Statistics Primer ORC Staff: Jayme Palka Peter Boedeker Marcus Fagan Trey Dejong 1
Quick Overview of Statistics 2
Descriptive vs. Inferential Statistics Descriptive Statistics: summarize and describe data (central tendency, variability, skewness) Inferential Statistics: procedure for making inferences about population parameters using sample statistics Sample Population 3
Measures of Central Tendency Mode: the most frequently occurring value in a distribution Select the value(s) with the highest frequency Median: the value representing the middle point of a distribution Order data Determine the median position = (n + 1) / 2 Locate the median based on step 2 Mean: the arithmetic average of a distribution Sum all the data values and divide by the number of values Σx n 4
Measures of Variability Range: difference between the largest and smallest values in the data 5 (X H X L ) Mean deviation: measure of the average absolute deviations from the mean uncommonly used Σ x n x These measures are not very descriptive of a distribution s variability, need better measures 5
Measures of Variability Cont. Sum of squares: sum of the squared deviation scores, used to compute variance and standard deviation SS = Σ x x 2 Variance: the average squared deviations from the mean x 2 s 2 = Σ x n 1 Standard deviation: square root of the variance - commonly used s = Σ x n 1 x 2 6
Variance and Sum of Squares Student x (x x) (x x) 2 SS x x 2 Girl #1 90 Girl #2 23 Girl #3 26 Boy #1 83 S 2 x x n 1 2 Boy #2 48 Boy #3 24 Average = Sum = S x x n 1 2 7
Empirical Rule The empirical rule states that symmetric or normal distribution with population mean μ and standard deviation σ have the following properties. 8
Sampling Distribution Theoretical distribution of sample statistics (e.g., the mean, standard deviation, Pearson s r), as opposed to individual scores NOT the same thing as a sample distribution or a population distribution Used to help generalize the findings of our sample statistics back to our populations Tough to understand, concrete example on next slide 9
Sampling Distribution All possible outcomes are shown below in Table 1. Table 1. All possible outcomes when two balls are sampled with replacement. Outcome Ball 1 Ball 2 Mean 1 1 1 1.0 2 1 2 1.5 3 1 3 2.0 4 2 1 1.5 5 2 2 2.0 6 2 3 2.5 7 3 1 2.0 8 3 2 2.5 9 3 3 3.0 10
Sampling Error As has been stated before, inferential statistics involve using a representative sample to make judgments about a population. Lets say that we wanted to determine the nature of the relationship between county and achievement scores among Texas students. We could select a representative sample of say 10,000 students to conduct our study. If we find that there is a statistically significant relationship in the sample we could then generalize this to the entire population. However, even the most representative sample is not going to be exactly the same as its population. Given this, there is always a chance that the things we find in a sample are anomalies and do not occur in the population that the sample represents. This error is referred as sampling error. 11
Sampling Error A formal definition of sampling error is as follows: Sampling error occurs when random chance produces a sample statistic that is not equal to the population parameter it represents. Due to sampling error there is always a chance that we are making a mistake when rejecting or failing to reject our null hypothesis. Remember that inferential procedures are used to determine which of the statistical hypotheses is true. This is done by rejecting or failing to reject the null hypothesis at the end of a procedure. 12
Sampling Distribution and Standard Error (SE) https://www.youtube.com/watch?v=hviduemwt2k 13
Hypothesis Testing Null Hypothesis Significance Testing (NHST) Testing p-values using statistical significance tests (image from cnx.org) Effect Size Measure magnitude of the effect (e.g., Cohen s d) 14
Null Hypothesis Significance Testing Statistical significance testing answers the following question: Assuming the sample data came from a population in which the null hypothesis is exactly true, what is the probability of obtaining the sample statistic one got for one s sample data with the given sample size? (Thompson, 1994) Alternatively: Statistical significance testing is used to examine a statement about a relationship between two variables. 15
Hypothetical Example Is there a difference between the reading abilities of boys and girls? Null Hypothesis (H 0 ): There is not a difference between the reading abilities of boys and girls. Alternative Hypothesis (H 1 ): There is a difference between the reading abilities of boys and girls. Alternative hypotheses may be non-directional (above) or directional (e.g., boys have a higher reading ability than girls). 16
Testing the Hypothesis Use a sampling distribution to calculate the probability of a statistical outcome. p calc = likelihood of the sample s result p calc < p critical : reject H 0 p calc p critical : fail to reject H 0 17
Level of Significance (p crit ) Alpha level (α) determines: The probability at which you reject the null hypothesis The probability of making a Type I error (typically.05 or.01) Observed Outcome True Outcome in Population Reject H 0 is true H 0 is false Reject H 0 Type I error (α) Correct Decision Fail to reject H 0 Correct Decision Type II error (β) 18
Example: Independent t-test Research Question: Is there a difference between the reading abilities of boys and girls? Hypotheses: H 0 : There is not a difference between the reading abilities of boys and girls. H 1 : There is a difference between the reading abilities of boys and girls. 19
Dataset Reading test scores (out of 100) Boys Girls 88 88 82 90 70 95 92 81 80 93 71 86 73 79 80 93 85 89 86 87 20
Significance Level α =.05, two-tailed test df = n 1 + n 2 2 = 10 + 10 2 = 18 Use t-table to determine t crit t crit = ±2.101 21
Decision Rules If t calc > t crit, then p calc < p crit Reject H 0 If t calc t crit, then p calc p crit Fail to reject H 0 p =.025 p =.025-2.101 2.101 22
Computations Boys Girls Frequency (N) 10 10 Sum (Σ) 807 881 Mean ( X) 80.70 88.10 Variance (S 2 ) 55.34 26.54 Standard Deviation (S) 7.44 5.15 23
Computations cont. Pooled variance = 40.944 Standard Error = 2.862 24
Computations cont. Compute t calc t = X 1 X 2 SE X 1 X 2 = -2.586 Decision: Reject H 0. Girls scored statistically significantly higher on the reading test than boys did. 25
Confidence Intervals Sample means provide a point estimate of our population means. Due to sampling error, our sample estimates may not perfectly represent our populations of interest. It would be useful to have an interval estimate of our population means so we know a plausible range of values that our population means may fall within. 95% confidence intervals do this. Can help reinforce the results of the significance test. CI 95 = x ± t crit (SE) = -7.4 ± 2.101(2.862) = [-13.412, -1.387] 26
Statistical Significance vs. Importance of Effect Does finding that p <.05 mean the finding is relevant to the real world? Not necessarily https://www.youtube.com/watch?v=5ol1rqhrzq8 Effect size provides a measure of the magnitude of an effect Practical significance Cohen s d, η 2, and R 2 are all types of effect sizes 27
Cohen s d Equation: = -1.16 Guidelines: d =.2 = small d =.5 = moderate d =.8 = large Not only is our effect statistically significant, but the effect size is large. 28