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Statistics. Statistics may be defined as a body of methods for making wise decisions in the face of uncertainty. W. Allen Wallis Economist & Statistician. Statistics Terms. • Statistics: Procedures used to summarize and analyze quantitative data.
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Statistics may be defined as a body of methods for making wise decisions in the face of uncertainty.W. Allen WallisEconomist & Statistician
Statistics Terms • • Statistics: Procedures used to summarize and analyze quantitative data. • • Descriptive statistics: Procedures used to summarize a set of numbers in terms of central tendency, variation, or relationships. • • Inferential statistics: Procedures used to determine the error when estimating a value for a population based upon the measurement of the same value for a sample of that population.
Types of Descriptive Statistics • • Central Tendency: The typical score (best bet). • • Variability: How different the scores are. • • Correlation Coefficient: A measure of the relationship between two variables. • • z-Score: The relationship of one score to the norm group in terms of standard units. • • Effect Size: A measure of the magnitude and difference of the means of two groups.
Descriptive Statistics • • Measures of Central Tendency • - Mean: The arithmetic average, sensitive to outliers • - Median: The middle score, reduces effect of outliers • - Mode: The most frequent score • • Measures of Variability • - Range: The difference between the largest and smallest. • - Standard Deviation: The average distance of all scores • from the mean. • • Correlation Coefficient • - How related two variables are, predictability. • - Sensitive to outliers (moving R closer to zero).
z-Score • The quantity z represents the distance between the raw score (of an individual’s score, for instance) and the group mean in units of the standard deviation. z is negative when the raw score is below the mean and positive when above.
Effect Size • The quantity ES represents the difference between the mean of the experimental group and the mean of the control group in units of the standard deviation.
Inferential Statistics • • The purpose of inferential statistics is to make conclusions about some value of a population on the basis of that same value measured for a sample. • • Inferential statistics allow us to estimate the magnitude of our error—the difference between the sample value and the population value—even though we don’t know what the population value is. • • One estimate of error is the “confidence interval”—a range within which the true value is likely (%) to be. The wider the range, the higher the confidence level.
Sampling Error • • It’s always easier and quicker to measure a sample drawn from a population than it is to measure every person in the population (a census). • • Unfortunately, the value for the sample is never exactly equal to the true population value. This is called sampling error (error due to sampling). • • The larger the percentage of the population that is sampled, the smaller the sampling error. (Think about the increase in accuracy by moving from a sample of 50% of the population to 99%.)
Sampling Fluctuation • • Sampling fluctuation occurs when we measure a value for samples repeatedly drawn from the same population. The value for each sample is different from the others (and different from the true value of the population). — b = 71” Population x = 70” — — a = 66” c = 73” —
Sampling Fluctuation Example • • Five people each grab a fistful of coins from a bucket. • • You would expect each to grab a different amount. • • When one person’s amount is much different from another’s, you could say there is a statistically significant difference between their “grab” and the other’s “grab.” • The difference between the two is larger than you would expect than from sampling fluctuation alone. $5.42 $8.23 $5.25 $5.58 $5.12
Statistical Significance • • Statistical significance is a mathematical test that gives a yes/no answer to the question: “Are the differences we see larger than we would expect than from sampling fluctuation alone?” • - It doesn’t tell us which value is larger. • - It doesn’t tell us how big the difference is. • - It doesn’t tell us how important the difference is. • - And because statistical significance is based on the size of the sample, one experiment may have statistically significant results while another may not simply because the sample sizes were different.
Practical Significance • • Practical significance answers the all-important question of “So what?” • • Statistical significance tells us whether the differences are larger than we would expect to see than from sampling fluctuation alone. • • Effect size tells us the magnitude and the direction of the differences. • • Practical significance tells us how important the differences are in terms of what people value.
Do SAT Scores Predict College GPA? • • Based on your experience at Vanderbilt: • – Does it seem like the students with the highest SAT scores have the highest GPA? • • Think about the kids you knew in high school: • – Did you know smart kids with low SAT scores? • – Did you know kids that weren’t that bright who were able to achieve high SAT scores? • • Do you think that high school SAT scores predict college GPA?
