Download Education 793 Class Notes

Education 793 Class Notes
Normal Distribution
24 September 2003
Today’s agenda
• Class and lab announcements
• What questions do you have?
• The normal distribution
– Its properties
– Identifying area under the curve
2
Properties of the
normal distribution
• Symmetrical, with one mode, (mean, median
and mode are all equal) – the classic bellshaped curve
• Really a family of distributions with similar
shape, but varying in terms of two parameters
mean and standard deviations
• Of most use to us is the standard normal
distribution, with a mean of zero and a
standard deviation of 1
3
Standard scores
Transformation of raw scores to a
standard scale that reflects the position
of each score relative to the distribution
of all scores being considered
Standard score = Raw score - mean score
Standard deviation
z = X s- X
4
Standard score properties
1. Shape of distribution unchanged
2. Mean of z-score distribution equals
zero
3. Variance of z-score distribution equals
one
5
Calculating Standard Scores
Sum
Raw SAT
score
340
450
510
550
580
600
620
660
670
710
5,690
Deviation
score
Std.
Deviation
111.8
111.8
111.8
111.8
111.8
111.8
111.8
111.8
111.8
111.8
0
X
X
z= s
Z
0
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Graphing scores
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SAT Math scores
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200 300 400 500 600 700 800
Standardized
SAT Math scores
-3
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-2
-1
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0
1
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2
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Family Traditions
1. Unimodal, symmetrical, and bell-shaped
2. Continuous
3. Asymptotic
Standard Normal Distribution
8
Area under the
standard normal
Defined by mathematical equation, that indicates:
50% of the area falls below the mean
34% falls between the mean and one standard deviation above
16% falls beyond one standard deviation above the mean
9
Moving beyond eyeballing
• Direct calculation / calculators
• Table look-ups
10
Navigating a Standard Normal
Probability Table
11
Probability questions
about the normal distribution
What percent of a standard normal distribution falls between one
and two standard deviations below the mean?
What percent falls above three standard deviations above the
mean?
If there were 100,000 people in a sample, how many
would be expected to fall more than three standard
deviations above the mean on any normally distributed
characteristic?
What percent of the normal distribution falls below a
point .675 standard deviations above the mean?
What percent of the normal distribution falls above a point 1.96 standard
deviations above the mean?
12
Group exercise
• See handout
13
Some final points about the
normal distribution
Standard scores can be calculated for any distribution
of numerical scores. In short, if we can calculate
meaningful values for mean and standard deviation
we can calculate standard scores.
Standard scores, regardless of other factors (such as
shape, skewness, and kurtosis), reflect the position of
each score relative to the distribution of all scores
being considered.
We cannot, however, make precise statements about
percentages associated with certain regions of a
given distribution unless it represents a standard
normal curve.
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Next week
• Chapter 6 p. 145-179, Correlation
• Chapter 7 p. 181-204 Linear Regression
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