1.
Applied 40S Standard Scores (z-scores)
A standard score indicates how many standard deviations a specific score is from the
mean of a distribution.
For example, the following table shows some standard scores for a distribution with:
a mean of 150
a standard deviation of 10
actual
score
130 140 150 160 170 174 180
standard
score
-2.0 -1.0 0.0 1.0 2.0 2.4 3.0
The term used to describe the above standardized score is z-score. The z-score may also
be described by the following formula:
where:
z = the z-score (standardized score)
x = a number in the distribution
μ= the population mean
σ= standard deviation for the population
Example 1:
Student A from Parkland High and Student B from Metro Collegiate both had a final
math mark of 95 percent. The awards committee must select the student with the 'highest'
mark for the annual math award. The table shows the mean mark and standard deviation
for each school.
School Mean
Standard
Deviation
Parkland High 75 8
Metro Collegiate 77 6
Calculate the z-score for each student. Which student should receive the award?
2.
Student A:
Student B:
Student A's 95 percent mark was 2.5 standard deviations above the class mean at
Parkland High, and Student B's 95 percent mark was 3.0 standard deviations above the
class mean at Metro Collegiate. Therefore, Student B has a higher z-score, and should
receive the award.
Example 2:
Numerous packages of raisins were weighed. The mean mass was 1600 grams, and the
standard deviation was 40 grams. Trudy bought a package that had a z-score of -1.6.
What was the mass of Trudy's package of raisins?
For the formula,
z = -1.6
m = 1600
s = 40
x = ?
Use algebra to solve for 'x':
x = 1600 + (40)(-1.6)
x = 1536
Therefore, the mass of Trudy's package is 1536 grams.
3.
Assignment
1. The contents in the cans of several cases of soft drinks were tested. The mean
contents per can was 356 mL, and the standard deviation was 1.5 mL.
a. Two cans were randomly selected and tested. One can held 358 mL, and
the other can 352 mL. Calculate the z-score of each.
b. Two other cans had z-scores of -3 and 1.85. How many mL did each
contain?
2. North American women have a mean height of 161.5 cm and a standard deviation
of 6.3 cm.
a. A car designer designs car seats to fit women taller than 159.0 cm. What is
the z-score of a woman who is 159.0 cm tall?
b. The manufacturer designs the seats to fit women with a maximum z-score
of 2.8. How tall is a woman with a z-score of 2.8?
3. The following information concerning grades is posted on the bulletin board.
Test Grade Z-Score
55 -2
65 -1
75 0
85 1
95 2
a. What is the average test grade?
b. What test score has a z-score of -2.76?
4. Tammy and Jamey both applied for the same job. Tammy scored 80 on a
provincial aptitude test where the mean was 70 and the standard deviation was
4.2. Jamey scored 510 on the company exam where the mean was 490 and the
standard deviation was 10.3. Assuming the company uses these test results as the
only criteria for hiring new employees and that both tests are considered to be
equal by company officials, who might get the job? Explain your answer.
4.
5. Mary Reed recently entered college and has not decided on a major. She decides
to take an aptitude test in order to help her select a major. Her results, as well as
the results of the other candidates, are as follows:
Talent Mean
Standard
Deviation
Mary Reed's
Score
Writing 58 3.7 46
Acting 84 7.9 85
Medicine 37 2.8 44
Law 49 4.7 41
a. Rewrite each of Mary Reed's scores as z-scores.
b. In which area does she have the most talent?
c. In which area does she have the least talent?
Answers:
1. a) 1st
can 1.33, 2nd
can -2.67 b)1st
can 351.5mL, 2nd
can 358.8 mL
2. a) -0.40 b)179.1 cm 3. a) 75 b)47.4
4. Tammy 2.38, Jamey 1.94, Tammy should get the job.
5. a) writing -3.24, acting 0.13, medicine 2.50, law -1.70
b) medicine c) writing
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