The document provides information about different measures of position for ungrouped data, including quartiles, deciles, and percentiles. It defines each measure and explains how to calculate them using methods like the Mendenhall and Sincich method and linear interpolation. Specifically, it discusses how quartiles divide a distribution into four equal parts at the 25th, 50th, and 75th percentiles, deciles divide into ten equal parts, and percentiles divide into 100 equal parts. Formulas and steps are provided to find the value of each measure for a given data set.

1. MEASURES OF POSITION
FOR UNGROUPED DATA:
QUARTILES , DECILES , &
PERCENTILES
WEEK 1

2. Objectives:
a. Illustrate the different measures of
positions for ungrouped data
b. Calculate the different measures of
position such as quartile, decile and
percentile.
c. interpret and apply the different
measures of position in real life situation
Measures of Position for Ungrouped Data

3. PRE- ASSESSMENT
1. The median score is also the
_____________.
A. 75th percentile C. 5th decile
B. 3rd decile D. 1st quartile

4. PRE- ASSESSMENT
2. When a distribution is divided into
hundred equal parts, each score point
that describes the distribution is
called___________.
A. percentile B. decile
C. quartile D. median

5. PRE- ASSESSMENT
3. The lower quartile is equal to
______________.
A. 50th percentile B. 25th percentile
C. 2nd decile D. 3rd quartile

6. PRE- ASSESSMENT
4. Rochelle got a score of 55 which is equivalent to
70th percentile in a mathematics test. Which of
the following is NOT true?
A. She scored above 70% of her classmates.
B. Thirty percent of the class got scores of 55 and
above.
C. If the passing mark is the first quartile, she
passed the test.
D. Her score is below the 5th decile.

7. PRE- ASSESSMENT
5. In a 100-item test, the passing mark is the 3rd
quartile. What does it imply?
A. The students should answer at least 75 items
correctly to pass the test.
B. The students should answer at least 50 items
correctly to pass the test.
C. The students should answer at most 75 items
correctly to pass the test.
D. The students should answer at most 50 items
correctly to pass the test.

8. Assume you are
a tourist who
wishes to visit
Sampaloc Lake
in San Pablo
City.
How will you
find the lake's
exact location?

9. Have you
thought of
comparing
academic
performance
with your
classmates?

10. Have you
wondered what
score you need
for each
subject area to
qualify for
honors?

11. Find the median.
1. 4, 7, 1, 8, 10
2. 8, 6, 6, 10, 85
3. 34, 43, 45, 1, 30, 4

12. Measures of Position
It shows where a certain data point or
value falls in a sample or array of
distribution. It can state whether a value
is about the average, or whether it is
unusually high or low compared to the
data as a whole. There are a lot of
measures of position, but the most
common are the quartiles, deciles, and
percentiles.

13. Measures of Position

14. Quartiles for ungrouped data
 The QUARTILES are the score points which divide a distribution into four
equal parts.
Q1 Q2 Q3
 25% of the data has a value ≤ Q1
 50% of the data has a value ≤ Q2
 75% of the data has a value ≤ Q3

15. Quartiles for ungrouped data
Q1 is called the LOWER QUARTILE
Q2 is nothing but the MEDIAN
Q3 is the UPPER QUARTILE

16. Quartiles for ungrouped data
First Quartile (Q1) is a number
such that at most one-fourth or 25
% of the data are smaller in value
than Q1 and at most three-fourths
or 75% are larger than Q1. It is also
called as lower quartile.

17. Quartiles for ungrouped data
Second Quartile (Q2) is a number
such that at most one-half or 50 %
of the data are below and above in
value than Q2. Second Quartile is
also known as median.

18. Quartiles for ungrouped data
Third Quartile (Q3) is a number
such that at most three-fourths or
75 % of the data are smaller in
value than Q3 and at most one-
fourth or 25% are larger than Q3. It
is also called as upper quartile.

19. Quartiles for ungrouped data
Interquartile range is the
difference between the upper
quartile (Q3) and the lower
quartile (Q1) in a set of data.

20. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
This method is being developed
by William Mendenhall and Terry
Sincich to find the position of the
quartile in the given data.

21. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 Formula:
Lower Quartile (L) = Position of Q1= ¼ (n+1)th
Q2= 2(n+1) = (n+1)th observation
4 2
Upper Quartile (U) = Position of Q3 = ¾ (n+1)th

22. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
N is the number of elements in the data
Example: The manager of a food chain
recorded the number of customers who came to
eat the products in each day. The results were
10,15,14,13,20,19,12 and 11.
In this example N=8

23. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 Q1= (n+1) th observation
4
 Q2= 2(n+1) = (n+1) th observation
4 2
 Q3= 3(n+1) th observation
4

24. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
Steps to solve the quartile of the given data
1. Arrange the data in ascending order or from the lowest
value to the highest value.
2. Find the N or the total number of elements presented in
the data.
3. Find the least value of the data and the greatest value of
the data.
4. find the lower quartile of the given data using the
Mendenhall & Sincich Method.
Lower Quartile (L) = Position of Q1= ¼ (n+1)

25. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
5. Find the middle value of the data or the
Median. Use this formula
Q2= 2(n+1) = (n+1) th observation
4 2
6. Find the upper quartile of the given data. Use
this formula
Upper Quartile (U) = Position of Q3 = ¾ (n+1)

26. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 Example:
The owner of the coffee shop
recorded the number of customers
who came into his café each hour
in a day. The results were 14, 10,
12, 9, 17, 5, 8, 9, 14, 10, and 11.

27. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
Solution
 Ascending order
{5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
N=11
Least value= 5
Greatest value= 17

28. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 Lower Quartile (L) = Position of Q1= ¼ (n+1)
Q1= ½ (n+1)th
Q1= ½ (11+1)th
Q1= ½ (12)th
Q1= 12/4 th (divide)
Q1= 3rd
{5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
 Therefore, the Q1 is the 3rd element in the data which is 9.

29. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 Median Value or the middle value
Q2= 2/4 (n+1) = n+1/2 th observation
Q2= 2/4 (11+1)th
Q2= 2/4 (12)th
Q2= 24/4
Q2= 6th {5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
 Therefore, the Q2 is the 6th element in the data which is 10

30. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 Upper Quartile (U)= Position of Q3= ¾ (n+1)th
Q3= ¾ (11+1)th
Q3= ¾ (12)th
Q3= 36/4 th
Q3= 9th {5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
 Therefore the Q3 is the 9th element in the data which is 14.

31. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 Example:
The following are the scores of Gabriel
in his 7 quizzes in Mathematics 10.
87, 88, 89, 87, 95, 86, 94
Find Q1, Q2, and Q3.

32. MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 Example:
Mrs. Ana is a veterinarian. One morning,
she asked her secretary to record the
service time for 15 customers. The
following are service times in minutes.
20, 35, 55, 28, 46, 32, 25, 56, 55, 28, 37,
60, 47, 52, 17
Find Q1, Q2, and Q3.

33. Linear interpolation
 A method of finding the quartile value.
 Is a method of constructing new data points within the range of
a discrete set of known data points.
 It is often required to interpolate, i.e., estimate the value of
that function for an intermediate value of the independent
variable.
 We need to use the interpolation if the value of the position is in
decimal form.

34. Linear interpolation
 Using the formula
Qk= k/4 (n+1)
k= position of the quartile n= total no. of the data

35. Linear interpolation
 Steps in interpolation method
1. Arrange the scores in ascending order
2. Locate the position of the score in the distribution
3. Since the result is in decimal number, proceed to linear
interpolation
4. Find the difference between the two values wherein Q1 is situated
5. Multiply the result in step 2 by the decimal part obtained in step 4
6. Add the result in step 5 to the second smaller number in step 4

36. Linear interpolation
 Example:
Find the first quartile (Q1), and the third quartile (Q3), Given
the scores of 9 students in their mathematics activity using linear
interpolation
{1, 27, 16, 7, 31, 7, 30, 3, 21 }

37. Linear interpolation
 Step 1 : Arrange the scores in ascending order
{ 1, 3, 7, 7, 16, 21, 27, 30, 31 }
 Step 2: Locate the position of the score in the distribution
Position of Q1= ¼(n+1)
Q1= ¼ (9+1)
Q1= 0.25(10)
Q1= 2.5

38. Linear interpolation
 Step 3: Since the result is in decimal number, proceed to linear
interpolation
 Step 4: Find the difference between the two values wherein Q1
is situated
{ 1, 3, 7, 7, 16, 21, 27, 30, 31 }
2.5 position
 Q1 is between the values 3 and 7, therefore
= 7-3
= 4

39. Linear interpolation
 Step 5: Multiply the result in step 2 by the decimal part obtained
in step 4
= 4 (0.5)
= 2
 Step 6: Add the result in step 5 to the second smaller number in
step 4
= 2+3
= 5
 Therefore the value of Q1 is equal to 5

40. Deciles for ungrouped data
 The deciles are the nine score points which divide a distribution into
ten equal parts.
D1 D2 D3 D4 D5 D6 D7 D8 D9

41. Deciles for ungrouped data
 Calculating the position deciles
1. Formula
Position of D1 = k/10 (n+1)
 Example:
Find the 7th decile (D7), Given the scores of 11 students in their
mathematics activity.
{ 1, 27, 16, 7, 31, 7, 30, 31, 3, 4, 21 }

42. Deciles for ungrouped data
 Step 1: Arrange the scores in ascending order
{ 1, 3, 4, 7, 7, 16, 21, 27, 30, 31, 31 }
 Step 2: Locate the Position of the score in the distribution
Position of D7= 7/10 (n+1)
D7= 7/10 (11+1)
D7= 8.4 = 8 (rounded off)
 D7 is the 8th element therefore D7=27

43. Percentile for ungrouped fata
 The percentiles are the ninety-nine score points which
divide a distribution into one hundred equal parts, so
that each part represents the data set.

44. Percentile for ungrouped fata
 Calculating the position of percentile
 Example:
Find the 58th percentile (P58), Given the scores of 10
students in their mathematics activity using linear
interpolation
{ 1, 27, 16, 7, 31, 7, 30, 3, 4, 21 }

45. Percentile for ungrouped fata
 Step 1: Arrange the scores in ascending order
{ 1, 3, 4, 7, 7, 16, 21, 27, 30, 31 }
 Step 2: Locate the position of the scores in the
distribution
Position of P58 = 58/100 (n+1)
P58 = 58/100 (10+1)
P58 = 6.38 = 6
 P58 is the 6th element therefore, P58 = 16

46. Thank you !