Measures of Relative Position
Measures of Relative Position
Measures of Relative Position
RELATIVE POSITION
MEASURES OF RELATIVE POSITION
Definition: - 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 z-score for a given data value x is the number of standard deviations
that x is above or below the mean of the data. The following formulas show
how to calculate the z-score for a data value x in a population and in a
sample.
Population:
Sample:
Example 9. COMPARE Z-SCORES
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 a 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 test or the
second test?
Solution:
Raul scored 0.875 standard deviation above the mean on
Given: First
test: x = 72, 65, standard
deviation (s) =8 the first test and 1.25 standard deviation above the mean on the
second test. These z-scores indicate that, in comparison to his
Second test: x = 60, 45, standard classmates, Raul scored better on the second test than he did
deviation (s) =12
on the first test.
First test Second test
ANSWER!!
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 Dura Bright Company had a z-score of 1.2.
what was the light span of this light bulb?
Solution:
Given: mean ( ) =842 , standard deviation (s) =90, z-score (z) = 1.2
Using the formula: If the standard deviation is missing:
Using the formula,
𝑧 = 𝑥 − ´𝑥
1.2= 𝑥 − 842
𝑥
𝑠
90
108=𝑥
−842 1.2
𝑠=108
950 = x 1.2
𝑠 = 108
1.2 1.2
The light bulb had a life span of 950 h.
𝒔=90
ANSWER!!
Solution:
a. By definition, the median is the 50th percentile. Therefore, 50% of
the physical therapists earned more than $57,720 per year.
b. Because $71,500 is the 85th percentile, 85% of all physical
therapists made less than $71,500.
c. From parts a and b, 85% - 50%= 35% of the physical therapists
earned between $57,720 and $71,500.
ANSWER!!
On a reading examination given to 900 students, Elaine’s score of 602 was higher
than a score of 576 of students who took the examination. What is the percentile for
Elaine’s score?
Solution:
Percent
ile =
Percent
ile =
Percent
ile =
Elaine’s score of 602 places her at the 64th percentile.
ANSWER!!
Quartiles
The three numbers Q1, Q2, and Q3, that partition a ranked data set into four
(approximately) equal groups are called the quartiles of the data.
For instance, for the data set below, the values Q1= 11,Q2 = 29 and Q3 = 104 are the
quartiles of the data.
2 5 5 8 11 12 19 22 23 29 31 45 83 91 104 159 181 312 354
Q1
Q2
Q3
The following table lists the calories per 100 milliliters of 25 popular sodas. Find the
quartiles for the data.
Step 2: The median of these 25 data values has a rank of 13. Thus the
median is 43. The second quartile Q2 is the median of the data, so Q2 = 43.
Step 3: There are 12 data values less than the median and 12 data values
greater than the median. The first quartile is the median of the data values less
than the median. Thus, Q1 is the mean of the data values with ranks of 6 and 7.
39 + 39
𝑄 1= =39
2
The third quartile is the median of the data values greater than the median. Thus, Q3
is the mean of the data values with ranks of 19 and 20.
50+53
𝑄 3= =51.5
2
ANSWER!!
PREPARED BY
MS. EDESA L. JARABEJO