The document discusses measures of position for ungrouped data including quartiles, deciles, and percentiles. It specifically describes quartiles, which divide a distribution into four equal parts (Q1, Q2, Q3). The Mendenhall and Sincich method is presented for finding quartile values using a formula based on the number of data points. The method involves arranging data in order and determining the quartile positions. Linear interpolation is described for estimating quartile values that fall between data points. An example applies these methods to calculate quartiles for a set of student test scores.

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

2. 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

3. QUARTILES FOR UNGROUPED DATA
Q1 is called the LOWER QUARTILE
Q2 is nothing but the MEDIAN
Q3 is the UPPER QUARTILE

4. 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.

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

6. 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

7. 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

8. 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)

9. 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)

10. 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.

11. 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

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

13. 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)
Q2= 2/4 (12)
Q2= 24/4
Q2= 6 {5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
 Therefore the Q2 is the 6th element in the data which is 10

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

15. 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.

16. LINEAR INTERPOLATION
 Using the formula
Qk= k/4 (n+1)
k= position of the quartilen= total no. of the data

17. 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

18. 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 }

19. 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

20. 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

21. 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