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Normal Distributions
1
Continuous Distribution
• For a discrete distribution, for example
Binomial distribution with n=5, and p=0.4, the
probability distribution is
x
0
1
2
3
4
5
f(x) 0.07776 0.2592 0.3456 0.2304 0.0768 0.01024
A probability histogram
0.3
0.2
P(x)
0.1
0.0
0
1
2
3
x
4
5
How to describe the distribution of a
continuous random variable?
• For continuous random variable, we also represent
probabilities by areas—not by areas of rectangles, but by
areas under continuous curves.
• For continuous random variables, the place of histograms will
be taken by continuous curves.
• Imagine a histogram with narrower and narrower classes.
Then we can get a curve by joining the top of the rectangles.
This continuous curve is called a probability density (or
probability distribution).
Continuous distributions
• For any x, P(X=x)=0. (For a continuous
distribution, the area under a point is 0.)
• Can’t use P(X=x) to describe the probability
distribution of X
• Instead, consider P(a≤X≤b)
Density function
0.20
0.15
0.00
• P(a≤X≤b) is the area
between a and b
0.05
0.10
y
• The area under the
curve is 1
0.25
• A curve f(x):
f(x) ≥ 0
0
2
4
6
x
8
10
0.00
0.05
0.10
y
0.15
0.20
0.25
P(2≤X≤4)= P(2≤X<4)= P(2<X<4)
0
2
4
6
x
8
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Properties Of Normal Curve
•
•
•
•
Normal curves are symmetrical.
Normal curves are unimodal.
Normal curves have a bell-shaped form.
Mean, median, and mode all have the same
value.
• Total area = 1
• Defined by mean and standard deviation
(centered at mean)
8
Percent of Values Within One
Standard Deviations
68.26% of Cases
9
Percent of Values Within Two
Standard Deviations
95.44% of Cases
10
Percent of Values Within Three
Standard Deviations
99.72% of Cases
11
Percent of Values Greater than
1 Standard Deviation
12
Data in Normal Distribution
(X  1S ) contains about 68% of the scores
(X  2S ) contains about 95% of the scores
(X  3S ) contains about 99% of the scores
13
Standard Scores (Z-scores)
• Expressed in standard deviations from mean
• There are many kinds of Standard Scores. The most
common standard score is the ‘z’ scores.
• A ‘z’ score states the number of standard deviations by
which the original score lies above or below the mean
of a normal curve.
z
x

14
Commonly used probabilities and z-scores:
Middle 90%: between -1.645 and +1.645
Middle 95%: between -1.96 and +1.96
Middle 99%: between -2.576 and +2.576
90th percentile: 1.28
95th percentile: 1.645
99th percentile: 2.33
15
Area When Score is Known
• For a normal distribution with mean of 100
and standard deviation of 20, what
proportion of cases fall below 80?
• ~16%
16
Calculator:
Normal cumulative density function: normalcdf(left bound,
right bound, mean, st. dev.)
(mean and st. dev. default to 0 and 1)
To find z-scores given the area in the left tail of a normal
distribution:
invNorm(area, mean, standard deviation)
17
Score When Area Is Known
• For a normal distribution with mean of 100 and
standard deviation of 20, find the score that
separates the upper 20% of the cases from the
lower 80%
• Answer = 116.8
18
Ways to Assess Normality
• Use graphs (dotplots,
boxplots, or histograms)
• Normal probability
(quantile) plot
Normal Probability (Quartile) plots
• The observation (x) is plotted against known
normal z-scores
• If the points on the quantile plot lie close to a
straight line, then the data is normally
distributed
• Deviations on the quantile plot indicate
nonnormal data
• Points far away from the plot indicate outliers
• Vertical stacks of points (repeated observations
of the same number) is called granularity
Are these approximately
distributed?
50 48 54 47 51 52
52 51 48 48 54 55
53 50 47 49 50 56
normally
46 53
57 45
53What
52is this
called?
Both the histogram & boxplot
are approximately
symmetrical, so these data
are approximately normal.
The normal probability
plot is approximately
linear, so these data are
approximately normal.