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Why Statistics?

- TwoPurposes
- Descriptive
- Finding ways to summarize the important characteristics of a dataset
- Inferential
- How (and when) to generalize from a sample dataset to the larger population

Firsthand Impression

Secondhand Impression

3.88 3.38

1.88 .63

2.00 3.13

3.88 4.25

2.50 .50

3.25 3.75

3.13 1.50

1.50 1.88

3.75 .88

2.00 2.25

2.38 1.13

3.25 3.38

2.88 1.00

.88 -.25

3.50 1.63

4.13 1.50

.38 2.00

4.63 2.13

Firsthand Impression

Secondhand Impression

3.88 3.38

1.88 .63

2.00 3.13

3.88 4.25

2.50 .50

3.25 3.75

3.13 1.50

1.50 1.88

3.75 .88

2.00 2.25

2.38 1.13

3.25 3.38

2.88 1.00

.88 -.25

3.50 1.63

4.13 1.50

.38 2.00

4.63 2.13

Looking for Linear Relationships

Anger during Conflict

Looking for Linear Relationships

Current relationship satisfaction

Comparing Linear Relationships

Current relationship satisfaction

Anger during Conflict

How could you summarize the differences?

Current relationship satisfaction

- Complex Linear Relationships

Descriptive Statistics

Provides graphical and numerical ways to organize, summarize, and characterize a dataset.

Types of Studies

- Experimental:
- The predictor variable is manipulated by the researcher.
- Observational:
- The predictor variables are merely observed and recorded by the researcher.

Types of Variables

- Predictor variable:
- The antecedent conditions that are going to be used to predict the outcome of interest. If an experimental study, then called an “independent variable”.
- Outcome variable:
- The variable you want to be able to predict. If an experimental study, then called a “dependent variable”.

Ordinal

Categorical

A set of categories that are organized in an ordered sequence

A set of categories that have different names

Types of Variables

- Continuous variable:
- There are an infinite number of possible values that fall between any two observed values.
- Discrete variable:
- Consists of separate, indivisible categories

Frequency Tables

Hours of Sleep

Frequency

1

3

6

14

16

5

3

2

3 - 4 hrs

4 - 5 hrs

5 - 6 hrs

6 - 7 hrs

7 - 8 hrs

8 - 9 hrs

9 - 10 hrs

10 - 11 hrs

Relative Frequency

2%

6%

12%

28%

32%

10%

6%

4%

Frequency Tables

Hours of Sleep

Frequency

3 - 4 hrs

4 - 5 hrs

5 - 6 hrs

6 - 7 hrs

7 - 8 hrs

8 - 9 hrs

9 - 10 hrs

10 - 11 hrs

1

3

6

14

16

5

3

2

Cumulative Frequency

2%

8%

20%

48%

80%

90%

96%

100%

Frequency Tables

Hours of Sleep

Relative Frequency

Frequency

3 - 4 hrs

4 - 5 hrs

5 - 6 hrs

6 - 7 hrs

7 - 8 hrs

8 - 9 hrs

9 - 10 hrs

10 - 11 hrs

1

3

6

14

16

5

3

2

2%

6%

12%

28%

32%

10%

6%

4%

Visual Depictions of Distributions Summary

Discrete Data

Frequency Tables

Bar Graphs

Continuous Data

Frequency Tables

Bar Graphs

Stem and Leaf Plots

Visual Depictions of Relationships

IV -- categorical; DV -- continuous

- Bar Graphs

Look at the same graph differently!

Person attributions

Implicit Theory

Visual Depictions of Relationships

IV -- categorical; DV -- continuous

- Error-bar plots

Current relationship satisfaction

Visual Depictions of Relationships

IV -- continuous; DV -- continuous

- Scatterplots

Current relationship satisfaction

Visual Depictions of Relationships

IV -- continuous; DV -- continuous

- Scatterplots

Visual Depictions of Relationships

IV -- continuous; DV -- continuous

- Scatterplots with regression lines

Current relationship satisfaction

IV -- categorical; DV -- continuous

- Charts, bar graphs, line graphs, box plots, error bar plots

IV -- continuous; DV -- continuous

- Scatterplot (regression line)

IV -- categorical; DV -- categorical

- Contingency table

Visual Depictions of Relationships

Inferential statistics

Inferential Statistics

- Population:
- The set of all individuals of interest (e.g. all women, all college students)
- Sample:
- A subset of individuals selected from the population from whom data is collected

Some important terms

- Parameter:
- A characteristic of the population. Denoted with Greek letters such as or .
- Statistic:
- A characteristic of a sample. Denoted with English letters such as X or S.
- Sampling Error:
- Describes the amount of error that exists between a sample statistic and the corresponding population parameter.

We want to know whether Joe is an above average free-throw shooter. We collect some data

B

B

B

M

M

M

B

B

B

B

B

B

B

B

M

M

% baskets = .75

M

M

B

B

M

% baskets = .63

B

B

M

% baskets = .58

Would you bet $10.00 that he makes the next shot?

H

T

H

H

Sample proportion = .75

Inferential Statistics helps us answer the question:

Given a fair coin tossed four times, how often would we get the result 75% heads by chance alone?

Answer: If we took a fair coin and repeated this procedure many times, we’d get this result one out of every four times. Pretty often!

So differences we see between samples might not be reliable

(especially when the differences are small or the samples are small)

Inferential statistics can tell us whether or not our results are likely to be due to chance alone

Inferential statistics separates

- Important Point of Clarification

Statistics asks: Was this observed “effect” caused by (lumpy) chance alone?

Random Causes:

Fluctuations of chance

Non-random causes:

True differences in the population

Bias in the design of the study

A statistically significant result doesn’t mean the results have to be “true”. Just that they are non-random.

Types of Analyses

- IV -- categorical (groups); DV -- continuous
- One Sample T-test. Inferences about the mean of one group
- Two Sample T-test. Differences between the means of two groups.
- ANOVA. Differences between the means of three or more groups.

Types of Analyses

- IV -- continuous; DV -- continuous
- Correlation. The linear association between two continuous variables
- Regression. The best fit line of prediction.

Types of Analyses

- IV -- categorical (groups); DV -- categorical
- Z-test for proportions. The difference between two sample proportions.
- Chi-square test. The distribution of counts in each category, compared across groups.

Why Statistics?

Fallibility of Everyday Reasoning

Everyday Statistical Reasoning

Something out of nothing: the misperception of random data.

Too much from too little: the misinterpretation of incomplete data

Seeing what you expect: biased evaluation of ambiguous data

Misperceiving Random Data

“The human understanding supposes a greater degree of order and equality in things than it really finds; and although many things in nature be most irregular, will yet invest parallels and conjugates and relatives where no such thing is.” -Francis Bacon

The clustering illusion

People do not intuitively expect chance to be lumpy. They reject the possibility that clustering can be random.

“Hot hand” in basketball. “Winning streak” or “hot table” in gambling.

Gilovich et al., 1985

- Interviewed 100 basketball fans
- 91% thought a player has a better chance of making a shot after having just made his last 2-3 shots than he does after having just missed his last 2-3 shots.
- They estimated that a player’s shooting percentage would be 61% after having just made a shot and 42% after having just missed a shot.
- 84% of the respondents thought that it is important to pass the ball to someone who has just made several shots in a row.

Gilovich et al., 1985

The data

- On average, players made 51% of shots after making their previous shot, 54% of shots after missing their previous shot.
- They made 50% of shots after making their previous two shots, 53% after missing their previous two shots.
- They made 46% of shots after making their previous three shots, 56% of shots after missing three in a row.
- There were no more streaks of 4, 5, or 6 hits in a row than chance would have predicted.

The players, however, believed that they tended to shoot in streaks.

Gilovich et al., 1985

The data

- A group of college b-ball players were asked to take 100 shots. Before each shot they chose either a risky or conservative bet on their ability to make the shot.
- They tended to make risky bets after hitting their previous shot and conservative bets after missing their previous shot.
- However, there was no correlation between the outcome of consecutive shots. No correlation between bets and outcomes.

“Who is this guy? So he makes a study. I couldn’t care less.” -Red Auerbach, Celtics

“There are so many variables involved in shooting the basketball that a paper like this doesn’t mean anything.” -Bobby Knight

Gilovich et al., 1985

The response

LAW of LARGE NUMBERS

The correct proportion of heads and tails or hits and misses will be present globally in a long sequence.

It will NOT, however, always be present locally, in each of its parts.

Misinterpreting Incomplete Data

“They still cling stubbornly to the idea that the only good answer is a yes answer. If they say, “Is the number between 5,000 and 10,000” and I say yes, they cheer; if I say no, they groan, even though they get exactly the same amount of information in either case.” -John Holt

Absent-Minded

Not Absent-Minded

Professors

600

400

Not Professors

300

200

“Are professors particularly likely to be absent-minded?”

“Does the Cosmo horoscope predict the future?”

Event happens

Event doesn’t happen

Cosmo predicts event

Cosmo doesn’t predict event

700

3800

Can alternative medical technique X help cancer patients who have been diagnosed as “incurable”?

Patient recovers

Patient fails to recover

Patient gets alternative med

500

4000

Patient does notget alternative med

A

B

2

3

“All cards with a vowel on one side have an even number on the other.”

WHY?

- Selective attention
- Available information
- Positive test strategy

WHY?

- Selective attention
- Available information
- Positive test strategy
- Under-appreciation of base rates

Watch out for incomplete data!

Event occurs

Event does not occur

Event hypothesized

No event hypothesized

III. Projecting onto Ambiguous Data

“I’ll see it when I believe it.” -Thane Pittman

Illusory correlations

When people “see” an association that is not present in the data.

“Arthritis pain is influenced by the weather.”

“Most women get bad moods before their menstrual periods.”

Chapman et al., 1967

- Why do clinical psychologists continue to use projective tests even though dozens of studies have shown these tests are not valid indicators of personality?
- Showed clinicians a series of Rorschach cards as well as the patient’s response to the card and some info describing the patient’s characteristics. (including sometimes sexual orientation).
- Examined the correlations that clinicians “saw” between particular responses and homosexuality.

Chapman et al., 1967

- In truth, there are some counter-intuitive relationships. Homosexuals are more likely to see a monstrous figure on one card and an ambiguous animal-human figure on another card.
- Many of the intuitive relationships do not hold. Homosexuals are not more likely to see anal content, feminine clothing, or humans of uncertain gender.

Chapman et al., 1967

- In Study 1, researchers designed the materials so that there was no correlation between any of the responses and homosexuality.
- Clinicians did, however, believe the highly intuitive -- but invalid -- correlations.

Chapman et al., 1967

- In followup studies, researchers designed the materials so that there was a negative correlation between the intuitive responses and homosexuality.
- The size of the illusory correlation was not reduced.

Clinicians may “see” non-existent correlations between test responses and diagnoses

- Managers may “see” non-existent correlations between employees’ race or gender and performance
- Parents may “see” nonexistent correlations between children’s sugar consumption and unruly behavior
- Students may “see” nonexistant correllations between their peers’ college majors and personalities.

Much of what we “learn” from experience may reflect our prior theories about reality rather than the actual nature of reality.

Everyday Statistical Reasoning

- Something out of nothing: the misperception of random data.

- Drawing strong conclusions from small “lumpy” samples

- Too much from too little: the misinterpretation of incomplete data

- Inadequate comparison groups

- Seeing what you expect: biased evaluation of ambiguous data

- Illusory correlation based on confirmation bias

But there’s hope …

Following training in probability and statistics, people are less likely to make these errors.

Why This Course?

Fallibility of Statistical Reports

Everyday Reasoning

Statistical Reports

Something out of nothing: the misperception of random data.

Too much from too little: the misinterpretation of incomplete data (~control groups)

Seeing what you expect: biased evaluation of ambiguous data

One thing out of something else: overgeneralization from biased samples and measures

Too much from too little: the misinterpretation of incomplete data (~control groups)

Getting what you expect: biased presentation of ambiguous data

Overgeneralizing from Biased Samples

1934 Election Poll

- In 1934, the Literary Digest predicted that Alf Landon would beat Franklin D. Roosevelt in the presidential election, based on approx 2 million survey responses

- How could a study with such a large sample be so wrong? Selection bias? But participants were selected randomly from phone books…

- Other polling agencies with smaller samples but more representative methods accurately predicted Roosevelt’s win

Overgeneralizing from Biased Samples

Sperm Study

- In early 1996, media raised the alarm about declining sperm counts, as a result of a book published by Colburn, an environmentalist

- The book relied heavily on a 1992 Danish meta-analysis reviewing 61 papers published between 1938 and 1991, in which a total of 14,947 men had their sperm tested.

- Found a “significant” decline in sperm count: from 113 m sperm per ml in 1940 to 66 m sperm per ml in 1990

one study!

Pre-1950596

19511000

1952-1970184

1970-199113,167

- Overgeneralizing from Biased Samples

Sperm Study

- Sample:

- The entire “decline” was carried by the single 1951 sample
- From 1970-1991, sperm counts actually increased

Misinterpretation of Incomplete Data

Crime Study

- Murders significantly fell in NYC in the last decade: from 2,245 in 1990 to 596 in 2003

presumed cause: Giuliani

- Murders significantly fell all across the country from 1990 to 2003

- Crime started dropping in NYC in 1990, four years before Giuliani became mayor.

Misinterpretation of Incomplete Data

Unwed Mothers Study

- In October 1996, NCHS issued data showing that the rate of births to unwed mothers had declined from 46.9 per thousand in 1994 to 44.9 per thousand in 1995. The first decline in 20 years. Front page coverage in the NYtimes and LAtimes.

- Clinton trumpets the results as a success for his new welfare policies (instituted in 1996)

- Not mentioned: from 1993-1994 there was the largest one-year increase in out-of-wedlock births since national figures have been kept

Biased Presentation of Ambiguous Results

- Selective presentation of results

Day Care Study

- In 1996, media publicized the results of a study presented at an NICHD conference claiming that the bond between mothers and babies is not weakened when the child is placed in day care
- Study measured the presence or absence of “secure attachment” in infants
- Overall no difference in day care versus home care babies

Biased Presentation of Ambiguous Results

- Selective presentation of results

Day Care Study

- What the media did not highlight: a more confusing picture emerges when the averages are broken out
- Baby boys were most likely to be insecurely bonded when they were in day care for more than thirty hours a week
- Baby girls were most likely to be insecurely bonded when they were in day care for less than 10 hours a week

Biased Presentation of Ambiguous Results

- Selective presentation of results

Psychology Research

- Researchers sometimes present significant results and fail to present null (or opposing) results.
- Sometimes you can catch them – look at their methods section and see how many tests they must have run and how many they reported.

Biased Presentation of Ambiguous Results

- “Spin” or specialized emphasis of particular results

Mortgage Study

- In 1995 a study by the Federal Reserve Bank of Chicago, showed that among people with bad credit ratings, 10% of white applicants are denied mortgages while 20% of black and Hispanic applicants are denied mortgages.
- In the same study, however, it was found that compared to past years, approved mortgages rose by 55% for black applicants rose by 55%, 45% for Hispanic applicants, and 16% for white applicants.

Biased Presentation of Ambiguous Results

- “Spin” or specialized emphasis of particular results

Mortgage Study

- In 1995 a study by the Federal Reserve Bank of Chicago, showed that among people with bad credit ratings, 90% of white applicants are granted mortgages while 80% of black and Hispanic applicants are granted mortgages.
- In the same study, however, it was found that compared to past years, approved mortgages rose by 55% for black applicants, by 45% for Hispanic applicants, and by only 16% for white applicants.

Biased Presentation of Ambiguous Results

- “Spin” or specialized emphasis of particular results

Mortgage Study

- The NYtimes did not report the second finding until the fourth paragraph of the article
- They also reported denial rates rather than approval rates.
- In approval terms, the comparison is 90% versus 80%. In denial terms, the comparison is 10% versus 20%.
- “Twice as likely to be denied”. (makes you think people of color were half as likely to be accepted, but actually they were 88% as likely to be accepted)

Biased Presentation of Ambiguous Results

- “Spin” or specialized emphasis of particular results

Psychology Research

- Sometimes a p-value of 0.10 is treated as “not significant” (especially if the researcher did not predict the effect)
- Other times the same p-value is emphasized as “marginally significant” (esp if the researcher predicted the effect)

What to Do?

Can’t always trust intuition

Can’t always trust statistical reports

- Learn more about possible pitfalls in intuitive decision making

- Learn more about how to evaluate statistical reports and research findings

Practice

Washington Post April 12, 2000

“Government-funded medical surveys since 1960 have shown higher rates of at least one type of cancer – varying from thyroid tumors to leukemia – at most of the major facilities that produced nuclear weapons.”

What’s problematic about this statement?

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