Transcript Measures of Dispersion & The Standard Normal Distribution

Measures of Dispersion
&
The Standard Normal Distribution
2/5/07
The Semi-Interquartile Range (SIR)
• A measure of dispersion
obtained by finding the
difference between the
75th and 25th percentiles
and dividing by 2.
• Shortcomings
– Does not allow for precise
interpretation of a score
within a distribution
– Not used for inferential
statistics.
Q3  Q1
SIR 
2
Calculate the SIR
6, 7, 8, 9, 9, 9, 10, 11, 12
• Remember the steps for finding quartiles
– First, order the scores from least to greatest.
– Second, Add 1 to the sample size.
– Third, Multiply sample size by percentile to find location.
– Q1 = (10 + 1) * .25
– Q2 = (10 + 1) * .50
– Q3 = (10 + 1) * .75
» If the value obtained is a fraction take the average of
the two adjacent X values.
Q3  Q1
SIR 
2
Variance (second moment about
the mean)
• The Variance, s2, represents the amount of variability of the
data relative to their mean
• As shown below, the variance is the “average” of the
squared deviations of the observations about their mean
s
2
( x  x)


i
n 1
2
SS

n 1
• The Variance, s2, is the sample variance, and is used to
estimate the actual population variance, s 2
s 
2
2
(
x


)
 i
N
SS

N
Standard Deviation
• Considered the most useful index of variability.
– Can be interpreted in terms of the original metric
• It is a single number that represents the spread of a
distribution.
• If a distribution is normal, then the mean plus or minus 3
SD will encompass about 99% of all scores in the
distribution.
Definitional vs. Computational
• Definitional
– An equation that
defines a measure
• Computational
– An equation that
simplifies the
calculation of the
measure
s
2
( x  x)


s2 
2
i
n 1
n X 2  ( X ) 2
n(n  1)
Calculating the Standard Deviation
s s
2
Interpreting the standard deviation
• We can compare the standard deviations
of different samples to determine which
has the greatest dispersion.
– Example
• A spelling test given to third-grader children
10, 12, 12, 12, 13, 13, 14
xbar = 12.28
s = 1.25
• The same test given to second- through fourthgrade children.
2, 8, 9, 11, 15, 17, 20
xbar = 11.71
s = 6.10
• Interpreting the standard
deviation
– Remember
• Fifty Percent of All Scores
in a Normal Curve Fall on
Each Side of the Mean
Probabilities Under the Normal Curve
The shape of distributions
• Skew
– A statistic that describes
the degree of skew for a
distribution.
• 0 = no skew
– + or - .50 is sufficiently
symmetrical
• + value = + skew
• - value = - skew
• You are not expected to
calculate by hand.
– Be able to interpret
3
(
X

X
)

N
3
s
( X  X ) 1.5

[
]
N
Kurtosis
• Mesokurtic (normal)
– Around 3.00
• Platykurtic (flat)
– Less than 3.00
• Leptokurtic (peaked)
– Greater than 3.00
• You are not expected
to calculate by hand.
– Be able to interpret
4
(
X

X
)

N
4 
2
s
(
X

X
)

[
]2
N
The Standard Normal Distribution
• Z-scores
– A descriptive statistic
that represents the
distance between an
observed score and
the mean relative to
the standard deviation
xi  x
z
s
z
xi  
s
Standard Normal Distribution
• Z-scores
– Convert a distribution to:
• Have a mean = 0
• Have standard deviation = 1
– However, if the parent distribution is not
normal the calculated z-scores will not be
normally distributed.
Why do we calculate z-scores?
• To compare two different measures
– e.g., Math score to reading score, weight to
height.
• Area under the curve
– Can be used to calculate what proportion of
scores are between different scores or to
calculate what proportion of scores are
greater than or less than a particular score.
• Used to set cut score for screening instruments.
Class practice
6, 7, 8, 9, 9, 9, 10, 11, 12
Calculate z-scores for 8, 10, & 11.
What percentage of scores are greater than
10?
What percentage are less than 8?
What percentage are between 8 and 10?
Z-scores to raw scores
• If we want to know
what the raw score of
a score at a specific
%tile is we calculate
the raw using this
formula.
• With previous scores
what is the raw score
– 90%tile
– 60%tile
– 15%tile
x  z (s)  x
Transformation scores
• We can transform
scores to have a
mean and standard
deviation of our
choice.
• Why might we want to
do this?
x  z (s)  x
With our scores
• We want:
– Mean = 100
– s = 15
• Transform:
– 8 & 10.
x  z (s)  x
Key points about Standard Scores
• Standard scores use a common scale to indicate how an
individual compares to other individuals in a group.
• The simplest form of a standard score is a Z score.
• A Z score expresses how far a raw score is from the
mean in standard deviation units.
• Standard scores provide a better basis for comparing
performance on different measures than do raw scores.
• A Probability is a percent stated in decimal form and
refers to the likelihood of an event occurring.
• T scores are z scores expressed in a different form (z
score x 10 + 50).
Examples of Standard Scores