Statistics 10
Professor Esfandiari
Summer2008
Week 2 session 1
The objective of this handout is to:
•
Review the concept of standard deviation,
•
Discuss the effects of linear transformation on mean, variance, and standard
deviation,
•
Introduce you to standardization as an example of linear transformation,
•
Show you how standard scores (Z scores) enable you to compare scores that were
measured with different units,
•
Introduce you to the normal model and the standard normal curve,
•
Show you how to calculate and interpret Z,
•
Show you how to transform percentile ranks into Z scores and Z scores into raw
scores,
•
Show you how to check for normality through the 68%, 95%, and 99.7% rule,
•
Show you how to check for normality through the pplot,
•
Discuss Pplot for normal and skewed data, and
•
Warn you not to use the normal approximation for distributions that are not
normal.
Variance
=
the sum of the square of the average deviations from the mean divided by N –
1:
S
Y
2
=
(
Y

Y
)
2
N
 1
You subtract each score from the mean (Y –
Y
)
The ∑(Y –
Y
) = 0; some scores are higher and some lower than the mean and the
sum is zero
To make the sum nonzero, you square it
To make the sum independent of sample size , you divide by N 1. In the future
discussions, we will explain why we divide by N 1 rather than N.
Standard deviation
= The square root of variance
The measure of scatter needs to be in terms of the same unit as X, since we
squared the deviations around the mean to find the variance, we will take the
square root of the variance to find standard deviation which is the measure of
spread.
Linear transformation could be one of the following or a combination of them:
•
Adding a constant to all the scores,
•
Subtracting all the scores from a constant,
•
Multiplying all the scores by a constant, and
•
Dividing all the scores by a constant.
1
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Table 1. In the following table we will summarize the effect of the above transformations
on the mean, variance, and standard deviation, and then take a practical example:
Linear
transformation
Effect on the mean
Or Xbar
Effect on variance
Or S square
Effect on standard
deviation or S
Adding a constant to
all the scores
The constant will be
added to the mean.
Variance will not
change.
SD will not change.
Subtracting all the
scores from a
constant.
The constant will be
subtracted from the
mean.
Variance will not
change.
SD will not change.
Multiplying all the
scores by a constant
The mean will be
multiplied by the
constant.
Variance will be
multiplied by the
square of the
constant.
SD will be
multiplied by the
absolute value of the
constant.
Dividing all the
scores by a constant
The mean will be
divided by the
constant.
Variance will be
divided by the
square of the
constant.
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 Summer '08
 Ioudina
 Statistics, Normal Distribution, Standard Deviation, Variance, Mean

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