This document provides a lecture on quantiles, including quartiles, deciles, and percentiles. It defines quantiles as statistics that divide data into equal proportions. Common quantiles include quartiles (dividing data into 4 equal parts), deciles (dividing into 10 parts), and percentiles (dividing into 100 parts). The document gives formulas for calculating quantiles from both grouped and ungrouped data and provides examples of calculating quartiles, deciles, and percentiles. It also explains the relationships between different quantile types.

2. 2
MATH – 361
Introduction to Probability and Statistics
Lecture No. 07
Quantiles
Reference: Ch # 1, Sec 1.3, Text Book
No. Of Slides: 34

3. 3
Desired Learning Objectives
After this lecture, students will be able to
 Interpret Quantiles
 Differentiate among Quartiles, Deciles and Percentiles
 Compute Quartiles, Deciles and Percentiles
 Apply these measures in the fields of engineering particularly while
conducting technical investigations

4. 4
Definition
 The data can be divided into more than two parts if it is very
large. This extension of divisions is called Quantiles
Quantiles

5. 5
Types
 Quantiles are statistics that describe various subdivisions of a
frequency distribution into equal proportions. Generally data is
either divided into 4 or 10 or 100 equal parts called
1. Quartiles
2. Deciles
3. Percentiles
Quantiles

6. 6
Definition
 Splitting the ranked or ordered data into 4
equal groups. Each group is called a
quarter
 Each quarter has a share of 25% in it
 Quartiles are represented as Q1,Q2 & Q3
 Quartile is a measure that tells us about
the share of dataset falling below it
Quartiles

7. 7
 Q1 mean 25% or ¼ of the data is equal to or below it for Q3 75%
or ¾ of the data is equal to or below it
 The first quartile is also known as Lower Quartile and third is
known as Upper Quartile
25% 25% 25% 25%
Q1 Q2 Q3
Quartiles

8. 8
Interquartiles Range (IQR)
 Quartile range also known as mid-spread is the difference
between third and first quartile
𝒊. 𝒆. 𝑰𝑸𝑹 = 𝑸𝟑 − 𝑸𝟏
25% 25% 25% 25%
Q1 Q2 Q3
Quartiles

9. 9
Formula
Un-grouped Data 𝑸𝒊 = 𝒊
𝒏+𝟏
𝟒
th term
Grouped Data 𝑸𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟒
− 𝒄 𝒘𝒉𝒆𝒓𝒆 𝒊 = 𝟏, 𝟐, 𝟑
Quartiles

10. 10
Example : Un-grouped Data
The Marks obtained by 9 students are given by 45,32,37,46,39, 36,
41,48,36. Calculate lower and upper quartile
Here it is important to note that
 Lower Quartile Means Q1
 Upper Quartile Mean Q3
 Whereas Q2 is the Median
Quartiles

11. 11
Example : Un-grouped Data
 Here the data is ungrouped therefore formula for this example is
𝑄𝑖 = 𝑖
𝑛+1
4
th term
 The term will be used to identify the required quartile
Quartiles

12. 12
Example : Un-grouped Data
 We need to find lower Quartile i.e. Q1 implies for this 𝑖 = 1 using
this value in the formula we get our required term number
 𝑄𝑖 = 𝑖
𝑛+1
4
𝑡ℎ
𝑡𝑒𝑟𝑚 = 1
9+1
4
𝑡ℎ
𝑡𝑒𝑟𝑚 = 2.5𝑡ℎ 𝑡𝑒𝑟𝑚 implies our Q1
lies between 2nd and 3rd term of the data after the data is
arranged in ascending order
Quartiles

13. 13
Arranging the data in ascending order
32,36,36,37,39,41,45,46,48
 𝑄𝑖 = 𝑖
𝑛+1
4
= 1
9+1
4
= 2.5𝑡ℎ
𝑡𝑒𝑟𝑚 implies our Q1 lies
between 2nd and 3rd term implies the required quartile is
36+36
2
= 36
Quartiles

14. 14
Example : Quartiles (Grouped Data)
Calculate lower and upper quartile
Quartiles
Classes Class Boundaries f C f
65 – 84 64.5 – 84.5 9 9
85 – 104 84.5 – 104.5 10 19
105 – 124 104.5 – 124.5 17 36
125 – 144 124.5 – 144.5 10 46
145 – 164 144.5 – 164.5 5 51
165 – 184 164.5 – 184.5 4 55
185 – 204 184.5 – 204.5 5 60
Total 60

15. 15
Example : Quartiles (Grouped Data)
Requirements
 Class boundaries
 Cumulative frequencies
 Sum of frequencies = n
 Identify desired quartile
 Value of
i𝑛
4
This formula works in the same way as formula for Median
𝑸𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟒
− 𝒄 𝒘𝒉𝒆𝒓𝒆 𝒊 = 𝟏, 𝟐, 𝟑
Quartiles

16. 16
 Cumulative frequency just over
i𝑛
4
value
(that class becomes reference class)
 f = Frequency of reference class
 l = Lower class boundary of reference class
 h = Class width of reference class
Quartiles

17. 17
Quartiles
Classes Class Boundaries f C f
65 – 84 64.5 – 84.5 9 9
85 – 104 84.5 – 104.5 10 19
105 – 124 104.5 – 124.5 17 36
125 – 144 124.5 – 144.5 10 46
145 – 164 144.5 – 164.5 5 51
165 – 184 164.5 – 184.5 4 55
185 – 204 184.5 – 204.5 5 60
Total 60 = n
Example : Quartiles (Grouped Data)
Calculate lower Quartile
For Q1
𝑖𝑛
4
= 1*60/4 = 15 Cumulative
frequency just over it is 19 that falls in 85 -
104 class so this is our ref class
C = cf before cf
of ref class = 9
f= Freq of ref class = 10
l= L C B
of ref class =
84.5
h=Class
width of ref
class = 20

18. 18
Example : Quartiles (lower quartile Q1)
𝑸𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟒
− 𝒄 𝒉𝒆𝒓𝒆 𝒊 = 𝟏
Using the values form the table we get
𝑸𝟏 = 𝟖𝟒. 𝟓 +
𝟐𝟎
𝟏𝟎
𝟔𝟎
𝟒
− 𝟗 = 𝟗𝟔. 𝟓
Quartiles

19. 19
Quartiles
Classes Class Boundaries f C f
65 – 84 64.5 – 84.5 9 9
85 – 104 84.5 – 104.5 10 19
105 – 124 104.5 – 124.5 17 36
125 – 144 124.5 – 144.5 10 46
145 – 164 144.5 – 164.5 5 51
165 – 184 164.5 – 184.5 4 55
185 – 204 184.5 – 204.5 5 60
Total 60 = n
Example : Quartiles (Grouped Data)
Calculate upper Quartile
For Q3
𝑖𝑛
4
= 3*60/4 = 45 Cumulative
frequency just over it is 46 that falls in 125 -
144 class so this is our ref class
C = cf before cf
of ref class =
36
f= Freq of ref class = 10
l= L C B
of ref class =
124.5
h=Class
width of ref
class = 20

20. 20
Example : Upper Quartile Q3
𝑸𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟒
− 𝒄 𝒉𝒆𝒓𝒆 𝒊 =3
Using the values form the table we get
𝑸𝟑 = 𝟏𝟐𝟒. 𝟓 +
𝟐𝟎
𝟏𝟎
𝟑 ∗ 𝟔𝟎
𝟒
− 𝟑𝟔 = 𝟏𝟐𝟒. 𝟓 + 𝟏𝟖 = 𝟏𝟒𝟐. 𝟓
Quartiles

21. It is clear from the formula of that if we take i = 2 this becomes
median formula. In other words the point which represents 50% of
the data is median of that data. May it be quartile, decile or
percentile
21
Quartiles
𝑸𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟒
− 𝒄 𝒉𝒆𝒓𝒆 𝒊 =2

22. 22
Deciles
Divide the data into ten equal parts. The 5th Decile is Median.
Type of Quantiles

23. 23
Deciles
Formula
Ungrouped Data 𝑫𝒊 = 𝒊
𝒏+𝟏
𝟏𝟎
𝐭𝐡
𝐭𝐞𝐫𝐦
Grouped Data 𝑫𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟏𝟎
− 𝒄 𝒘𝒉𝒆𝒓𝒆 𝒊 = 𝟏, 𝟐, … , 𝟗

24. 24
Example Find 8th Decile for the given distribution

Deciles
Classes Class Boundaries f C f
30 – 39 29.5 – 39.5 8 8
40 – 49 39.5 – 49.5 87 95
50 – 59 49.5 – 59.5 190 285
60 – 69 59.5 – 69.5 304 589
70 – 79 69.5 – 79.5 211 800
80 – 89 79.5 – 89.5 85 885
90 – 99 89.5 – 99.5 20 905
Total 905
For D8
in
10
= 8 ∗
905
10
= 724 Cumulative
frequency just over 724 is 800 that falls in 70 -
79 class so 70-79 is our ref class
C = cf before cf
of ref class = 589
f= Freq of ref class = 211
l= L C B
of ref class =
69.5
h=Class
width of ref
class = 10

25. 25
Example : (8th Decile)
𝑫𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟏𝟎
− 𝒄 𝒘𝒉𝒆𝒓𝒆 𝒊 = 𝟏, 𝟐, … , 𝟗
Using the values from the table we get
𝑫𝟖 = 𝟔𝟗. 𝟓 +
𝟏𝟎
𝟐𝟏𝟏
𝟖 ∗ 𝟗𝟎𝟓
𝟏𝟎
− 𝟓𝟖𝟗 = 𝟕𝟒. 𝟓 𝒘𝒉𝒆𝒓𝒆 𝒊 = 𝟖
Deciles

26. 26
Percentiles
Divide the data into hundred equal parts. P50 will be equal to
median of the data
Type of Quantiles

27. 27
Percentiles
Formula
Ungrouped Data 𝑷𝒊 = 𝒊
𝒏+𝟏
𝟏𝟎𝟎
𝒕𝒉
𝒕𝒆𝒓𝒎
Grouped Data 𝑷𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟏𝟎𝟎
− 𝒄 𝒘𝒉𝒆𝒓𝒆 𝒊 = 𝟏, 𝟐, … , 𝟗𝟗

28. 28
Example Find P25 and P75 for the given distribution (P25 do yourself)
Classes Class Boundaries f C f
30 – 39 29.5 – 39.5 8 8
40 – 49 39.5 – 49.5 87 95
50 – 59 49.5 – 59.5 190 285
60 – 69 59.5 – 69.5 304 589
70 – 79 69.5 – 79.5 211 800
80 – 89 79.5 – 89.5 85 885
90 – 99 89.5 – 99.5 20 905
Total 905
Percentiles
For P75 In/100 = 75*905/100 = 678.75
Cumulative frequency just over it is 800 that
falls in 70 - 79 class so this is our ref class
C = cf before cf
of ref class = 589
f= Freq of ref class = 211
l= L C B
of ref class =
69.5
h=Class
width of ref
class = 10

29. 29
Example : (75th Percentile)
𝑷𝒊 = 𝒍 +
𝒉
𝒇
𝒊𝒏
𝟏𝟎𝟎
− 𝒄 𝒘𝒉𝒆𝒓𝒆 𝒊 = 𝟏, 𝟐, … , 𝟗𝟗
Using the values from the table we get
𝑷𝟕𝟓 = 𝟔𝟗. 𝟓 +
𝟏𝟎
𝟐𝟏𝟏
𝟕𝟓 ∗ 𝟗𝟎𝟓
𝟏𝟎𝟎
− 𝟓𝟖𝟗 = 𝟕𝟑. 𝟕𝟓 𝒘𝒉𝒆𝒓𝒆 𝒊 = 𝟕𝟓
Percentiles

30. Relationship
Q1 = P25
Q2 = P50
Q3 = P75
Q2 = D5
Q2 = D5 = P50 (Median)
D1 = P10
D2 = P20
D3 = P30
•
Deciles
30
Partition Values or Quantiles Relationship
Q1 = 25%
Q2 = 50%
Q3 = 75%
Quartiles
D3 = 30%
D5 = 50%
D9 = 90%
Percentiles
P10 = 10%
P25 = 25%
P75 = 75%

31. For the following frequency distribution calculate the followings
Class 10-19 20-29 30-39 40-49 50-59 60-69
f 7 13 22 11 6 5
31
Practice Problem 1
 Find class boundaries
 Find cumulative frequencies
 Find the median class
 Identify lower class boundary of median
class
 Find width of median class
 Identify the frequency of median class
 Identify the cumulative frequency that
comes just before median class frequency
 Find Q1,Q2,Q3,D5,D7,P50,P70 & P95 and show
that Q2 = D5 = P50 = Median

32. For the following frequency distribution calculate the followings
class 40-49 50-59 60-69 70-79 80-89 90-99 100-109 110-119 120-129 130-139 140-149 150-159 160-169
f 1 2 3 5 17 65 69 79 37 19 7 3 2
32
Practice Problem 2
 Find class boundaries
 Find cumulative frequencies
 Find the median class
 Identify lower class boundary of median
class
 Find width of median class
 Identify the frequency of median class
 Identify the cumulative frequency that
comes just before median class frequency
 Find Q1, D5, P90 & P95 and show that the
D5 = Median

33. 33
Practice Problem 3
The Marks obtained by 9 students are given by
45,32,37,46,39,36,41,48,36. Calculate Q3, D7 and P75 also calculate
interquartile range

34. 34
https://www.youtube.com/watch?v=Wy7w_5N7vLI
https://uomustansiriyah.edu.iq/media/lectures/5/5_2018_12_10!09_06_45_PM.
pdf
http://www.tihe.org/courses/it133/IT%20133%20Lectures/IT133%20-
%20Lecture%2004.pdf
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