# STATISTICAL METHODOLOGY T-Test, Chi-Squared, Mean, Median, Mode.

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STATISTICAL METHODOLOGY T-Test, Chi-Squared, Mean, Median, Mode

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

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

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

Descriptive Statistics – Central Tendency – 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

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?

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

Descriptive Statistics – bimodal – 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

– Bimodalunimodal – Bimodal or unimodal? A B

Descriptive Measures 2 ways we measure: Range 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 6 Range: 8 - 2 = 6 Ex: Age Range 15-17, Difference 2 7-17, Difference 10 Child prodigies, Dougie Houser, Chess, sci, art, music

Descriptive Measures The other way to measure is: Standard Deviation 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. Savant, 220 1 in 30 million Einstein 160 Bill Gates Stephen Hawking Hillary Clinton Madonna 140 Clinton 137 Bush Rumor: 85 Actual: 125 Obama 145-148 -1 SD-2 SD-3 SD+1 SD+2 SD+3 SD

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

Marilyn vos Savant 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)

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

Standard Deviation To calculate standard deviation (SD): mean4 1. find the mean of the distribution 4 subtractmean 4-2, 4-2, 4-3, 4-5, 4-8 2. subtract each score from the mean 4-2, 4-2, 4-3, 4-5, 4-8 square4-2=2 2 squared=4 3. square each result – “deviations” 4-2=2 2 squared=4 add4 + 4 + 1 + 1 + 16 = 26 4. add the squared deviations 4 + 4 + 1 + 1 + 16 = 26 dividenumber variance 26 / 4 (5 – 1) = 6.5 (V) 5. divide by the total number (n - 1) of scores; this result is called the variance 26 / 4 (5 – 1) = 6.5 (V) square root variancestandard deviation (SD)2.55 (SD) 6. find the square root of the variance; this is the standard deviation (SD)2.55 (SD) n = biased sample 5 7. n = biased sample – does not accurately represent population being tested (out of the norm, get rid of out-liers) 5 (n - 1) unbiased sample4 8. (n - 1) = unbiased sample4 (ex: 3 different class scores, 78, 80, 92) 9. now you can compare distributions with different means and standard deviations (ex: 3 different class scores, 78, 80, 92) Set: 2, 2, 3, 5, 8

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

Z-Score Z-scores Z-scores: a way of expressing a score’s distance from the mean in terms of the standard deviation (SD) subtract meandivide standard deviation 8 – 4 (M)= 4 / 2.55 (SD) = 1.56 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 positive a positive Z-score shows that the number is higher than the mean (You’re OK, IQ, health average or higher) negative 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 POS Z NEG Z

Skewed Results highlow skewed 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 – Single tailed = extreme score on either side – Which direction are the “outliers?” – Called Right/Left Skew – Also Pos./Neg. Skew Majority Outliers: fringe, oddball, genius, bad egg

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

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

Inferential Statistics Tests of Significance experimentalcontrol groups independent variable random chance 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 p <.05 (95%) If p <.05 (95%) the outcome (or the difference between experimental and control groups) has a probability of occurring by random chance less than 5 x per 100 – Researchers conclude the effect of the independent variable is significant (real).

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

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

Inferential Statistics Statistically Significant 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)

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

Year 2 IB Psych Only

Null Hypothesis Practice Accept or Reject the Null? 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%. 5% 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

Null Hypothesis Practice Accept or Reject the Null? 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. 5% 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!

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

Which letters belong together? AGHOLPEWQMC AGHOLPEWQMC ANSWER: AHLEWM GOPQC

When to Use the T-Test? T-Test1variable2 situations T-Test – when 1 variable 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

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

How to Use the T-Test? T-Test T-Test Subtractmean 1. Subtract mean from each score 2. Rank items Positive 3. Sum of Positive Ranks Negative 4. Sum of Negative Ranks 5. Smallest score = T t > 1.96 1.96 or < - 1.96, then p <.05 (Test is Valid) GIRLSBOYS 74 52 23 14 41 53 64 3021

How to Use the T-Test? T-Test T-Test Subtractmean 1. Subtract mean 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 Positive 3. Sum of Positive Ranks 1+1+1+0=3 Negative 4. Sum of Negative Ranks -1 + -2 = -3 5. Smallest score = t (-3) t > 1.96 1.96 or < - 1.96, then p <.05 BOYS 4 2 3 4 1 3 4 21 7 scores Test is VALID

We have to redo our hypothesis ??? That bites

When do I Use Chi-Square? A common situation in psychology is when a researcher is interested in the relationship between 2 nominal 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. 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

Example Data: Observed (Actual Data) Males dogs Males are more likely to have dogs as opposed to cats Females cats Females are more likely to have cats than dogs CatDog Male203050 Female302050 100 NHST 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?

Example Data: Observed (Actual Data) Are females more emotional? Are females more emotional? EmotionalNot Emotional Female203050 Male302050 100

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.

Example Data: Expected Data expected value To find expected value for 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 CatDog Male25 50 Female25 50 100

Example Data: Expected vs. Observed expected values (25) observed values(see boxes) The differences between these (E) expected values (25) and the (O) observed values (see boxes) are aggregated according to the Chi-square formula: CatDog Male203050 Female302050 100

chi-square chi-square sampling distribution Once you have the chi-square statistic, it can be evaluated against a chi-square sampling distribution if the null hypothesis is true 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. 4.0p- value was >.05 In our example in which the chi-square was 4.0, the associated p- value was >.05 Null Hypothesis Accept the Null Hypothesis, need an Alternative Hypothesis, You did NOT prove your experiment Do We Accept or Reject the Null?

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

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

How To Use Mann-Whitney U Test – Ex: Experimental & Control – Lay out all of your scores (in both groups) Rank 1Rank 15 – Rate them Rank 1 (lowest) - Rank 15 (highest) Experimental GroupControl Group Experimental GroupControl Group Time (min)RankTime (min)Rank 1 14041301 14761352 15381383 160101445 165111487 170131559 1711416812 15 19315

How To Use Mann-Whitney U Test Add up the sum of both groups (+) Experimental GroupControl Group Experimental GroupControl Group Time (min)RankTime (min)Rank 14041301 14761352 15381383 160101445 N1=8 16511 N1=8 1487 170131559 1711416812 19315________________________________ R1 =81, N1=8 R2 =39, N2=7 R1 =81, N1=8 R2 =39, N2=7 N2=7

How To Use Mann-Whitney U Test Experimental GroupControl Group Experimental GroupControl Group R1 =81, N1=8 R2 =39, N2=7 R1 =81, N1=8 R2 =39, N2=7 Formula to find U 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

How To Use Mann-Whitney U Test Experimental GroupControl Group Experimental GroupControl Group R1 =81, N1=8 R2 =39, N2=7U=11 R1 =81, N1=8 R2 =39, N2=7U=11 Is 11 in between the N1-N2 range of #s on the chart? (6-50) YES Is 11 in between the N1-N2 range of #s on the chart? (6-50) YES Go to the Mann-Whitney Chart (Table 1) N12345678 N2 2 3 4 5 6 76/50

How To Use Mann-Whitney U Test Experimental GroupControl Group Experimental GroupControl Group R1 =81, N1=8 R2 =39, N2=7U=11 R1 =81, N1=8 R2 =39, N2=7U=11 Is 11 in between the N1-N2 range of #s on the chart? (6-50) Is 11 in between the N1-N2 range of #s on the chart? (6-50) If YES, If YES, reject the Null Hypothesis, your data is acceptable to use Your distribution and population is acceptable, even though a skew has occurred, you are within the acceptable range If NO If NO, accept the Null Hypothesis, your data is not acceptable Something has contaminated your population or your data, you must go to a Null, or Alternate Hypothesis.

Video Clips Stossel, Media Scare (Statistics) ThePsychFiles, Faces, Metacafe, 14 min Eddie Izzard, Part 9, Tea & Cake or Death?