S519: Evaluation of
Information Systems
Social Statistics
Chapter 7: Are your curves
normal?
Last week
This week
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Why understanding probability is important?
What is normal curve
How to compute and interpret z scores.
What is probability?
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The chance of winning a lotter
The chance to get a head on one flip of a
coin
Determine the degree of confidence to state
a finding
Normal curve
Symmetrical: (bellshaped)
mean=median=mode
Asymptotic:
tail closer to the
horizontal axis, but
never touch.
Normal distribution
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Figure 7.4 – P157
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Almost 100% of the scores fall between (-3SD,
+3SD)
Around 34% of the scores fall between (0, 1SD)
Normal distribution
The distance between
contains
Range (if mean=100,
SD=10)
Mean and 1SD
34.13% of all cases
100-110
1SD and 2SD
13.59% of all cases
110-120
2SD and 3SD
2.15% of all cases
120-130
>3SD
0.13% of all cases
>130
Mean and -1SD
34.13% of all cases
90-100
-1SD and -2SD
13.59% of all cases
80-90
-2SD and -3SD
2.15% of all cases
70-80
< -3SD
0.13% of all cases
<70
Z score – standard score
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If you want to compare individuals in different
distributions
Z scores are comparable because they are
standardized in units of standard deviations.
Z score
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Standard score
X X
z
s
X: the individual score
X : the mean
S: standard deviation
Z score
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Z scores across different distributions are
comparable
Z scores represent a distance of z score
standard deviation from the mean
Raw score 12.8 (mean=12, SD=2)  z=+0.4
Raw score 64 (mean=58, SD=15)  z=+0.4
Equal distances from the mean
Excel for z score
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Standardize(x, mean, standard deviation)
(a2-average(a2:a11))/STDEV(a2:a11)
What z scores represent?
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Raw scores below the mean has negative z
scores
Raw scores above the mean has positive z
scores
Representing the number of standard
deviations from the mean
The more extreme the z score, the further it
is from the mean,
What z scores represent?
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84% of all the scores fall below a z score of
+1 (why?)
16% of all the scores fall above a z score of
+1 (why?)
This percentage represents the probability of
a certain score occurring, or an event
happening
If less than 5%, then this event is unlikely to
happen
Lab
Exercise
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In a normal distribution with a mean of 100
and a standard deviation of 10, what is the
probability that any one score will be 110 or
above?
16%
Table B.1 (s-p357)
Lab
If z is not integer
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Table B.1 (S-P357-358)
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Exercise
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The probability associated with z=1.38
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41.62% of all the cases in the distribution fall between
mean and 1.38 standard deviation,
About 92% falls below a 1.38 standard deviation
How and why?
Between two z scores
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What is the probability to fall between z score
of 1.5 and 2.5
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Z=1.5, 43.32%
Z=2.5, 49.38%
So around 6% of the all the cases of the
distribution fall between 1.5 and 2.5 standard
deviation.
Lab
Exercise
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What is the percentage for data to fall
between 110 and 125 with the distribution of
mean=100 and SD=10
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Answer: 15.25%
Excel
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NORMSDIST(z)
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To compute the probability associated with a
particular z score
Lab
Exercise
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The probability of a particular score occurring
between a z score of +1 and a z score of
+2.5
15%
What can we do with z score?
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Research hypothesis presents a statement of
the expected event
We use statistics to evaluate how likely that
event is.
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Z tests are reserved for populations
T tests are reserved for samples
Lab
Exercise
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Compute the z scores where mean=50 and
the standard deviation =5
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55
50
60
57.5
46
Lab
Exercise
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Based on a distribution of scores with
mean=75 and the standard deviation=6.38
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What is the probability of a score falling between
a raw score of 70 and 80?
What is the probability of a score falling above a
raw score of 80?
What is the probability of a score falling between
a raw score of 81 and 83?
What is the probability of a score falling below a
raw score of 63?