The document provides examples and explanations of how to calculate and use z-scores. Some key points: - A z-score indicates how many standard deviations a score is from the mean of a distribution. - It can be calculated using the formula: z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation. - Examples are given of z-scores for different data sets and distributions. - Z-scores can be used to standardize scores from different data sets or scales onto a common scale. - Several word problems demonstrate calculating z-scores and using them to determine actual values or scores.

1. Working with z-scores
z by sandcastlematt

2. Let's apply what we've learned ...
mark interval mark # of students
The frequency distribution table 29 to 37 33 1
at right shows the midterm marks 38 to 46 42 4
47 to 55 51 12
of 85 Grade 12 math students.
56 to 64 60 18
The first column shows the mark
65 to 73 69 24
interval, the second column the
74 to 82 78 16
average mark within each mark 83 to 91 87 7
interval, and the third column the 92 to 100 96 3
number of students at each mark. Total 85
This is the
(a) Calculate the mean and std. dev. to two decimal places. correct way
Mean = 66.99 σ = 13.27 to do this.
(b) Calculate the number of students that have marks within one
std. dev. of the mean.
58 or 70
(c) What percent of students have marks within one std. dev. of
the mean?
68% or 82%

3. HOMEWORK
An experiment was performed to determine the approximate mass of
a penny. Three hundred pennies were weighed, and the weights
were recorded in the frequency distribution table shown below.
Mass (grams) 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4
Frequency 2 4 34 71 94 74 17 4
Determine the mean, median, mode, range, and standard deviation
of the data. Create a histogram that shows the frequencies of
different masses of this set of pennies.

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

5. HOMEWORK weight interval mean interval # of infants
The table shows the
3.5 to 4.5 4 4
weights (in pounds) of 125 4.5 to 5.5 5 11
newborn infants. The first 5.5 to 6.5 6 19
6.5 to 7.5 7 33
column shows the weight
7.5 to 8.5 8 29
interval, the second column
8.5 to 9.5 9 17
the average weight within 9.5 to 10.5 10 8
each weight interval, and 10.5 to 11.5 11 4
the third column the TOTAL 125
number of newborn infants
at each weight.
(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?

6. 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

7. z-scores
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

8. 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?

9. 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?

10. 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 C 15 R 45
D 21 S 50
taste appeal. Each
E 28 T 70
consumer group used
a different rating
HOMEWORK
system.
Use z-scores to determine which cereal has the higher taste
appeal rating.

11. 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
HOMEWORK
(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?

12. 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
HOMEWORK
package of raisins?

13. 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?