SOCI 1005 – Introductory Statistics for the Behavioural Sciences
Z – Score Worked Examples
Note 1: When doing Z-score questions, if the mean and standard deviation have not been given,
your first two steps are to calculate these values as they are necessary for finding the z-score of
any data point. If you are unsure about these two steps, please see the worked examples posted.
Note 2/VERY IMPORTANT: To find the proportion, probability and percentile related to any
raw score, you must first find the z-score of the raw score and then follow through with the other
steps.
PLEASE REFER TO THE “NORMAL DISTRIBUTION AND
MEASURES OF POSITION” HANDOUT BEFORE
ATTEMPTING!
The table below shows the weekly earnings of 10 part-time waitresses at a restaurant. Please
note that columns 1 and 2 of the table (shaded blue) show the waitresses and their
corresponding earnings. If the mean and standard deviation have not been provided to you,
you will be required to insert column 3 (the yellow shaded column) as part of calculating
the standard deviation.
Weekly earnings
x2
($) (x)
1
195
38025
2
250
62500
3
130
16900
4
145
21025
5
95
9025
6
200
40000
7
175
30625
8
170
28900
9
183
33489
10
200
40000
∑ = 1743
∑ = 320489
The mean (x̅) for the data set above = 174.30
The standard deviation (S) for = 43.06 (Note: S= 43.055  43.06 to 2 d.p)
(Please see the handouts for mean and standard deviation for steps in calculating the above
values)
Waitress
1
1. What proportion of waitresses earned less than $150?
Firstly, find the Z-score of 150.
Z=x–x
S
z of $150 = 150 – 174.30
43.06
z of $150 = -24.3
43.06
z of $150 = -0.564  -0.56 (2 d.p)
(Note: All Z-scores are rounded off to 2 decimal places)
Secondly, draw the normal curve and position the z-score you have calculated onto the curve
(the sign of the z-score, that is, positive or negative, indicates which side of the curve the zscore should be placed)
Thirdly, look at the 3rd column on the z table for z= 0.56 to find the corresponding
proportion.
(Note: Because you are looking for the proportion less than, you will look for the area to
the left of wherever the z-score lies. Do not forget to pay attention to the sign before the
z-score to know if it falls on the left or right hand side of the curve).
3rd Column = .2877
Z= -0.56
The proportion of waitresses who earned less than $150 = .2877
(Note: When asked to find proportion, your final answer will be a decimal.)
2
2. What proportion of waitresses earned above $150?
First, find the z-score of 150:
Z of $150 = -0.56 (found in part 1 of the question)
(Note: When asked to find proportion above, or more than x, you are required to find the
area to the right of the z-score calculated)
In the question below, the shaded area to the right of the z-score covers two areas: i. the area
between the z-score and the mean; ii. The half of the curve to the right of the mean)
Therefore, on the Z table, you must first look at the 2nd column for z = 0.56 to find the area
between the z and the mean: Area = .2123
Then, add 0.5 which is the area of the other half of the curve to the right of the mean:
.2123 + 0.5 = .7123
i) Area between z and the mean
ii) Half of the curve to the right of the mean
x̅
0.5
Z = -0.56
.2123
Proportion of waitresses who earned above $150 = .7123
3
3. What is the probability of a waitress earning above $200.00?
First, calculate the z-score for 200:
Z=x–x
S
z of $200 = 200 – 174.30
43.06
z of $200 = 25.7
43.06
z of $200 = 0.597  0.60 (2 d.p)
Secondly, draw the normal curve and position the calculated z-score. Pay attention to the
sign before the z-score to know if it falls on the left or right hand side of the curve.
Thirdly, look at the 3rd column on the z table for z= 0.60 to find the corresponding
proportion.
(Note: Because you are looking for the proportion more than, you will look for the area
to the right of wherever the z-score lies).
3rd Column = .2743
x̅
Z = 0.60
The proportion of waitresses who earned more than $200 = .2743
To find the probability, convert the proportion to a percentage by multiplying by 100
Probability of a waitress earning more than $200.00 = .2743 x 100 = 27.43%
(Note: When asked to find probability, your final answer will be a percentage.)
4
4. What is the proportion of waitresses earning less than $200.00?
First, find the z-score of 200:
Z of $200 = 0.60 (found in part 1 of the question)
(Note: When asked to find proportion below, or less than x, you are required to find the
area to the left of the z-score calculated)
In the question below, the shaded area to the right of the z-score covers two areas: i. the area
between the z-score and the mean; ii. The half of the curve to the right of the mean)
Therefore, on the Z table, you must first look at the 2nd column for z = 0.60 to find the area
between the z and the mean: Area = .2257
Then, add 0.5 which is the area of the other half of the curve to the right of the mean:
.2257 + 0.5 = .7257
ii) Half of the curve to the left of the mean
i) Area between z and the mean
x̅
Z = 0.60
0.5
0.2257
5
5. What is the probability of earning between $140 and $200?
First, find the z-scores of 140 and 200 using the steps shown in part 1 of the question:
Z of $140 = -0.796 -0.80 (2 d.p.)
Z of $200 = 0.596 0.60 (2 d.p.)
Secondly, draw the normal curve and correctly position the z-scores (both must be placed on the
same curve). Pay attention to sign of both z-scores!
Thirdly, look in column 2 of Table Z to the find the area between the z and the mean for both Z
scores and add the areas (see normal distribution handout for rules for finding areas between zscores) to find the total proportion between scores:
z = -0.8 = 0.2881 (2nd column)
z = 0.60 = 0.2257 (2nd column)
i) Area between z and the mean
Z = -0.8
0.2881
i) Area between z and the mean
x̅
Z = 0.60
0.2257
(Note: when you have added the two areas, what you have found is the proportion between.
The question asks for probability, therefore to convert from proportion to probability,
multiply by 100)
The probability of earning between $140 and $200 = 0.2881 + 0.2257 = 0.5138
= 0.5138 x 100
= 51.38%
6
6.
Approximately how many persons earned between $140 and $200?
First, find the proportion earning between $140 and $200 = 0.5138 (see part 3)
Then multiply the proportion found by the total number (n):
= 0.5138 x 10
= 5.138
Therefore, approximately five persons earned between $140 and $200 per week.
7. What is the proportion of the sample who earned between $180 and $225?
First find the z-scores:
Z of $180 = 0.13
Z of $225 = 1.177  1.18 (2 d.p)
Secondly, draw the normal curve and correctly position the two z-scores. Pay attention to the
sign of each z-score to ensure you place the z-scores correctly.
Thirdly, go to the 2nd column on the Z table for both Z scores to find the area:
Z 0.13 = 0.0517
Z 1.8 = 0.4641
In this case, subtract the smaller area from the bigger area (see normal distribution handout for
rules on finding the area between two scores) to find the proportion between:
= 0.4641 – 0.0517
= 0.4124
i) Area between mean and z=1.8
Area = 0.4641
i) Area between mean and z=0.13
Area = 0.0517
i) Area to be found
x̅
Z = 0.13
Z = 1.8
The proportion which earned between $180 and $225 is 0.4124
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8. What weekly wage must be earned to fall into the 95th percentile?
i.
95/100 = 0.95 (returns the percentile to a proportion)
ii.
0.95-0.5 = 0.45 (subtracts half the curve leaving only the area between mean and
z)
iii.
Go to the 2nd column of the Z table and find the area closest 0.45 without being
less = 0.4505
iv.
Read back to the 1st column to find the Z score which corresponds to the area
0.4505 = Z 1.65
(Note: steps i-iv allow you to find the z-score to plug into the formula below as the mean
and standard deviation are already known)
v.
Plug the Z score, mean and standard deviation into the following formula to find
for x:
x = x + (z . S)
x = 174.3 + (1.65 x 43.06)
x = 174.3 + 71.04
x = 245.34
A weekly wage of $245.34 must be earned to fall in the 95th percentile.
9. What weekly wage must be earned to fall into the 40th percentile?
i.
40/100 = 0.4 (converts percentile to a proportion)
ii.
0.5 – 0.4 = 0.1 (leaves only the area between the mean and z)
iii.
Go to the 2nd column of the Z table and find the area closest to 0.1 without being
less = 0.1026
iv.
Read back to the 1st column to find the Z score which corresponds to the areas
0.1026 = Z -0.26 (negative Z score as the percentile is below the 50th percentile and
therefore the z-score will be on the left side of the curve)
v.
Plug the z score, mean and standard deviation into the following formula to find
for x:
x = x + (z . S)
x = 174.3 + (-0.26 x 43.06)
x = 174.3 + (-11.19)
x = 174.3 – 11.19
x = 163.11
A weekly wage of $163.11 must be earned to fall into the 40th percentile.
8