Practice Page 65 –2.1 Positive Skew Note Slides online

Practice

• Page 65– 2.1

1 2.0 2.0 2.01 2.0 2.0 4.01 2.0 2.0 6.03 6.0 6.0 12.04 8.0 8.0 20.06 12.0 12.0 32.010 20.0 20.0 52.07 14.0 14.0 66.07 14.0 14.0 80.03 6.0 6.0 86.02 4.0 4.0 90.02 4.0 4.0 94.01 2.0 2.0 96.01 2.0 2.0 98.01 2.0 2.0 100.050 100.0 100.050 100.0

10.0011.0012.0015.0016.0017.0018.0019.0020.0021.0022.0023.0024.0031.0040.00Total

Valid

Total

Frequency PercentValidPercent

CumulativePercent

VAR00001

• Positive Skew

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40 45

Note

• Slides online

12 13 9 15 18 18 13 7

16 8 22 15 19 15 9 14

16 10 6 16 14 20 17 13

12 15 15 17 16 11 24 20

19 15 14 24 22 17 21 14

Neuroticism Score

f Neuroticism Score

f

24 2 14 4

23 0 13 3

22 2 12 2

21 1 11 1

20 2 10 1

19 2 9 2

18 2 8 1

17 3 7 1

16 4 6 1

15 6 N = 40

Class Interval

Midpoint f Cumulative f

25-23 24 2 40

22-20 21 5 38

19-17 18 7 33

16-14 15 14 26

13-11 12 6 12

10-8 9 4 6

7-5 6 2 2

Histogram

0

24

68

10

1214

16

6 9 12 15 18 21 24

Neuroticism

Fre

qu

ency

Frequency Polygon

0

24

68

10

1214

16

6 9 12 15 18 21 24

Neuroticism

Fre

qu

ency

Stem-and-Leaf Display

• Stem-and-leaf display with a bigger data set

• Note: The stem-and-leaf is like a histogram turned sideways!

Describing Distributions

• Bell-shaped distribution



0

2

4

6

8

10

12

0 2 4 6 8 10 12 14

Happiness Score

Fre

quen

cy

Kurtosis

• The relative concentration of scores in the center of the distribution

• Mesokurtic

Kurtosis


• Platykurtic

Kurtosis


• Leptokurtic

12 13 9 15 18 18 13 7

16 8 22 15 19 15 9 14

16 10 6 16 14 20 17 13

12 15 15 17 16 11 24 20

19 15 14 24 22 17 21 14

Measures of Central Tendency

• Give one value that represents an entire group of scores

• Mean

• Median

• Mode

Mean

• On your tests you get: 70%, 80%, 80%, 90%

• The mean is 80%

• You know how to do this!

Mean

X

XN

= the mean

= an instruction to add (sigma) “the sum of”

= a score

= the number of scores

Practice

• What is the mean of:

5, 8, 6, 3, 2, 2, 9

Mean = 35 / 7 = 5

10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5

Mean = 107.8 / 8 = 13.48

Characteristics of the mean

• If the mean is subtracted from each score and the differences added, the sum will equal zero


70, 80, 80, 90 Mean = 80

70 - 80 = -1080 - 80 = 080 - 80 = 090 - 80 = 10 = 0


• The mean is the point about which the sum of the squared deviations is minimized


70, 80, 80, 90 Mean = 80

70 - 80 = -102= 10080 - 80 = 02 = 080 - 80 = 02 = 090 - 80 = 102 = 100 = 200

Population vs. Sample

_x = The mean of a sample

= The mean of a population

*They are both calculated the same way!

Population vs. Sample

The Median

• The point that divides a distribution of scores into two parts that are equal in size

The Median

10, 5, 13, 6, 14, 17, 2, 6, 9

2, 5, 6, 6, 9, 10, 13, 14, 17

The Median

10, 5, 13, 6, 14, 17, 2, 6, 9

2, 5, 6, 6, 9 10, 13, 14, 17

The Median

5, 8, 9, 15, 20, 25, 50

The Median

5, 8, 9, 15 20, 25, 50

The Median

5, 8, 9, 12, 15, 18, 22, 30, 32, 40

The Median

5, 8, 9, 12, 15, 18, 22, 30, 32, 40

The Median

5, 8, 9, 12,15, 18,22, 30, 32, 40

16.5

Practice

• What is the median of:

5, 8, 6, 3, 2, 2, 9

2, 2, 3, 5, 6, 8, 9

(7+1) / 2 = 4

Median = 5

Practice

• What is the median of:

10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5

10.5, 10.5, 11.6, 11.9, 12.9, 14.7, 15.5, 20.2

(8+1) / 2 = 4.5

(11.9 + 12.9) / 2 = 12.4

Median = 12.4

The Mode

• The most frequently occurring score

5, 6, 8, 9, 10, 10, 10, 12, 14, 14

Mode = 10

Practice

• What is the mode of:

5, 8, 6, 3, 2, 2, 9

Mode = 2

10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5

Mode = 10.5

Determining Skewness


• For example:

Mean = 4

Median = 10

Mean Median


• For example:

Mean = 10

Median = 4

Median Mean


• Mean < Median = Negative Skew

• Mean > Median = Positive Skew

• Mean = Median = No Skew

Which should you use?

Mean Median Mode

Nominal NO NO OK

Ordinal NO OK OK

Interval OK OK OK

Ratio OK OK OK

Happiness Score

f

15 9

14 12

13 10

11 2

10 3

5 1

2 1

What is the mean, median, and mode?

Happiness Score

f score(f)

15 9 135

14 12 168

13 10 130

11 2 22

10 3 30

5 1 5

2 1 2

Mean = 492 / 38 = 12.95

Happiness Score

f

15 9

14 12

13 10

11 2

10 3

5 1

2 1

Mode = 14

Happiness Score

f

15 9

14 12

13 10

11 2

10 3

5 1

2 1

(38 + 1)/ 2 = 19.5

Median = 14

The Test Scores of 3 Students

• Joe = 78, 60, 92, 80, 80

• Bob = 47, 100, 98, 45, 100

• Mary = 78, 79, 77, 78, 78


• Joe = 78, 60, 92, 80, 80– Mean = 78

• Bob = 47, 100, 98, 45, 100– Mean = 78

• Mary = 78, 79, 77, 78, 78– Mean = 78

Variability

• Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together

Range

• The highest score minus the lowest score

• Joe = 78, 60, 92, 80, 80

• Range = 92 - 60 = 32

Range


• Bob = 47, 100, 98, 45, 100

• Range = 100 - 45 = 55

Range


• Mary = 78, 79, 77, 78, 78

• Range = 79 - 77 = 2


• Joe = 78, 60, 92, 80, 80– Mean = 78 Range = 32–

• Bob = 47, 100, 98, 45, 100– Mean = 78 Range = 55

• Mary = 78, 79, 77, 78, 78– Mean = 78 Range = 2

Range

• In general - the larger the range score, the more variance

• Pro: Easy to calculate

• Con: The range only depends on two extreme scores; can be misleading

Range

20, 62, 54, 32, 28, 44, 72, 69, 50

1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,5, 5, 5, 5, 5, 99

Range

20, 62, 54, 32, 28, 44, 72, 69, 50

Range = 49

1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,5, 5, 5, 5, 5, 99

Range = 98!!

Interquartile Range

• The range of scores that make up the middle 50 percent of the distribution

• Need to find the 25th percentile score and the 75th percentile score

Interquartile Range

50%

Interquartile Range

.25 (N) = The location of the 25th percentile score counting from the bottom

.25 (N) = The location of the 75th percentile score counting from the top

N = the number of cases

*If the answer is not even simply average

*Similar to how you found the median!!

Interquartile Range

IQR = 75th percentile - 25th percentile

Interquartile Range

2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Interquartile Range

2, 5, 66, 10, 14, 16, 29, 40, 56, 62, 82, 99

.25 (12) = 3

Counting 3 from the bottom the 25th percentile score = 6

Interquartile Range

2, 5, 6, 10, 14, 16, 29, 40, 56, 62,62, 82, 99

.25 (12) = 3

Counting 3 from the top the 75th percentile score = 62

Interquartile Range

2, 5, 6,6, 10, 14, 16, 29, 40, 56, 62,62, 82, 99

IQR = 75th percentile - 25th percentile

56 = 62 - 6

Score f

80 4

72 5

70 9

69 10

68 6

54 5

52 1N = 40

Practice

Score f

80 4

72 5

70 9

69 10

68 6

54 5

52 1

N = 40

70 - 68 = 240(.25) = 10

Practice

• Find the range for:

• 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22

• 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2

• 102.58, 51.25, 58.00, 96.34, 54.43

Practice

• Find the range for:

• 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22– Range =76 - 4 = 72

• 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2– Range = 78.6 - 3.2 = 75.4

• 102.58, 51.25, 58.00, 96.34, 54.43– Range = 102.58 - 51.25 = 51.33

Practice

• Find the interquartile range for:

• 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22

• 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2

• 102.58, 51.25, 58.00, 96.34, 54.43

Practice


• 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22

• 4, 4, 8, 10, 15, 22, 25, 43, 56, 56, 64, 76

• (12) .25 = 3

• 56 - 8 = 48

Practice


• 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2

• 3.2, 8.5, 12.9, 59.5, 68.2, 75.2, 78.3, 78.6• (8).25 = 2• 78.3 - 8.5 = 69.8

Practice• Find the interquartile range for:

• 102.58, 51.25, 58.00, 96.34, 54.43

• 51.25, 54.43, 58.00, 96.34, 102.58• (5).25 = 1.25• (51.25+54.43)/2 = 52.84• (96.34+102.58)/2 = 99.46• 99.46-52.84 = 46.62

Boxplots

• The boxplot graphically displays three different characteristics of the distribution– Extreme scores– Interquartile range– Median

Boxplot

100N =

NEUR

40

30

20

10

0

Boxplot

100N =

NEUR

40

30

20

10

0

Interquartile range

25th - 75th percentile

Boxplot

100N =

NEUR

40

30

20

10

0

Extreme Scores

Boxplot

100N =

NEUR

40

30

20

10

0

Median

Boxplot

100N =

NEUR

40

30

20

10

0

Skew -- Look at the “whiskers” to determine if the distribution is skewed

Create a boxplot

• Create a boxplot with this data set

2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Create a boxplot


2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Median =

25th =

75th =

Lowest =

Highest =

Create a boxplot


2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Median = 22.5

25th = 6

75th = 62

Lowest = 2

Highest = 99

12N =

VAR00004

120

100

80

60

40

20

0

-20

Neuroticism

130130N =

MNEURCNEUR

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

Extraversion

130130N =

MEXTRACEXTRA

5.0

4.5

4.0

3.5

3.0

2.5

2.0

106120

15272

Conscientiousness

128128N =

MCONCCON

5

4

3

2

1

0

99999N =

VAR00006VAR00005VAR00004VAR00003VAR00002

120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution has a positive skew?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution has a negative skew?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution is most compact?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution has a median close to 25?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution is most symmetrical?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution has has the largest range?

Review

Measures of central tendency– Mean – Median– Mode

• Measures of variability– Range– IQR

Variability

• Range

• IQR

• Problem with range and IQR– variability is still measured with only two

numbers!

Deviation Score Formula

• Deviation scores

Name Score x - DeviationScore

Joe 15

Bill 12

Mary 17

Sue 20


Joe 15

Bill 12

Mary 17

Sue 20

x = 64


Joe 15 15 - 16

Bill 12 12 - 16

Mary 17 17 - 16

Sue 20 20 - 16

x = 64


Joe 15 15 - 16 -1

Bill 12 12 - 16 -4

Mary 17 17 - 16 1

Sue 20 20 - 16 4

x = 64 (x- ) = 0


Joe 26

Bill 6

Mary 1

Sue 31


Joe 26

Bill 6

Mary 1

Sue 31

= 16


Joe 26 26 - 16 10

Bill 6 6 - 16 -10

Mary 1 1 - 16 -15

Sue 31 31 - 16 15

= 16

Sample 1 vs. Sample 2

• Sample 1:

Raw scores: 15, 12, 17, 20

• Sample 2:

Raw scores: 26, 6, 1, 31

Sample 1 vs. Sample 2

• Sample 1:

Raw scores: 15, 12, 17, 20

Deviation scores: -1, -4, 1, 4

• Sample 2:

Raw scores: 26, 6, 1, 31

Deviation scores: 10, -10, -15, 15

Deviation Scores

• As variability increases the absolute value of the deviation scores also goes up!

• How can we use this information to create a measure of variability?

How about?

How about?

Formula

Formula

- 1

Why “ – 1” ?

• Without it the answer is biased -- its answer tends to be too small

• Page 53 – 56 explain

• Don’t worry about why -- unless you want too!!

Name x

Bart 5

Lisa 8

Homer 2

Ned 10

Maggie 5

( )

Name x

Bart 5

Lisa 8

Homer 2

Ned 10

Maggie 5

= 6

( )

Name x

Bart 5 -1

Lisa 8 2

Homer 2 -4

Ned 10 4

Maggie 5 -1

= 6

( )

Name x

Bart 5 -1 1

Lisa 8 2 4

Homer 2 -4 16

Ned 10 4 16

Maggie 5 -1 1

= 6

( )

Name x

Bart 5 -1 1

Lisa 8 2 4

Homer 2 -4 16

Ned 10 4 16

Maggie 5 -1 1= 6 = 38

( )

Formula

- 1

Formula

- 1

38

Formula

- 1

38

5

Formula

- 1

38

59.5

Formula

- 1

38

59.53.08

Practice

• For the sample data below calculate s

6, 8, 4, 3, 4, 5

Obs x

Score1 6 1 1

Score2 8 3 9

Score3 4 -1 1

Score4 3 -2 4

Score5 4 -1 1

Score6 5 0 0

( )

= 5

= 16

-1

16

61.79

-1

Practice

2.462.47

Documents

Practice Page 65 –2.1 Positive Skew Note Slides online