The document discusses measures of location and position, specifically quartiles, deciles, and percentiles. It defines quartiles as the values that divide an ordered data set into four equal parts. The first, second, and third quartiles (Q1, Q2, Q3) represent the lower quartile, median, and upper quartile, respectively. Formulas are provided to calculate quartiles from both ungrouped and grouped data by determining the median value of each quarter of the data. An example calculation is shown for finding the quartiles of marks obtained by 20 students in a test of statistics using ungrouped data and for finding quartiles from grouped frequency data.

1. Measures of Location and position
Quartile, Decile, and percentile
Quartile:
The values which divide an array (a set of data arranged in ascending or
descending order) into four equal parts are called quartiles. The first, second
and third quartiles are denoted by respectively. The first and third
quartiles are also called the lower and upperquartiles respectively.The second
quartile represents the median, the middle value.
Quartiles for Ungrouped Data:
[ ( 1)]
P
Quartile n
L
 
1
2
3
1
[ ( 1)]
4
2
[ ( 1)]
4
3
[ ( 1)]
4
Q n
Q n
Q n
 
 
 
For Example:
Following is the data is of marks obtained by 20 students in a test of statistics
find quartiles Q1, Q2, Q3 :
53 74 82 42 39 20 81 68 58 28
67 54 93 70 30 55 36 37 29 61
In order to apply formulae we need to arrange the above data into ascending
order i.e. in the form of an array.
20 28 29 30 36 37 39 42 53 54
55 58 61 67 68 70 74 81 82 93
Here, n = 20

2. 1
2
3
1 1
[ ( 1)] [ (20 1)] (5.25) 5 36
4 4
2 2
[ ( 1)] [ (20 1)] (10.5) 54.5
4 4
3 3
[ ( 1)] [ (20 1)] (15.75) (16) 70
4 4
th th th th
th th th
th th th th
Q n value value value value
Q n value value value
Q n value value value
      
     
      
Quartiles for Grouped Data:
The quartiles may be determined from grouped data in the same way as the
median except that in place of n/2 we will use n/4. For calculating quartiles from
grouped data we will form cumulative frequency column. Quartiles for grouped
data will be calculated from the following formulae;
For ascending distribution
1
[ ].
n
i K
i
k
k
P
f F
L
Quartile L w
f

 

For Descending distribution:
*
[ ].
n
k i
i
k
k
P
F f
L
Quartile L w
f

 

= Median.
Where,
Lk = lower class containing the , i.e. the class corresponding to the
cumulative frequency in which n/4 or 3n/4 lies
n = number of values, or the total frequency.
F k
*
= cumulative frequency of the class preceding the class containing
L= sum of quartiles and
p =position of quartile

3. For Example:
Find q1, q2, q3,for the following data:
class fi Fi F*
10-19 5 5 38
20-29 7 12 33
30-39 10 22 26
40-49 8 30 16
50-59 4 34 8
60-69 3 37 4
70-79 1 38 1
total 38
38
9.5
4 4
f
 

1 5 9.5 12
4
i
K K
f
F F   

p p
1
1
1
9.5 5
4[ ]. 19.5 [ ].10 25.9
7
n
i K
i
k
k
P
f F
L
Q L w
f
 
    
