MEASURES OF
RELATIVE POSITION
MEASURES OF RELATIVE POSITION
• are conversions of values, usually standardized
test scores, to show where a given value stands
in relation to other values of the same grouping.
• The commonly used measures of relative
position are quartiles, deciles and percentiles.
MEASURES OF RELATIVE POSITION
• QUARTILES
• DECILES
• PERCENTILES
QUARTILES
• These are the numbers Q1, Q2, and Q3 that
partition a ranked data set into four equal
groups.
Example:
Find the first, second, and third quartiles in the
givens set of data: 45, 46, 48, 51, 53, 54, 55, 58, 59
THE MEDIAN PROCEDURE IN FINDING
QUARTILES
1. Rank the data.
2. Find the median of the data. This is Q2.
Example: 45, 46, 48, 51, 53, 54, 55, 58, 59
NOTE: Median is the middlemost score for odd-numbered data. It is the average of the two
middlemost score for even-numbered data.
Median = Q2 Q2 = 53
THE MEDIAN PROCEDURE IN FINDING
QUARTILES
3. For Q1 , find the median of the values less than Q2
and for Q3 , find the median of the values greater than
Q2.
Example: 45, 46, 48, 51, 53, 54, 55, 58, 59
Q1 = (46+48)/2 Q3 = (55+58)/2
Q1 = 47 Q3 = 56.5
USING RANK IN FINDING QUARTILES
Example: 45, 46, 48, 51, 53, 54, 55, 58, 59
2.5th 5th 7.5th
Qk = k(n + 1)/4
Q1 = 1(9 + 1)/4 Q2 = 2(9 + 1)/4 Q3 = 3(9 + 1)/4
= 2.5th = 5th = 7.5th
Q1 = 47 Q2 = 53 Q3 = 56.5
FINDING QUARTILES
Example: Find the first, second, third quartiles of the
ages of 10 middle-management employees of AST
Company. The ages are 2, 5, 5, 8, 11, 12 ,19, 23, 40,
54 Q = 3(10+1)/4
3
= 8.25th
Answers: Q3 = 23 + 0.25(40-23)
= 27.25
Q1 = 5 Q2 = 11.5 Q3 = 27.25
DECILES
• These are the numbers that partition a ranked
data set into TEN equal groups.
Example:
Find the third, fifth, and eight deciles in the givens
set of data: 2, 5, 5, 8, 11, 12 ,19, 23, 40, 54
FINDING DECILES
Example:
Find the third, fifth, and eight deciles in the givens
set of data: 2, 5, 5, 8, 11, 12 ,19, 23, 40, 54
D3 = 3(10+1)/10 D5 = 5(10+1)/10 D8 = 8(10+1)/10
= 3.3rd = 5.5th = 8.8th
FINDING DECILES
Example: 2, 5, 5, 8, 11, 12 ,19, 23, 40, 54
D3 = 3.3rd D5 = 5.5th D8 = 8.8th
D3 = 5+0.3(8-5) D5 = 11+0.5(12-11) D8 = 23+0.8(40-23)
= 5.9 = 11.5 = 36.6
D3 = 5.9 D5 = 11.5 D8 = 36.6
FINDING DECILE
Example: Find the first, second, third deciles of the
ages of 9 middle-management employees of AST
Company. The ages are 45, 46, 48, 51, 53, 54, 55, 58,
and 59.
Answers:
D1 = 45 D2 = 46 D7 = 55
PERCENTILES
• Most standardized examination provide scores in terms of
percentiles.
• A value is called the pth percentile of a data set provided p%
of the data values are less than x.
• PERCENTILE for a GIVEN DATA VALUE
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑥
Percentile of score x = ∙ 100
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠
PERCENTILES
Examples:
1. On a reading examination given to 900 students. Elaine’s score of 602 was higher
that the scores of 576 of the students who took the examination. What is the
percentile for Elaine’s score?
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑥
Percentile of score x = ∙ 100
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 602
Percentile of Elaine’s score = ∙ 100
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠
576 Elaine’s score of 602 places
Percentile of Elaine’s score = ∙ 100
900
her at 64th percentile.
= 64
PERCENTILES
Examples:
2. Find P25, P50 and P75 in the givens set of data: 2, 5, 5,
8, 11, 12 ,19, 23, 40, 54
P25 = 25(10+1)/100 P50 = 50(10+1)/100 P75 = 75(10+1)/100
= 2.75th = 5.5th = 8.25th
FINDING PERCENTILES
Example: 2, 5, 5, 8, 11, 12 ,19, 23, 40, 54
P25 = 2.75th P50 = 5.5th P75 = 8.25th
P25 =5+0.75(5-5) P50 =11+0.5(12-11) P75 =23+0.25(40-23)
P25 = 5 P50 = 11.5 P75 = 27.25
FINDING DECILE
Example: Find the P25, P50 and P75 of the ages of 9
middle-management employees of AST Company. The
ages are 45, 46, 48, 51, 53, 54, 55, 58, and 59.
Answers:
P25 = 47 P50 = 53 P75 = 56.5
Z-SCORES
• Or standard score
• It is the number of standard deviations between a
data value and the mean.
𝑥 − 𝑥ҧ where: zx = z-score value
𝑧𝑥 = x = data value
𝑠
𝑥ҧ = sample mean
s = standard deviation
Z-SCORES
Example:
1. Raul has taken two tests in his chemistry class. He scored 72 on the first test for which the
mean of all scores was 65 and the standard deviation was 8. He received 60 on a second test,
for which the mean of all scores was 45 and the standard deviation was 12. In comparison to
the other students, did Raul do better on the first or the second?
1st Test
𝑧𝑥 =
𝑥 − 𝑥ҧ 2nd Test
𝑧𝑥 =
𝑥 − 𝑥ҧ Raul scored 0.875 standard deviation above
𝑠 𝑠 the mean on the first test and 1.25 standard
𝑧72 =
72 − 65
𝑧60 =
60 − 45 deviation above the mean on the second test.
8 12 These z-scores indicates that he scored
𝑧72 = 0.875 𝑧72 = 1.25
better in thhe 2nd test than in his 1st test.
Z-SCORES
Example:
1. A consumer group tested a sample of 100 light bulbs. It found that the mean life expectancy
of the bulbs was 842 h, with a standard deviation of 90. One particular light bulb from the
DuraBright Company had a z-score of 1.2.What was the life span of this light bulb?
𝑥 − 𝑥ҧ 𝑥 − 842
𝑧𝑥 = 1.2 =
𝑠 90
12(90) = x - 842
x = 842 + 108
x = 950
The light bulb has a life span of 950h.
EXERCISES:
A. Find Q3, D7, and P50 of the following data.
1. 89, 90, 85, 88, 91, 84, 78, 80, 88, 97
2. 92, 93, 89, 87, 88, 90, 91, 85, 86, 87
B. A data set has a mean of 75 and a standard deviation of 11.5. Find the z-scores for each of the
following:
1. 85
2. 50
3. 95
4. 75
5. 89
REFERENCES
Daligdig, Romeo. (2019). Mathematics in the modern wordl. Quezon City: Lorimar Publishing,
Inc.
_____. (2017). Mathematics in the modern world. Philippine Edition. Philippines: Rex Bookstore,
Inc.
Sirug, Winston S. (2018).Mathematics in the modern world.Manila: Mindshapers Co., Inc.
Medallon, Merlita C. & F. M. Calubaquib. (2018). Mathematics in the modern world. Manila:
Mindshapers Co., Inc.
• https://www.anthonypicciano.com/education-research-methods/measures-of-relative-position/
