Frequency Distributions
FSE 200
Why Probability?
• Basis for the normal curve
– Provides basis for understanding
probability of a possible outcome
• Basis for determining the degree of
confidence that an outcome is “true”
– Example: Are changes in student
scores due to a particular intervention
that took place or by chance alone?
The Normal Curve
(a.k.a. the Bell-Shaped Curve)
• Visual representation of a distribution of
scores
• Three characteristics…
– Mean, median, and mode are equal to one
another
– Perfectly symmetrical about the mean
– Tails are asymptotic (get closer to
horizontal axis but never touch)
The Normal Curve
The normal or bell shaped curve
Hey, That’s Not Normal!
• In general, many events occur in the middle
of a distribution with a few on each end.
How scores can be distributed
More Normal Curve 101
• For all normal distributions…
– Almost 100% of scores will fit between –3 and +3
standard deviations from the mean
– So…distributions can be compared
– Between different points on the x-axis, a certain
percentage of cases will occur
What’s Under the Curve?
Distribution of cases under the normal curve
The z Score
• A standard score that is the result of
dividing the amount that a raw score
differs from the mean of the distribution by
the standard deviation.
(X  X )
z
s
• What about those symbols?
The z Score
• Scores below the mean are negative (left
of the mean), and those above are positive
(right of the mean)
• A z score is the number of standard
deviations from the mean
• z scores across different distributions are
comparable
Using Excel to Compute z Scores
What z Scores Represent
• The areas of the curve that are covered by
different z scores also represent the
probability of a certain score occurring.
• So try this one…
– In a distribution with a mean of 50 and a
standard deviation of 10, what is the
probability that one score will be 70 or above?
How to Figure Out the Probability
How to Figure Out the Probability
What z Scores Really Represent
• Knowing the probability that a z score will
occur can help you determine how
extreme a z score you can expect before
determining that a factor other than
chance produced the outcome
• Keep in mind…z scores are typically
reserved for populations. 
Example
• Normal Distribution
– Mean = 0
– Standard Deviation = 1
• Answer the following:
A. What is the probability that a randomly selected
value will be less than -0.1?
B. What is the probability that a randomly selected
value will be greater than 0.6?
C. What is the probability that a randomly selected
value will be between 0.6 and 0.9?
Question A
Z-Score = (-0.1-0)/1 = -0.1
From chart on Slide 12, the area between the mean and the z-score is 3.98
Therefore, the probability is equal to 100-50-3.98 = 46.02% or 0.4602
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-6
-4
-2
0
2
4
6
Question B
Z-Score = (0.6-0)/1 = 0.6
From chart on Slide 12, the area between the mean and the z-score is 22.24.
Therefore, the probability is equal to 100-50-22.24 = 27.76% or 0.2776
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-6
-4
-2
0
2
4
6
Question C
We have to compute two z-scores for this problem since we are finding the probability
in between two values.
We know that the probability of a value being above 0.6 is 27.76%.
What is the probability of the value being above 0.9?
Z-Score = (0.9-0)/1 = 0.9
Pick value from Table on Slide 13 which yields 31.59%.
The odds of the value being above 0.9 is 100-50-31.59 = 18.41% or 0.1814
To figure the odds of being between:
0.45
Subtract the two values.
0.4
0.35
27.76-18.41 = 9.35% or 0.0935
0.3
0.25
0.2
This is essentially the area shaded in the
diagram
0.15
0.1
0.05
0
-6
-4
-2
0
2
4
6
Acknowledgement
The majority of the content of these slides were
from the Sage Instructor Resources Website