Review Ways to “see” data –Simple frequency distribution –Group frequency distribution –Histogram –Stem-and-Leaf Display –Describing distributions –Box-Plot

Review

• Ways to “see” data– Simple frequency distribution– Group frequency distribution– Histogram– Stem-and-Leaf Display– Describing distributions– Box-Plot

• Measures of central tendency– Mean – Median– Mode

Review

• Measures of variability– Range– IQR– Standard deviation

Compute a standard deviation with the Raw-Score Method

• Previously learned the deviation formula– Good to see “what's going on”

• Raw score formula – Easier to calculate than the deviation formula– Not as intuitive as the deviation formula

• They are algebraically the same!!

Raw-Score Formula

-1

Step 1: Create a table

Coffee X

X2

4 10 22 2 6

Step 2: Square each value

CoffeeX

X2

4 1610 10022 4842 46 36

Step 3: Sum

CoffeeX

X2

4 1610 10022 4842 46 36

X = 44 X2 = 640

Step 4: Plug in values

N = 5

X = 44

X2 = 640

-1


N = 5

X = 44

X2 = 640

5 - 15


N = 5

X = 44

X2 = 640

5 - 1544


N = 5

X = 44

X2 = 640

5 - 1544

640

Step 5: Solve!

5 - 1544

6401936

Step 5: Solve!

4544

6401936387.2

Step 5: Solve!

5544

6401936387.263.2

Answer = 7.95

Practice

• You are interested in how citizens of the US feel about the president. You asked 8 people to rate the president on a 10 point scale. Describe how the country feels about the president -- be sure to report a measure of central tendency and the standard deviation.

8, 4, 9, 10, 6, 5, 7, 9

Central Tendency

8, 4, 9, 10, 6, 5, 7, 9

4, 5, 6, 7, 8, 9, 9, 10

Mean = 7.25

Median = (4.5) = 7.5

Mode = 9

Standard Deviation X X2

8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

-1


8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

-1

45258 8

8 - 1


8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

-1

58452 8

8 - 1


8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

-1

452 420.5

7


8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

-1

2.12

Variance

• The last step in calculating a standard deviation is to find the square root

• The number you are fining the square root of is the variance!

Variance

S 2 =

Variance

- 1S 2 =

Practice• Below are the test score of Joe and Bob.

What are their means, medians, and modes? Who tended to have the most uniform scores?

• Joe

80, 40, 65, 90, 99, 90, 22, 50• Bob

50, 50, 40, 26, 85, 78, 12, 50

Practice

• Joe

22, 40, 50, 65, 80, 90, 90, 99

Mean = 67

• Bob

12, 26, 40, 50, 50, 50, 78, 85

Mean = 48.88

Practice

• Joe

22, 40, 50, 65, 80, 90, 90, 99

Median = 72.5

• Bob

12, 26, 40, 50, 50, 50, 78, 85

Median = 50

Practice

• Joe

22, 40, 50, 65, 80, 90, 90, 99

Mode = 90

• Bob

12, 26, 40, 50, 50, 50, 78, 85

Mode = 50

Practice

• Joe

22, 40, 50, 65, 80, 90, 90, 99

S = 27.51; S2 = 756.80

• Bob

12, 26, 40, 50, 50, 50, 78, 85

S = 24.26; S2 = 588.55

Thus, Bob’s scores were the most uniform

Review

• Ways to “see” data– Simple frequency distribution– Group frequency distribution– Histogram– Stem-and-Leaf Display– Describing distributions– Box-Plot

• Measures of central tendency– Mean – Median– Mode

Review

• Measures of variability– Range– IQR– Standard deviation – Variance

What if. . . .

• You recently finished taking a test that you received a score of 90 and the test scores were normally distributed.

• It was out of 200 points

• The highest score was 110

• The average score was 95

• The lowest score was 90

Z-score

• A mathematical way to modify an individual raw score so that the result conveys the score’s relationship to the mean and standard deviation of the other scores

• Transforms a distribution of scores so they have a mean of 0 and a SD of 1

Z-score

• Ingredients:

X Raw score

Mean of scores

S The standard deviation of scores

Z-score

What it does

• x - Tells you how far from the mean you are and if you are > or < the mean

• S Tells you the “size” of this difference

Example

• Sample 1:

X = 8

= 6

S = 5

Example

• Sample 1:

X = 8

= 6

S = 5

Z score = .4

Example

• Sample 1:

X = 8

= 6

S = 1.25

Example

• Sample 1:

X = 8

= 6

S = 1.25

Z-score = 1.6

Example

• Sample 1:

X = 8

= 6

S = 1.25

Z-score = 1.6

Note: A Z-score tells you how many SD above or below a mean a specific score falls!

Practice

• The history teacher Mr. Hand announced that the lowest test score for each student would be dropped. Jeff scored a 85 on his first test. The mean was 74 and the SD was 4. On the second exam, he made 150. The class mean was 140 and the SD was 15. On the third exam, the mean was 35 and the SD was 5. Jeff got 40. Which test should be dropped?

Practice

• Test #1

Z = (85 - 74) / 4 = 2.75

• Test #2

Z = (150 - 140) / 15 = .67

• Test #3

Z = (40 - 35) / 5 = 1.00

Practice

Time(sec)

Distance(feet)

Rachel 30 6

Joey 40 8

Ross 25 4

Monica 45 10

Chandler 33 9

Which challenge did Ross do best? Which did Monica do best?

Time(sec)

Distance(feet)

Rachel 30 6

Joey 40 8

Ross 25 4

Monica 45 10

Chandler 33 9

Practice

Time (sec)

Distance (feet)

Rachel 30 6

Joey 40 8

Ross 25 4

Monica 45 10

Chandler 33 9

= 34.6 = 7.4

S = 7.96 S = 2.41

Practice

Time (sec)

Distance (feet)

Rachel 30 -.58 6

Joey 40 .68 8

Ross 25 -1.21 4

Monica 45 1.31 10

Chandler 33 -.20 9

= 34.6 = 7.4

S = 7.96 S = 2.41

Practice

Time (sec)

Distance (feet)

Rachel 30 -.58 6 -.58

Joey 40 .68 8 .25

Ross 25 -1.21 4 -1.66

Monica 45 1.31 10 1.08

Chandler 33 -.20 9 .66

= 34.6 = 7.4

S = 7.96 S = 2.41

Ross did worse in the throwing challenge than the endurance and Monica did better in the endurance than the throwing challenge.

Time (sec)

Distance (feet)

Rachel 30 -.58 6 -.58

Joey 40 .68 8 .25

Ross 25 -1.21 4 -1.66

Monica 45 1.31 10 1.08

Chandler 33 -.20 9 .66

= 34.6 = 7.4

S = 7.96 S = 2.41

Shifting Gears

Question

• A random sample of 100 students found:– 56 were psychology majors– 32 were undecided– 8 were math majors– 4 were biology majors

• What proportion were psychology majors?

• .56

Question


• What is the probability of randomly selecting a psychology major?

Question


• What is the probability of randomly selecting a psychology major?

• .56

Probabilities

• The likelihood that something will occur

• Easy to do with nominal data!

• What if the variable was quantitative?

Extraversion

BFISUR

4.88

4.63

4.38

4.13

3.88

3.63

3.38

3.13

2.88

2.63

2.38

2.13

1.88

1.63

1.38

1.13

Co

un

t50

40

30

20

10

0

BFIOPN

5.00

4.80

4.60

4.40

4.20

4.00

3.80

3.60

3.40

3.20

3.00

2.80

2.60

2.40

2.20

2.00

1.60

Co

un

t

40

30

20

10

0

Openness to Experience

BFISTB

4.88

4.50

4.25

4.00

3.75

3.50

3.25

3.00

2.75

2.50

2.25

2.00

1.75

1.50

1.25

Co

un

t40

30

20

10

0

Neuroticism

Probabilities

Normality frequently occurs in many situations of psychology, and other sciences

COMPUTER PROG

• http://www.jcu.edu/math/isep/Quincunx/Quincunx.html

• http://webphysics.davidson.edu/Applets/Galton/BallDrop.html

• http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

http://webphysics.davidson.edu/Applets/Galton/BallDrop.html

http://webphysics.davidson.edu/Applets/Galton/BallDrop.html

http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

Next step

• Z scores allow us to modify a raw score so that it conveys the score’s relationship to the mean and standard deviation of the other scores.

• Normality of scores frequently occurs in many situations of psychology, and other sciences

• Is it possible to apply Z score to the normal distribution to compute a probability?

Theoretical Normal Curve

-3 -2 -1 1 2 3


-3 -2 -1 1 2 3


-3 -2 -1 1 2 3


-3 -2 -1 1 2 3

Note: A Z-score tells you how many SD above or below a mean a specific score falls!


-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

We can use the theoretical normal distribution to determine the probability of an event. For example, do you know the probability of getting a Z score of 0 or less?

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.50

We can use the theoretical normal distribution to determine the probability of an event. For example, you know the probability of getting a Z score of 0 or less.

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.50

With the theoretical normal distribution we know the probabilities associated with every z score! The probability of getting a score between a 0 and a 1 is

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

What is the probability of getting a score of 1 or higher?

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

These values are given in Appendix Z

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

Mean to Z

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.3413 .3413

.1587 .1587

Smaller Portion

-3 -2 -1 1 2 3

Z-scores -3 -2 -1 0 1 2 3

.84

.1587

Larger Portion

Practice

• What proportion of the normal distribution is found in the following areas (hint: draw out the answer)?

• Between mean and z = .56?

• Above z = 2.25?

• Above z = -1.45

Practice


• Between mean and z = .56?.2123

• Above z = 2.25?

• Above z = -1.45

Practice



• Above z = 2.25?.0122

• Above z = -1.45

Practice



• Above z = 2.25?.0122

• Above z = -1.45.9265

Practice

• What proportion of this class would have received an A on the last test if I gave A’s to anyone with a z score of 1.25 or higher?

• .1056

Example: IQ

• Mean IQ = 100

• Standard deviation = 15

• What proportion of people have an IQ of 120 or higher?

Step 1: Sketch out question

-3 -2 -1 1 2 3


-3 -2 -1 1 2 3

120

Step 2: Calculate Z score

-3 -2 -1 1 2 3

120

(120 - 100) / 15 = 1.33

Step 3: Look up Z score in Table

-3 -2 -1 1 2 3

120

Z = 1.33

.0918

Example: IQ

• A proportion of .0918 or 9.18 percent of the population have an IQ above 120.

• What proportion of the population have an IQ below 80?


-3 -2 -1 1 2 3


-3 -2 -1 1 2 3

80


-3 -2 -1 1 2 3

80

(80 - 100) / 15 = -1.33


-3 -2 -1 1 2 3

80

Z = -1.33

.0918

Example: IQ

• Mean IQ = 100

• SD = 15

• What proportion of the population have an IQ below 110?


-3 -2 -1 1 2 3


-3 -2 -1 1 2 3

110


-3 -2 -1 1 2 3

(110 - 100) / 15 = .67

110


-3 -2 -1 1 2 3

Z = .67

110

.7486

Example: IQ

• A proportion of .7486 or 74.86 percent of the population have an IQ below 110.

Finding the Proportion of the Population Between Two

Scores• What proportion of the population have IQ

scores between 90 and 110?


-3 -2 -1 1 2 3

11090

?

Step 2: Calculate Z scores for both values

• Z = (X - ) /

• Z = (90 - 100) / 15 = -.67

• Z = (110 - 100) / 15 = .67

Step 3: Look up Z scores

-3 -2 -1 1 2 3

.67-.67

.2486 .2486

Step 4: Add together the two values

-3 -2 -1 1 2 3

.67-.67

.4972

• A proportion of .4972 or 49.72 percent of the population have an IQ between 90 and 110.

• What proportion of the population have an IQ between 110 and 130?


-3 -2 -1 1 2 3

110 130

?

Step 2: Calculate Z scores for both values

• Z = (X - ) /

• Z = (110 - 100) / 15 = .67

• Z = (130 - 100) / 15 = 2.0

Step 3: Look up Z score

-3 -2 -1 1 2 3

.67 2.0.4772

Step 3: Look up Z score

-3 -2 -1 1 2 3

.67 2.0.4772

.2486

Step 4: Subtract

-3 -2 -1 1 2 3

.67 2.0

.2286

.4772 - .2486 = .2286

• A proportion of .2286 or 22.86 percent of the population have an IQ between 110 and 130.

Finding a score when given a probability

• IQ scores – what is the range of IQ scores we expect 95% of the population to fall?

• “If I draw a person at random from this population, 95% of the time his or her score will lie between ___ and ___”

• Mean = 100• SD = 15


? 100 ?

95%


? 100 ?

95% 2.5%2.5%


? 100 ?

95% 2.5%2.5%

Z = 1.96Z = -1.96

Step 3: Find the X score that goes with the Z score

• Z score = 1.96

• Z = (X - ) / • 1.96 = (X - 100) / 15

• Must solve for X

• X = + (z)()

• X = 100 + (1.96)(15)


• Z score = 1.96• Z = (X - ) / • 1.96 = (X - 100) / 15

• Must solve for X• X = + (z)()• X = 100 + (1.96)(15)• Upper IQ score = 129.4


• Must solve for X

• X = + (z)()

• X = 100 + (-1.96)(15)

• Lower IQ score = 70.6


70.6 100 129.4

95% 2.5%2.5%

Z = 1.96Z = -1.96


• “If I draw a person at random from this population, 95% of the time his or her score will lie between 70.6 and 129.4”

Practice

• GRE Score – what is the range of GRE scores we expect 90% of the population to fall?

• Mean = 500

• SD = 100


? 500 ?

90% 5%5%

Z = 1.64Z = -1.64


• X = + (z)()• X = 500 + (1.64)(100)• Upper score = 664

• X = + (z)()• X = 500 + (-1.64)(100)• Lower score = 336


• “If I draw a person at random from this population, 90% of the time his or her score will lie between 336 and 664”

Practice

Practice

• The Neuroticism Measure

= 23.32

S = 6.24

n = 54

How many people likely have a neuroticism score between 18 and 26?

Practice

• (18-23.32) /6.24 = -.85

• area = .3023

• ( 26-23.32)/6.26 = .43

• area = .1664

• .3023 + .1664 = .4687

• .4687*54 = 25.31 or 25 people

SPSS

PROGRAM:

https://citrixweb.villanova.edu/Citrix/XenApp/auth/login.aspx

BASIC “HOW TO”

http://www.psychology.ilstu.edu/jccutti/138web/spss.html

SPSS “HELP” is also good





SPSS PROBLEM #1

• Page 65• Data 2.1

• Turn in the SPSS output for

• 1) Mean, median, mode• 2) Standard deviation• 3) Frequency Distribution• 4) Histogram

Documents

Review Ways to “see” data –Simple frequency distribution –Group frequency distribution –Histogram –Stem-and-Leaf Display –Describing distributions –Box-Plot