean be ?? = 900 and h = 20. 
Calculation of Mean 
Wage (in Rs) ?? ?? No. of workers ?? ?? ?? ?? = ?? ?? - ?? = ?? ?? - 900 
?? ?? =
?? ?? - 900
20
 
?? ?? ?? ?? 
800 7 -100 -5 -35 
820 14 -80 -4 -56 
960 19 -40 -2 -38 
920 25 0 1 0 
 20 100 5 20 
 
We have, 
 ?? = 100, ? ?
?
?
 ?? ?? ?? ?? = -44, ?? = 900 ?????? h = 20 ?   ???????? = ?? ? = ?? + h (
1
?? ? ?
?
?
 ?? ?? ?? ?? )
? ?? ? = 900 + 20 ×
-44
100
= 900 - 8.8 = 891.2  
Hence, mean wage = Rs. 891.2 
3. Geometric mean: 
If ?? 1
, ?? 2
, ?? 3
, … . . ?? 3
 are ?? positive values of variate then their geometric mean ?? is given by 
?? = ( ?? 1
?? 2
?? 3
… . . ?? ?? )
1/?? 
? ?? = ?????????????? [
1
?? ?
?? =1
?? ??????? ?? ?? ] 
4. Median: 
The median of a series is values of middle term of series when the values are written is 
ascending order or descending order. Therefore median, divide on arranged series in two 
equal parts 
(i) For ungrouped distribution: 
If ?? be number of variates in a series then 
 ???????????? = { (
?? + 1
2
)
?? h 
 ???????? , ( ?? h???? ?? ???? ?????? )   ???????? ???? (
?? 2
)
?? h 
 ?????? (
?? 2
+ 2)
?? h 
 ???????? ( ?? h???? ?? ???? ???????? )   
(ii) For ungrouped frequency distribution: 
First we calculate cumulative frequency (sum of all frequencies). Let it be ?? then 
Median { (
?? +1
2
)
?? h 
 ???????? ( ?? h???? ?? ?????? )   ???????? ???? (
?? 2
)& (
?? +2
2
) ( ?? h???? ?? ???? ???????? )   
 
5. Merits and demerits of median: 
The following are some merits and demerits of median: 
(i) Merits: 
(a) It is easy to compute and understand. 
(b) It is well defined an ideal average should be 
(c) It can also be computed in case of frequency distribution with open ended classes. 
(d) It is not affected by extreme values. 
(e) It can be determined graphically. 
(f) It is proper average for qualitative data where items are not measured but are scored. 
(ii) Demerits: 
(a) For computing median data needs to be arranged in ascending or descending order. 
(b) It is not based on all the observations of the data. 
(c) It cannot be given further algebraic treatment. 
(d) It is affected by fluctuations of sampling. 
(e) It is not accurate when the data is not large. 
(f) In some cases median is determined approximately as the mid-point of two 
observations whereas for mean this does not happen. 
Problem 4: Find the median of observations 4, 6, 9, 4, 2, 8, 10 
Solution:   Values in ascending order are 2,4,4,6,8,9,10 
here ?? = 7 so 
?? +1
2
= 4 
so median = 4
?? h 
 observaiton = 6 
Problem 5: Obtain the median for the following frequency distribution: 
?? 11 13 15 18 21 23 30 40 50 
?? 8 10 11 16 20 25 15 9 6 
 
 
 
Solution: 
?? ?? ???? 
11 8 8 
13 10 18 
15 11 29 
18 16 45 
21 20 65 
23 25 90 
30 15 105 
40 9 114 
50 6 120 
?? = 120 
 
Here,  ?? = 120 ? 
?? 2
= 60 
We find that the cummulative frequency just greater than 
?? 2
 i.e., 60 is 65 and the value of 
?? corresponding to 65 is 21 . Therefore, Median = 21. 
6. Harmonic Mean: 
If ?? 1
, ?? 2
, ?? 3
, … . . . ?? 3
 are ?? non-zero values of variate then their harmonic mean ?? is 
defined as 
?? =
?? 1
?? ?? +
1
?? 2
+ ? +
1
?? ?? =
?? ?
?? =1
?? ?
1
?? ?? 
7. Mode: 
If a frequency distribution the mode is the value of that variate which have the maximum 
frequency. Mode for 
(i) For ungrouped distribution: 
The value of variate which has maximum frequency. 
(ii) For ungrouped frequency distribution: 
The value of that variate which have maximum frequency. Relationship between mean, 
median and mode. 
? In symmetric distribution, mean = mode = median 
? In skew (moderately asymmetrical) distribution median divides mean and mode 
internally in 1: 2 ratio. 
?  median =
2( ???????? ) +( ???????? )
3
 
8. Merits and demerits of mode: 
The following are some merits and demerits of mode: 
(i) Merits: 
(a) It is readily comprehensible and easy to compute. In some case it can be computed 
merely by inspection.