Aron chpt 2

The Mean Variance Standard DeviationThe Mean Variance Standard Deviation

Chapter 2(part 1)

Copyright © 2011 by Pearson Education, Inc. All rights reserved

Chapter OutlineChapter OutlineRepresentative ValuesVariability


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)


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?


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


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



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


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.


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


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?


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?

10Copyright © 2011 by Pearson Education, Inc. All rights reserved

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


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


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.


Formulas for Standard Formulas for Standard DeviationDeviationStandard Deviation (SD):

◦√SD2 or √S2


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?

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Aron chpt 2