Review Ways to “see” data Measures of central tendency 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

Step 2: Square each value

Step 3: Sum

Step 4: Plug in values -1 N = 5 X = 44  X2 = 640

Step 4: Plug in values 5 5 - 1 N = 5 X = 44  X2 = 640

Step 4: Plug in values 44 5 5 - 1 N = 5 X = 44  X2 = 640

Step 4: Plug in values 44 640 5 5 - 1 N = 5 X = 44  X2 = 640

Step 5: Solve! 1936 44 640 5 5 - 1

Step 5: Solve! 1936 44 640 387.2 5 4

Step 5: Solve! 1936 44 63.2 640 387.2 5 5 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 -1

Standard Deviation 452 58 8 8 - 1 -1

Standard Deviation 58 452 8 8 - 1 -1

Standard Deviation 452 420.5 7 -1

Standard Deviation 2.12 -1

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 S 2 = - 1

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 67 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 Measures of central tendency 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 X = 8 = 6 S = 1.25 Sample 1: 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

Practice

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

Practice = 34.6 = 7.4 S = 7.96 S = 2.41

Practice = 34.6 = 7.4 S = 7.96 S = 2.41

Practice = 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. = 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 A random sample of 100 students found: 56 were psychology majors 32 were undecided 8 were math majors 4 were biology majors What is the probability of randomly selecting a psychology major?

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

Openness to Experience

Neuroticism

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

COMPUTER PROG http://www.mathsisfun.com/data/quincunx.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 

Theoretical Normal Curve -3 -2 -1  1 2  3 

Theoretical Normal Curve -3 -2 -1  1 2  3 

Theoretical Normal Curve -3 -2 -1  1 2  3  Note: A Z-score tells you how many SD above or below a mean a specific score falls!

Theoretical Normal Curve -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? .50 -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, you know the probability of getting a Z score of 0 or less. .50 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

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 .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

What is the probability of getting a score of 1 or higher? .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

These values are given in Appendix Z .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

Mean to Z -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

Smaller Portion -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

-3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3 .84 .1587 Larger Portion .84 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

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 What proportion of the normal distribution is found in the following areas (hint: draw out the answer)? Between mean and z = .56? .2123 Above z = 2.25? Above z = -1.45

Practice What proportion of the normal distribution is found in the following areas (hint: draw out the answer)? Between mean and z = .56? .2123 Above z = 2.25? .0122 Above z = -1.45

Practice What proportion of the normal distribution is found in the following areas (hint: draw out the answer)? Between mean and z = .56? .2123 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 

Step 1: Sketch out question 120 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (120 - 100) / 15 = 1.33 120 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table 120 .0918 -3 -2 -1  1 2  3 

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?

Step 1: Sketch out question -3 -2 -1  1 2  3 

Step 1: Sketch out question 80 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (80 - 100) / 15 = -1.33 80 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table 80 .0918 -3 -2 -1  1 2  3 

Example: IQ Mean IQ = 100 SD = 15 What proportion of the population have an IQ below 110?

Step 1: Sketch out question -3 -2 -1  1 2  3 

Step 1: Sketch out question 110 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (110 - 100) / 15 = .67 110 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table 110 .7486 -3 -2 -1  1 2  3 

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?

Step 1: Sketch out question 90 110 ? -3 -2 -1  1 2  3 

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 -.67 .67 .2486 .2486 -3 -2 -1  1 2  3 

Step 4: Add together the two values -.67 .67 .4972 -3 -2 -1  1 2  3 

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?

Step 1: Sketch out question 110 130 ? -3 -2 -1  1 2  3 

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 .67 2.0 .4772 -3 -2 -1  1 2  3 

Step 3: Look up Z score .67 2.0 .4772 .2486 -3 -2 -1  1 2  3 

Step 4: Subtract .4772 - .2486 = .2286 .67 2.0 .2286 -3 -2 -1  1 2  3 

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

Step 1: Sketch out question 95% ? 100 ?

Step 1: Sketch out question 95% 2.5% 2.5% ? 100 ?

Step 1: Sketch out question Z = -1.96 Z = 1.96 95% 2.5% 2.5% ? 100 ?

Step 3: Find the X score that goes with the Z score Z = (X -  ) /  1.96 = (X - 100) / 15 Must solve for X X =  + (z)() X = 100 + (1.96)(15)

Step 3: Find the X score that goes with the Z score Z = (X -  ) /  1.96 = (X - 100) / 15 Must solve for X X =  + (z)() X = 100 + (1.96)(15) Upper IQ score = 129.4

Step 3: Find the X score that goes with the Z score Must solve for X X =  + (z)() X = 100 + (-1.96)(15) Lower IQ score = 70.6

Step 1: Sketch out question Z = -1.96 Z = 1.96 95% 2.5% 2.5% 70.6 100 129.4

Finding a score when given a probability “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

Step 1: Sketch out question Z = -1.64 Z = 1.64 90% 5% 5% ? 500 ?

Step 3: Find the X score that goes with the Z score X =  + (z)() X = 500 + (1.64)(100) Upper score = 664 X = 500 + (-1.64)(100) Lower score = 336

Finding a score when given a probability “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 .3023 + .1664 = .4687 .4687*54 = 25.31 or 25 people

SPSS PROGRAM: http://www1.villanova.edu/villanova/unit/service/hwsw/SPSS.html BASIC “HOW TO” http://www.psychology.ilstu.edu/jccutti/138web/spss.html SPSS “HELP” is also good

SPSS PROBLEM #1 Data 2.1 (due Jan) Turn in the SPSS output for 1) Mean, median, mode 2) Standard deviation 3) Frequency Distribution 4) Histogram