PSY 1950Chance, Probability, and Sampling

September 24, 2008

vs

Probability: Perspectives• Analytic: possible outcomes

– theoretical

• Relative frequency: past performance– empirical

• Subjective: belief– Psychological

Probability: Applications• Is data-generating process random?– Yes

•Debunk or study psychological bias to see patterns – Chance is lumpy, brains are pattern-detectors

– e.g., bushy tiger vs. tigery bush– e.g., hot hand in basketball

•Debunk patterns vs. affirming randomness

– No•Pattern demands explanation

Probability as Area

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Probability as Area

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NORMDIST(x, mean, standard_dev, cumulative) NORMSDIST(z) NORMINV(probability, mean, standard_dev) NORMSINV(probability)

The Normal Distribution

Why The Normal Distribution?

• Variables are often (or are often assumed to be) normally distributed in population– e.g., Quetelet’s (1835) measurements of heights– e.g., IQ scores

• Errors are often (or are often assumed to be) normally distributed– Sampling error (cf. terminology: normal, error)

• Assuming (approximate) normality allows inference

• Assuming normality enables parametric statistics– Normal distributions have “amazing” mathematical properties• Linear combinations of scores from two normally distributed variables are themselves normally distributed!

Binomial Distribution• Two possible outcomes• Constant outcome probability• Trial-to-trial independence• If pn and qn 10:

Binomial Distribution

http://www.socr.ucla.edu/htmls/SOCR_Distributions.html

Example: Missing girls• 1049 males are born in the world for every 1000 females. From 2000-2005, there were approximately 17 million children born in China, approximately 7,730,000 of whom were female. What are the odds that this is by chance?

Sampling• Overconfidence abhors uncertainty

– Law of small numbers– Correspondence bias– Overconfidence bias

• Bias– e.g., Who likes statistics?

•Characteristics of sample: representativeness

•Measurement of sample: response, non-response

– e.g., How many children in your family•Sampling unit: people vs. families

– Only family with children are represented– Families with multiples children are overrepresented

– How would you obtain a representative sample?

Sampling Terms• Sampling error

– Variability of a statistic from sample to sample due to chance

• Sampling distribution– The distribution of a statistic over repeated sampling from a specified population

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X1 X2 X3 X4 X5

5 8 9 12 6

5 7 8 9 6

7 8 7 7 6

7 7 7 7 7

Standard Error (of the Mean)

• Standard deviation of the distribution of sample means

• Estimate of how well, on the average, a sample mean estimates its population mean

• Expected error• Depends on sample size

– Law of large numbers

• Depends on population variability• Is not standard deviation of sample (s) or distribution ()

Standard Error vs Standard Deviation

• Standard deviation– Descriptive statistic– Measure of dispersion– Standard distance between scores and their mean

– Does not depend on sample size

• Standard error (of the mean)– Inferential statistic*– Measure of precision– Standard distance between sample means and population mean

– Depends on sample size

• Standard error is type of standard deviation

• Equal when n = 1

Central Limit Theorem• For any population with mean and standard deviation , the distribution of sample means for sample size n will:– have a mean of – have a standard deviation of /√n – will approach a normal distribution as n approaches infinity

• Valid for any population– Explains normal distribution of many psychological variables

• Distribution of sample means approaches normal distribution very quickly (n 30)

Central Limit Theorem

Sampling distribution (CLT) experiment

http://www.socr.ucla.edu/htmls/SOCR_Experiments.html

• Examine mean, standard deviation, skewness, and kurtosis of sample mean

• Try different sample sizes• Try other sample statistics (e.g.,

variance)• Sampling from different (e.g.,

Poisson) distributions

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