1.
Introduction to Statistics for Built
Environment
Course Code: AED 1222
Compiled by
DEPARTMENT OF ARCHITECTURE AND ENVIRONMENTAL DESIGN (AED)
CENTRE FOR FOUNDATION STUDIES (CFS)
INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA
2.
Lecture 10
Normal Curve
Todayâ€™s Lecture:
ï‚§ Calculating the z-score
ï‚§ Interpreting the z-score
ï‚§ Areas under the normal curve
ï‚§ Skewness
3.
Z-Scores â€“ a Measure of Position
â€¢ The z -score is also called the standard score.
â€¢ It is an ordinary score converted so that it better describes
that score's place in its distribution.
â€¢ It denotes the number of standard deviations a data value is
above the mean (if +) or below the mean (if -).
â€¢ Why z-score?
â€“ As a scale.
â€“ Provides a generalized standard of comparison.
4.
Calculating the z-score
Z-score formula for sample data
Z-score formula for population data
Z =
s
x - x
Z = x - Âµ
s
This value is a
measure of
the distance
in standard
deviations of
a data value
from the
mean.
5.
5
Calculating Z â€“ Scores: an example
Mansur is interested to become a salesperson to gain basic
marketing skills.
During an interview, he took a sales skill test and scored 135.
Does this indicate he has the potential for sales?
If average score of this test is 100, his score (135) is above average.
But how much is above average? Slightly, moderately or
dramatically?
How high is Mansurâ€™s score as compared to anyone else who took
the same test?
6.
Given standard deviation is 25, convert Mansurâ€™s score to a z-
score.
Z = 1.4
Original score New score/Z-Score
Z =
x - Âµ
s
Z = 135 - 100
25
Calculating z-scores cont.
7.
7
The same case, continued
At the interview, Mansur noticed thereâ€™s another vacant
position in research analysis.
He thought why not having a try in case he does not make it in
the first interview. In a test to apply for this post, he scored 85.
The average score (mean) of this test is 60 and the standard
deviation is 10.
Calculating z-scores cont.
8.
Letâ€™s see Mansurâ€™s z-score in the second test>>
Z = x - Âµ
s
Z = 85 - 60
10
Z = 2.5
Original score New score/Z-Score
So Mansur scored 2.5 std. deviations above the mean in research
analysis test but he scored only 1.4 std. deviations above the
mean in sales skill test.
It seems that Mansurâ€™s inclinations is more towards research
analysis.
Calculating z-scores cont.
9.
Interpreting z-scores
z-score < 0
Is when a data value is less than the mean
z -score > 0
Is when a data value is greater than the mean
z -score = 0
Is when a data value is equal to the mean
10.
A normal distribution is a bell-shaped
distribution of data where the mean, median
and mode all coincide. Chebyshevâ€™s theorem
applies to any distribution regardless of its
shape. However, when a distribution is bell-
shaped, the following statements which make
up the empirical rule, are true.
11.
Approximately:
â€¢68% of the data values will fall within 1 sd. Of
the mean.
â€¢95% of the data values will fall within 2 sd. Of
the mean.
â€¢99.7% of the data values will fall within 3 sd. Of
the mean.
12.
For example:
The score on a national achievement exam have a mean
of 480 and a deviation of 90. If these scores are normally
distributed, then approximately 68% will fall between
390 and 570 (480+90=570 and 480-90=390)
480390 570
13.
Skewness
In probability theory and statistics, skewness is a
measure of the asymmetry of the probability
distribution of a real-valued random variable.
Mean Mode
14.
Positively skewed/ skewed to the right
If there are extreme values towards the positive
end of a distribution, the distribution is said to be
positively skewed. In a positively skewed
distribution, the mean is greater than the mode.
For example:
15.
Negatively skewed/ skewed to the left
A negatively skewed distribution, on the other
hand, has a mean which is less than the mode
because of the presence of extreme values at
the negative end of the distribution.
16.
Calculating skewness
There are a number of ways of measuring skewness:
Pearsonâ€™s coefficient of skewness = mean â€“ mode = 3(mean â€“ median)
Standard deviation Standard deviation
The value determines whether the data distribution is positively skewed or negatively
skewed. If the value is positive, it means the distribution is positively/rightly skewed.
If the value is negative, it means the distribution is negatively/ leftly skewed.
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