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Statistical Methodology

T-Test, Chi-Squared, Mean, Median, Mode. Statistical Methodology. 2 Types of Statistics. 2 types of analysis techniques: 1. Descriptive statistics : techniques that help summarize large amounts of info. Include measures of variability and measures of correlation (Describe the data)

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Statistical Methodology

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  1. T-Test, Chi-Squared, Mean, Median, Mode Statistical Methodology

  2. 2 Types of Statistics • 2 types of analysis techniques: • 1. Descriptive statistics: techniques that help summarize large amounts of info. Include measures of variability and measures of correlation (Describe the data) • Population, Bag of M&Ms • 2. Inferential statistics: techniques that help researchers make generalizations about a finding, based on a limited number of subjects • Sample, Handful of M&Ms

  3. M&M Sampling • 13% brown, red • 14% yellow • 24% blue • 20% orange • 16% green • What was yours?

  4. Descriptive Statistics • Frequency distribution - organizational technique that shows the number of times each score occurs, so that the scores can be interpreted • Graph depictions • frequency polygon - curve • frequency histogram - bars

  5. Descriptive Statistics • Central Tendency - a number that represents the entire group or sample • Tend to hover towards the center • Average IQ score, around 100 • 2 genius parents tend to have average IQ child • Politicians (Dem or Rep) dance in the center for max. votes • Weight distribution

  6. Descriptive Statistics • The Bell Curve • Grades, IQ, Poverty • Link between intelligence and salary • When did a C become an F? • Is a C acceptable? C=average • Does everyone get a trophy, ribbon? • Can everyone get an A?

  7. Descriptive Statistics • mean - the arithmetic average • median - middle score when arranged lowest to highest • mode - the most frequent score in a distribution • unimodal - one high point • bimodal - two high points Set: 2, 2, 3, 5, 8 Median: 3 Mode: 2 Mean: Add up (20), divide by 5= 4

  8. Descriptive Statistics • bimodal - two high points • The more overlap in the bimodal arches, the higher the variable link between the data • The less overlap, the lower the connection

  9. Bimodal or unimodal? B A

  10. Descriptive Measures • 2 ways we measure: • 1. Range: Highest score minus the lowest score--tells how far apart the scores are • simplest measures of variability to calculate. • (weakness of range: it can easily be influenced by one extreme score, Savant IQ of 220) • Set: 2, 2, 3, 5, 8 • Range: 8 - 2 = 6 Ex: Age Range 15-17, Difference 2 7-17, Difference 10 Child prodigies, Dougie Houser, Chess, sci, art, music

  11. Descriptive Measures • The other way to measure is: • 2. Standard Deviation: measure of variability that describes how scores are distributed around the mean. • (1 SD, 2 SD, -1, -2) • Central Tendency: tend to hover near the center. Obama 145-148 Savant, 220 1 in 30 million Bush Rumor: 85 Actual: 125 Clinton 137 -3 SD -2 SD -1 SD +1 SD +2 SD +3 SD Einstein 160 Bill Gates Stephen Hawking Hillary Clinton Madonna 140

  12. Standard Deviation 1% outliers Savant, 220 1 in 30 million 34% 34% 13.5% 13.5% 2% 2% 68% 95% 99%

  13. Case Study: Marilyn vos Savant • Born Aug 11, 1946 (63) Missouri • American magazine columnist “ask Marilyn”, Parade Magazine (logic, math puzzles), books • Guinness Book World records, Highest IQ (220+) • Age 10 (1957) scored 167-218 (1 in 30 million)

  14. Case Study: Rain Man • 1988 comedy-drama (Tom Cruise) • Dustin Hoffman portrays Raymond Babbitt • Autistic Savant • Based on 2 real people (Kim Peek) Video clip: Rain Man

  15. Set: 2, 2, 3, 5, 8 Standard Deviation To calculate standard deviation (SD): • 1. find the mean of the distribution 4 • 2. subtract each score from the mean 4-2, 4-2, 4-3, 4-5, 4-8 • 3. square each result – “deviations” 4-2=2 2 squared=4 • 4. add the squared deviations 4 + 4 + 1 + 1 + 16 = 26 • 5. divide by the total number (n - 1) of scores; this result is called the variance 26 / 4 (5 – 1) = 6.5 (V) • 6. find the square root of the variance; this is the standard deviation (SD) 2.55 (SD) • 7. n = biased sample – does not accurately represent population being tested (out of the norm, get rid of out-liers) 5 • 8. (n - 1) = unbiased sample 4 • 9. now you can compare distributions with different means and standard deviations (ex: 3 different class scores, 78, 80, 92)

  16. Sigma Σ • Σthe symbol for standard deviation (SD) is s. • Greek letter “sigma” (lower case form) • S upper case letter (other Greek “sigma”) • Standard meaning in mathematics, “add up a list of numbers.” • Represents Sum, i.e. add together

  17. Z-Score Z-scores: a way of expressing a score’s distance from the mean in terms of the standard deviation (SD) • to find a Z-score for a number in a distribution, subtract the mean from that number, and divide the result by the standard deviation 8 – 4 (M)= 4 / 2.55 (SD) = 1.56 • a positive Z-score shows that the number is higher than the mean (You’re OK, IQ, health average or higher) • a negative Z-score allows psychologists to compare distributions with different means and standard deviations (Below average, health, psych concerns) • Sometimes Z-scores are necessary to explain standard deviation in an experiment’s results/discussion NEG Z POS Z

  18. Skewed Results • When there are more scores at the high or low end of a distribution it is said to be skewed • tail signifies the extreme score • Single tailed = extreme score on either side • Which direction are the “outliers?” • Called Right/Left Skew • Also Pos./Neg. Skew Majority Majority Outliers: fringe, oddball, genius, bad egg

  19. A Skewed Distribution Are the results positively or negatively skewed? Positive Skew or Skewed Right

  20. Statistical Significance • Statistics & Data • T-test, CHI-square, Z-score • Psychometrics • Statistical Significance • “I want to prove that my independent variable causes my dependent variable 95% of the time” • 95% to be valid • Probability= P<.05(5%) chance, random, chaos theory

  21. Inferential Statistics Tests of Significance - used for determining whether the difference in scores between the experimental and control groups is really due to the effects of the independent variable or just due to random chance • If p < .05 (95%) the outcome (or the difference between experimental and control groups) has a probability of occurring by random chance lessthan 5 x per 100 • Researchers conclude the effect of the independent variable is significant (real).

  22. Confused about Significance? Tests of Significance – • You want brain surgery to work (at least) 95% of the time. • Your car? • Guns in military? • Prescription drugs? • Cancer? • Dr. House: knows what the results of a test/disease SHOULD be 95%, move on to the next test.

  23. There is a 5% chance for random results (chaos theory)

  24. Inferential Statistics • Statistically Significant – • It is concluded that the independent variable made a real difference between the experimental group and the control group • Ritalin really DOES help ADHD • Raising serotonin levels DOES help Depression (yoga)

  25. Null Hypothesis • Null Hypothesis: any alternative hypothesis, if yours is wrong! • Significance tests are used to accept or reject the null hypothesis. • If the probability of observing your result is < .05 (95%) • Your theory is true, reject the null hypothesis • Meaning that your original hypothesis is possible (without chance, random, chaos) • If the probability of observing your result is > .05accept the null. • Meaning that your original hypothesis is not possible (too much left to chance, random events) • You need a backup, alternative hypothesis

  26. Year 2 IB Psych Only

  27. Null Hypothesis Practice • Accept or Reject the Null? • My hypothesis: Drug X will stop sleep walking 95%. • Do the testing. Do the data. • Drug X has a probability of 63%. • Is it greater than or less than 5% chance? <>.05? • Do you accept the Null or reject the Null Hypothesis? • ACCEPT the NULL! My theory was wrong! • 37% chance, error, random • Maybe it’s the patients I chose? • Maybe too much caffeine before bed? • Maybe drug was contaminated in the lab? • Start over, new test, new drug, new data

  28. Null Hypothesis Practice • Accept or Reject the Null? • My hypothesis: Stress causes mice to gain weight. • Do the testing. Do the data. • The “stressed mice” gained weight 97%. • The “control group” of mice showed no weight gain. • Is it greater than or less than 5% chance? <>.05? • Do you accept the Null or reject the Null Hypothesis? • REJECT the NULL! My theory was right! • 3% chance, error, random • Good Job! Bonus and a raise!

  29. Types of Tests • 1. T-Test • 2. Chi-Square Test • 3. Mann-Whitney U • 4. Sign Test • 5. Wilcoxon Matched-Pairs Signed-Rank Test

  30. Which letters belong together? • AGHOLPEWQMC ANSWER: AHLEWM GOPQC

  31. When to Use the T-Test? • T-Test – when 1variable is used in 2 situations -- Ex: Ritalin effects in either ADHD males or ADHD females -- Ex: subject has to pick out a letter in a round list or a square list

  32. When to Use the T-Test? • Common situation in psychology: • Randomly assign people to an “experimental” group or a “control” group to study the effect • In this situation, we are interested in the mean difference between the 2 conditions. • The significance test used in this kind of scenario is called a t-test. • Used to determine whether the observedmean difference is within the range (less that.05) that would be expected if the null hypothesis were true.

  33. How to Use the T-Test? • T-Test • 1. Subtractmean from each score • 2. Rank items • 3. Sum of Positive Ranks • 4. Sum of Negative Ranks • 5. Smallest score = T • If t > 1.96 or < - 1.96, then p < .05 (Test is Valid) BOYS GIRLS 4 7 2 5 3 2 4 1 1 4 3 5 4 6 21 30

  34. How to Use the T-Test? • T-Test • 1. Subtractmean from each score Mean= 21 divided by 7 = 3 4-3, 2-3, 3-3, 4-3, 1-3, 3-3, 4-3 1 , -1, 0, 1, -2, 0, 1 • 2. Rank items 1, 1, 1, 0, -1, -2 • 3. Sum of Positive Ranks 1+1+1+0=3 • 4. Sum of Negative Ranks -1 + -2 = -3 • 5. Smallest score = t(-3) • If t > 1.96 or < - 1.96, then p < .05 BOYS 4 2 3 7 scores 4 1 3 4 21 Test is VALID

  35. We have to redo our hypothesis ??? That bites

  36. Awesome Calculators! • www.graphpad.com/quickcalcs/index.cfm • T-Test • Chi-Square

  37. When do I Use Chi-Square? • A common situation in psychology is when a researcher is interested in the relationship between 2nominal or categorical variables. • The significance test used in this kind of situation is called a chi-square (2). • Ex: We are interested in whether single men vs. women are more likely to own cats vs. dogs. • Notice that both variables are categorical. • Kind of pet • Gender male or female. Chai-squared

  38. Example Data: Observed (Actual Data) • Males are more likely to have dogs as opposed to cats • Females are more likely to have cats than dogs NHST (Null Hypothesis Significance Testing) Question: Are these differences best accounted for by the null hypothesis? Is there is a real relationship between gender and pet ownership?

  39. Example Data: Observed (Actual Data) • Are femalesmore emotional?

  40. Chi-Square Test – when there are 2 variables • The closer your results (Experimental and Control), the harder to prove if indep. variable (IV) really worked. • Further apart, you can see definite difference.

  41. Example Data: Expected Data • To find expected valuefor a cell of the table, multiply the corresponding row total by the column total, and divide by the grand total • For the first cell (and all other cells) • (50 x 50)/100 = 25 • Thus, if the two variables are unrelated, we would expect to observe 25 people in each cell

  42. Example Data: Expected vs. Observed • The differences between these (E) expected values (25) and the (O) observedvalues(see boxes) are aggregated according to the Chi-square formula:

  43. Do We Accept or Reject the Null? • Once you have the chi-square statistic, it can be evaluated against a chi-square sampling distribution • The sampling distribution characterizes the range of chi-square values we might observe if the null hypothesis is true, but sampling error is giving rise to deviations from the expected values. • In our example in which the chi-square was 4.0, the associated p-value was >.05 • Accept the Null Hypothesis, need an Alternative Hypothesis, You did NOT prove your experiment

  44. Prisoner’s Dilemma, Social Trap Game Matrix, Non-zero-Sum-Game, Game Theory (Nash) ALL CHI SQUARED

  45. Mann-Whitney U Test • Skewed results? Are they from the same distribution? • Use to determine if there were problems with sampling, population, contamination • Use for 2 groups (samples) • Sub. For T-Score (T-Test) • Ex: Experimental & Control

  46. How To Use Mann-Whitney U Test • Ex: Experimental & Control • Lay out all of your scores (in both groups) • Rate them Rank 1 (lowest) - Rank 15 (highest) • Experimental GroupControl Group • Time (min) Rank Time (min) Rank • 140 4 130 1 • 147 6 135 2 • 153 8 138 3 • 160 10 144 5 • 165 11 148 7 • 170 13 155 9 • 171 14 168 12 • 193 15

  47. How To Use Mann-Whitney U Test • Add up the sum of both groups (+) • Experimental GroupControl Group • Time (min) Rank Time (min) Rank • 140 4 130 1 • 147 6 135 2 • 153 8 138 3 • 160 10 144 5 • 165 11 N1=8 148 7 • 170 13 155 9 • 171 14 168 12 • 193 15________________________________ • R1 =81, N1=8 R2 =39, N2=7 N2=7

  48. How To Use Mann-Whitney U Test • Experimental GroupControl Group • R1 =81, N1=8 R2 =39, N2=7 • Formula to find U (Hypothetical Data Statistics) • U=N1N2 + N1(N1+1)-R1 2 • U=(8)(7) + 8(9) -81 2 • U= 56 + 36 – 81 • U= 11

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