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

OBJECTIVES
•ILLUSTRATE THE FOLLOWING MEASURES OF
POSITION: QUARTILES, DECILES AND PERCENTILES
•CALCULATE SPECIFIED MEASURE OF POSITION (E.G.
90TH PERCENTILE) OF A SET OF DATA.

QUARTILE 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 ≤ QI
•50% OF THE DATA HAS A VALUE ≤ Q2
•75% OF THE DATA HAS A VALUE ≤ Q3

•Q1 IS CALLED THE LOWER QUARTILE
•Q2 IS NOTHING BUT THE MEDIAN
•Q3 IS THE UPPER QUARTILE

MENDENHALL AND SINCICH METHOD
A METHOD OF FINDING THE QUARTILE VALUE.
1. CALCULATE THE POSITION OF THE LOWER QUARTILE
•IF L FALLS HALFWAY BETWEEN TWO INTEGERS, ROUND UP.
THE LTH ELEMENT IS THE LOWER QUARTILE VALUE (Q1).
Lower Quartile (L) = Position of Q1 =¼ ( n+1 )

2. CALCULATE THE POSITION OF THE LOWER QUARTILE
•IF U FALLS HALFWAY BETWEEN TWO INTEGERS, ROUND DOWN.
THE UTH ELEMENT IS THE UPPER QUARTILE VALUE (Q3).
Upper Quartile (U) = Position of Q3= ¾ ( n+1 )

EXAMPLE DATA SET
TO FIND Q1, LOCATE ITS POSITION USING THE FORMULA ¼ ( n+1 )
AND ROUND OFF TO THE NEAREST INTEGER
Position of Q1 =¼ ( n+1 )
=¼ ( 9+1 )
=2.5 (round up)
=3
{1, 3, 7, 7, 16, 21, 27, 30, 31} and n=9
THE LOWER QUARTILE VALUE Q1 IS THE 3RD DATA ELEMENT, SO Q1 = 7
7

EXAMPLE DATA SET
TO FIND Q3, LOCATE ITS POSITION USING THE FORMULA ¾ ( n+1 )
AND ROUND OFF TO THE NEAREST INTEGER
Position of Q3= ¾ ( n+1 )
= ¾ ( 9+1 )
=7.5 (round down)
=7
{1, 3, 7, 7, 16, 21, 27, 30, 31} and n=9
THE UPPER QUARTILE VALUE Q3 IS THE 7TH DATA ELEMENT, SO Q3 = 27
27

LINEAR INTERPOLATION
STEP A1. ARRANGE THE SCORES IN ASCENDING ORDER.
STEP A2. LOCATE THE POSITION OF THE SCORE IN THE DISTRIBUTION
Position of Q1=¼ ( n+1 )
STEP A3. IF THE RESULT IS A DECIMAL NUMBER, PROCEED FOR THE
INTERPOLATION

STEP B1. FIND THE DIFFERENCE BETWEEN THE TWO VALUES WHEREIN
QK IS SITUATED.
STEP B2. MULTIPLY THE RESULT IN STEP B1 BY THE
DECIMAL PART OBTAINED IN STEP A2
STEP B3. ADD THE RESULT IN STEP B2 TO THE SECOND SMALLER
NUMBER IN STEP B1

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

STEP A2. LOCATETHE POSITION OF THE SCORE IN THE DISTRIBUTION
1 3 7 7 16 21 27 30 31
Position of Q1=¼ ( n+1 )
=¼ ( 9+1 )
=2.5
LINEAR INTERPOLATION FOR QUARTER I
A1 A2 A3 B1 B2 B3

STEPA3. SINCE THE RESULT IS A DECIMAL NUMBER, PROCEED TO LINEAR INTERPOLATION
STEP B1. 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
A1 A2 A3 B1 B2 B3

STEPB2. MULTIPLY THE RESULT IN STEP B1 BY THE DECIMAL PART OBTAINED IN STEP A2
= 4 ( 0.5 )
= 2
STEP B3. ADD THE RESULT IN STEP B2 TO THE SECOND SMALLER NUMBER IN STEP B1
= 2 + 3
= 5
THEREFORE THE VALUE OF Q1 IS EQUAL TO 5
A1 A2 A3 B1 B2 B3

1 3 7 7 16 21 27 30 31
LINEAR INTERPOLATION FOR QUARTER 3
Position of Q3= ¾ ( n+1 )
= ¾ ( 9+1 )
=7.5

STEPA3. SINCE THE RESULT IS A DECIMAL NUMBER, PROCEED TO LINEAR INTERPOLATION
STEP B1. FIND THE DIFFERENCE BETWEEN THE TWO VALUES WHEREIN Q1 IS SITUATED.
1 3 7 7 16 21 27 30 31
7.5 POSITION
Q3 IS BETWEEN THE VALUES 27 AND 30, THEREFORE
= 30 – 27
= 3

STEPB2. MULTIPLY THE RESULT IN STEP B1 BY THE DECIMAL PART OBTAINED IN STEP A2
= 3 ( 0.5 )
= 1.5
STEP B3. ADD THE RESULT IN STEP B2 TO THE SECOND SMALLER NUMBER IN STEP B1
= 1.5 + 27
= 28.5
THEREFORE THE VALUE OF Q3 IS EQUAL TO 28.5

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

CALCULATING THE POSITION OF
DECILES
1. TO CALCULATE THE POSITION OF DECILES USE THE FORMULA
Position of Dk = ( n+1 )
k
10

CALCULATING THE POSITION OF DECILES
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

CALCULATING THE POSITION OF DECILES
1, 3, 4, 7, 7, 16, 21, 27, 30, 31, 31
Position of D7 =7/10 ( n+1 )
=7/10 (11+1 )
=8.4 ≈ 8
D7 is the 8TH ELEMENT
therefore D7 = 27

PERCENTILE FOR UNGROUPED DATA
•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.

CALCULATING THE POSITION OF PERCENTILE
A METHOD OF FINDING THE PERCENTILE VALUE.
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

CALCULATING THE POSITION OF PERCENTILE
A METHOD OF FINDING THE PERCENTILE VALUE.
1 3 4 7 7 16 21 27 30 31
Position of P58=58/100 ( n+1 )
=58/100 (10+1 )
=6.38 ≈ 6
P58 is the 6th element
Therefore, P58 = 16

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

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