The document provides examples of calculating z-scores from data with known means and standard deviations. It defines what a z-score is and how it indicates the number of standard deviations a score is from the mean. Examples include calculating z-scores for test scores, weights, lengths, and other measurements to determine which values fall within one or more standard deviations of the mean.

1. Working with z-scores
z by sandcastlematt

2. HOMEWORK
Four hundred people were surveyed No. of Videos Returned No. of Persons
to find how many videos they had 1 28
rented during the last month. 2 102
Determine the mean and median of the 3 160
frequency distribution shown below, 4 70
and draw a probability distribution 5 25
histogram. Also, determine the mode 6 13
by inspecting the frequency 7 0
distribution and the histogram. 8 2

3. HOMEWORK
The table shows the weights weight interval mean interval # of infants
(in pounds) of 125 newborn 3.5 to 4.5 4 4
infants. The first column 4.5 to 5.5 5 11
shows the weight interval, 5.5 to 6.5 6 19
the second column the 6.5 to 7.5 7 33
average weight within each 7.5 to 8.5 8 29
weight interval, and the third 8.5 to 9.5 9 17
column the number of 9.5 to 10.5 10 8
newborn infants at each 10.5 to 11.5 11 4
weight. Total 125
(a) Calculate the mean weight and standard deviation.
(b) Calculate the weight of an infant at one standard deviation below the
mean weight, and one standard deviation above the mean.
(c) Determine the number of infants whose weights are within one
standard deviation of the mean weight.
(d) What percent of the infants have weights that are within one
standard deviation of the mean weight?

4. HOMEWORK
The table shows the lengths in millimetres of 52 arrowheads.
16 16 17 17 18 18 18 18 19 20 20 21 21
21 22 22 22 23 23 23 24 24 25 25 25 26
26 26 26 27 27 27 27 27 28 28 28 28 29
30 30 30 30 30 30 31 33 33 34 35 39 40
(a) Calculate the mean length and the standard deviation.
(b) Determine the lengths of arrowheads one standard deviation below and
one standard deviation above the mean.
(c) How many arrowheads are within one standard deviation of
the mean?
(d) What percent of the arrowheads are within one standard deviation of the
mean length?

5. Standard Score AKA z-score: A z-score indicates how many standard
deviations a specific score is from the mean of a distribution.
For example, the following table shows some z-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
A z-score may also be described
by the following formula:
z = the z-score (standardized score)
x = a number in the distribution
μ = the population mean
σ = standard deviation for the population

6. 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?

7. The Canadian Armed Forces used to have a height requirement of 158 cm to
194 cm for men. The mean height of Canadian men at that time was 176 cm
with a standard deviation of 8 cm. What was the z-score range for allowable
heights?

8. Standardizing Two Sets of Scores
Two consumer groups,
Vancouver Halifax
one in Vancouver and
Cereal Brand Rating Cereal Brand Rating
one in Halifax, recently
A 1 P 25
tested five brands of
B 10 Q 35
breakfast cereal for taste
C 15 R 45
appeal. Each consumer
D 21 S 50
group used a different
E 28 T 70
rating system.
Use z-scores to determine which cereal has the higher taste appeal rating.

9. Using Z-Score, Mean, and Standard Deviation to Calculate the Real Score
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?

10. HOMEWORK
A survey was conducted at DMCI to determine the number of music CDs
each student owned. The results of the survey showed that the average
number of CDs per student was 73 with a standard deviation of 24. After the
scores were standardized, the people doing the survey discovered that DJ
Chunky had a z-score rating of 2.9. How many CDs does Chunky have?

11. HOMEWORK
The contents in the cans of several cases of soft drinks were tested. The
mean contents per can is 356 mL, and the standard deviation is 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?

12. HOMEWORK
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?