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Measures of Dispersion&
The Standard Normal Distribution
9/12/06
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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.
211 QQ
SIR
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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.
211 QQ
SIR
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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
1
)( 22
n
xxs i
• The Variance, s2, is the sample variance, and is used to estimate the actual population variance, 2
N
xi
22 )(
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Standard Deviation
• Considered the most useful index of variability.• 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.
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Definitional vs. Computational
• Definitional– An equation that
defines a measure
• Computational– An equation that
simplifies the calculation of the measure
1
)( 22
2
nN
XX
s
1
)( 22
n
xxs i
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Calculate the variance using the computational and definitional
formulas.• 6, 7, 8, 9, 9, 9, 10, 11, 12
1
)( 22
n
xxs i
1
)( 22
2
nn
XX
s
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Calculating the Standard Deviation
2ss
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• Interpreting the standard deviation– Remember
• Fifty Percent of All Scores in a Normal Curve Fall on Each Side of the Mean
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Probabilities Under the Normal Curve
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With our previous scores
• What score is one standard deviation above the mean?– Two standard deviations?– Three standard deviations?
• What score is one standard deviation below the mean?– Two standard deviations?– Three standard deviations?
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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 children10, 12, 12, 12, 13, 13, 14xbar = 12.28 s = 1.25
• The same test given to second- through fourth-grade children.
2, 8, 9, 11, 15, 17, 20xbar = 11.71 s = 6.10
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The shape of distributions
• Skew– A statistic that
describes the degree of skew for a distribution.
• 0 = no skew• + or - .50 is sufficiently
symmetrical
s
Medianxs
)(33^
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Kurtosis
• Mesokurtic (normal)– Around 3.00
• Platykurtic (flat)– Less than 3.00
• Leptokurtic (peaked)– Greater than 3.00 )(2
31090
13^4
PP
QQs
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From our previous scores
• Calculate the skew6, 7, 8, 9, 9, 9, 10, 11,
12
xbar = 9.00mdn = 9.00 s = 1.87
s
Medianxs
)(33^
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• Calculate Kurtosis
6, 7, 8, 9, 9, 9, 10, 11, 12
Q3 =10.5
Q1 = 7.5
P10 = 6
P90 = 12
)(23
1090
13^4
PP
QQs
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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
s
xxz
X
z
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Standard Normal Distribution
• Z-scores – Convert and 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.
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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.
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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?
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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. xszx )(
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Transformation scores
• We can transform scores to have a mean and standard deviation of our choice.
• Why might we want to do this?
xszx )(
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With our scores
• We want:– Mean = 100– s = 15
• Transform:– 8 & 10.
xszx )(