aron chpt 2
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Chapter 2 PowerPoint (revised)TRANSCRIPT
The Mean Variance Standard DeviationThe Mean Variance Standard Deviation
Chapter 2(part 1)
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Chapter OutlineChapter OutlineRepresentative ValuesVariability
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Measures of Central Measures of Central TendencyTendencyMeasures of central tendency
provide a typical score for a set of scores
Useful for making comparisons
◦Mean (average; ratio/interval data)◦Median (middle score; ratio/interval,
ordinal data)◦Mode (most frequent score;
ratio/interval, ordinal & nominal data)
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Which measure should you Which measure should you use to determine the use to determine the measure of central tendency measure of central tendency forfor
◦gender?◦age?◦high school rank?
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MeanMean
Average of a group of scores◦ sum of the scores divided by the number of
scoresMathematical formula for figuring the
mean:
M = ∑X or X = ∑X N N
M = mean∑ = sum (add up all of the scores following this symbol)X = scores in the distribution of the variable XN = number of scores in the distribution
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Example of Figuring the Example of Figuring the MeanMeanIf the scores for a particular
study were◦10, 5, 9, 8, 6, 5, 9, 8, 7, 6, 5, 6
Mean = 7 M = ∑X = 84 = 7
N 12
Let’s do this on the board
Copyright © 2011 by Pearson Education, Inc. All rights reserved
Copyright © 2011 by Pearson Education, Inc. All rights reserved
MedianMedianThe middle score when all of the scores are
lined up from lowest to highest◦ For an even number of scores, the median is the average
of the two middle scores.
To find the median:◦ Line up all the scores from lowest to highest.◦ Figure how many scores there are to the middle score by
adding 1 to the number of scores and dividing by 2.◦ Count up to the middle score or scores.
For this group of scores:◦ 9, 5, 7, 5, 6, 10, 8, 6, 5, 6, 9
Median = 6
Let’s do this on the board
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ModeModeThe most common single value in a
distributionThe value with the largest frequency in a
frequency table◦ the high point or peak of a distribution’s
histogramUsual way of describing the representative
value for a nominal variable ◦ rarely used with numerical variables
red, orange, blue, green, red, orange, yellow, red
Mode = redCopyright © 2011 by Pearson Education, Inc. All rights reserved
Types of DistributionsTypes of DistributionsIf the distribution is unimodal and
perfectly symmetrical, the mean, median and mode are the same.
Types of DistributionsTypes of DistributionsIf the distribution is unimodal but
skewed, the mean, median and mode will not be the same.
Comparing Representative Comparing Representative ValuesValuesThe median is better than the
mean or mode as a representative value when a few extreme scores would strongly affect the mean but not the median.◦An outlier is an extreme score that
can make the mean unrepresentative of most of the scores.
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Comparing Representative Comparing Representative ValuesValues
Student Salary
John $7,000
Aaron $5,000
Carrie $7,500
Maddie $2,500
Steve $750,000
Ashley $3,000
Brad $4,500
Sydney $3,000
Which measure of central tendency is most appropriate?
Comparing Representative Comparing Representative ValuesValuesYour turn
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9, 5, 7, 5, 6, 10, 8, 6, 5, 6, 9, 29
How Are You Doing?How Are You Doing?Find the mean, median, and
mode for the following scores:◦1, 4, 3, 2, 10, 2, 1, 3, 2, 4, 3, 2, 4, 1,
3 Which one of the above scores
would be considered an outlier?
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How Are You Doing?How Are You Doing?Find the mean, median, and
mode for the following scores:◦1, 4, 3, 2, 10, 2, 1, 3, 2, 4, 3, 2, 4, 1,
3 Mean = 3 Median = 3 Mode = 2 & 3 (bimodal)
Which one of the above scores would be considered an outlier?
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VariabilityVariabilityHow spread out the scores are in
a distribution◦In other words, how similar are the
scores in a particular set of scores?
Statistics Exam Scores
Classroom A Classroom B
95 80
63 82
82 81
90 83
75 79
M = 81 M = 81
Measures of VariabilityMeasures of VariabilityMeasures of variability describe
the differences among a set of scores◦
Range◦Variance◦Standard Deviation
RangeRangeSimplest measure of variability is the
range.◦ Range is the highest score in a distribution (H)
minus the lowest score in the distribution (L)
Classroom A Range = 95-63 = 32Classroom B Range = 83-79 = 4
Statistics Exam Scores
Classroom A Classroom B
95 80
63 82
82 81
90 83
75 79
Potential Problems with Potential Problems with RangeRangeRange becomes a problem when there are extreme scores.
Sales Representative A Range = $15,000 - $1,000 = $14,000Sales Representative B Range = $15,000 - $1,000 = $14,000
Sales Commissions
Sales Representative A
Sales Representative B
$15,000 $14,000
$1,000 $15,000
$2,500 $1,000
$3,000 $11,500
VarianceVarianceA better measure of variability is the
VarianceOne logical way to determine how these
scores vary from one another is to find how far each individual statistics exam score deviates from the mean of 81.
Classroom A
95
63
82
90
75
M = 81
95 - 81 = 14 points
63 - 81 = -18 points
82 - 81 = 1 point
90 - 81 = 9 points
75 - 81 = -6 points
0
0/5 = 0
VarianceVariance
We obtained 0 variability. We know that the variability is greater than zero. What went wrong?
Anytime you add up any deviation scores in a distribution, you will always obtain a variability score of 0.
Classroom A
95
63
82
90
75
M = 81
95 - 81 = 14 points
63 - 81 = -18 points
82 - 81 = 1 point
90 - 81 = 9 points
75 - 81 = -6 points
0
0/5 = 0
VarianceVarianceOne way to fix the problem is to get rid of
the negative scores.Can transform the data by squaring all the
scores.95 - 81 = 14
63 - 81 = -18
82 - 81 = 1
90 - 81 = 9
75 - 81 = -6
VarianceVarianceOne way to fix the problem is to get rid of
the negative scores.Can transform the data by squaring all the
scores.
VarianceVarianceThe average deviation should tell us the
overall variability of the set of scores.
Instead of using the actual N for denominator, we will subtract 1 from N.
Not in text
Formulas for the VarianceFormulas for the VarianceVariance:
◦SD2 = ∑(X-M)2 or S2 = ∑(X-M)2
N-1 N-1
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VarianceVariance
Measure of how spread out a set of scores are◦ average of the squared deviations from the mean
To calculate the variance of a distribution:◦ Find the deviation score for each score.
Subtract the mean from each score.◦ Find the squared deviation score for each score.
Square each of these deviation scores.◦ Find the sum of squared deviations.
Add up the squared deviation scores to get the sum of squared deviations.
◦ Find the average of the squared deviations. Divide the sum of squared deviations by the number of scores
minus 1 to get the average of the squared deviations.
Let’s do Let’s do classroom Bclassroom B
◦SD2 = ∑(X-M)2
N-1
Classroom B
80
82
81
83
79
M = 81
Still not doneStill not doneMust transform data back to
being “non-squared”
Classroom A √ 159.4 = 12.63
Classroom B √ 2.5 = 1.58
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Standard DeviationStandard DeviationMost widely used way of
describing the spread of a group of scores◦the positive square root of the
variance◦the average amount the scores differ
from the meanTo calculate the standard
deviation:◦Figure the variance.◦Take the square root of the variance.
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Formulas for Standard Formulas for Standard DeviationDeviationStandard Deviation (SD):
◦√SD2 or √S2
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How Are You Doing?How Are You Doing?What do the variance and
standard deviation tell you about a distribution of scores?
What are the formulas for finding the variance and standard deviation of a group of scores?