Statistics Primer ORC Staff: Xin Xin (Cindy) Ryan Glaman Brett Kellerstedt 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 SamplePopulation 3

Measures of Central Tendency Raw dataSimple frequency distribution Group frequency distribution Notations Mode Pick out the value (s) occurring more than any other value. Pick out the value (s) with the highest frequency. = Difference between the freq. of modal class and the freq. of the next lower class. = Difference between the freq. of modal class and the freq. of the next higher class. L 1 = Lower class boundary of the modal class c = class width of the modal class Median 1. Order data 2. Determine median position = (n+1)/2 3. Locate median based on step Order data 2. Determine median position = (n+1)/2 3. Locate median based on step 2 using the freq. column L m =lower class boundary of median class n = sample size C.F. = sum of all frequencies lower than the median class f med = frequency of the median class c = class width of the median class Mean Add up all the data values and divide by the number of values. Find the product of all the values and their frequencies ; then add all the products; and finally divide by the total frequency. Find the product of all the midpoints and their frequencies ; then add all the products; and finally divide by the total frequency. X = the actual values (for raw data and ungrouped freq. dist.) = midpoints (for group freq. dist.) f = frequency n = sample size N = population size = summation or sum of 4

DescriptionApplicabilityAdvantageDisadvantage Range Difference between the largest and the smallest value in the data. 1. Interval/ratio 2. No outliers exist 1. Simple to calculate 1. Highly influenced by outliers. 2. Does not use all data Mean deviation It measures the average absolute deviations from the mean. Uncommonly used 1. Interval/ratio 2. When no outliers exist 1. Use all the data 2. Easy to interpret 1. Not resistant to outliers 2. Does not yield any further useful statistical properties. Variance/ standard deviation Variance is the average squared deviations from the mean. Standard deviation is square root of the variance. Commonly used. 1. Interval/ratio 2. When no outliers exist 1. Provides good statistical properties, by avoiding the use of absolute values. 2. Use all the data 1. Not resistant to outliers. 2. Variance depends on the units of measurement, therefore not easy to make comparisons. Sum of Squares Measures variability of the scores, the total variation of all scores 1. Interval/ratio 2. When no outliers exist 1. Effect size calculation 1. Not resistant to outliers. 5 Measures of Variability 5

Variance and Sum of Squares 6

Empirical Rule  The empirical rule states that symmetric or normal distribution with population mean μ and standard deviation σ have the following properties. 7

OutcomeBall 1Ball 2Mean All possible outcomes are shown below in Table 1. Table 1. All possible outcomes when two balls are sampled with replacement. Sampling Distribution 8

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. 9

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. 10

Sampling Distribution and Standard Error (SE) 

Hypothesis Testing  Null Hypothesis Statistical Significance Testing (NHSST)  Testing p-values using statistical significance tests  Effect Size  Measure magnitude of the effect (e.g., Cohen’s d) 12

Null Hypothesis Statistical 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. 13

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). 14

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 15

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) True Outcome in Population Reject H 0 is trueH 0 is false Observed Outcome Reject H 0 Type I error (α)Correct Decision Fail to reject H 0 Correct DecisionType II error (β) 16

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. 17

Dataset  Reading test scores (out of 100) BoysGirls

Significance Level  α =.05, two-tailed test  df = n 1 + n 2 – 2 = – 2 = 18  Use t-table to determine t crit  t crit = ±

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 p =

Computations BoysGirls Frequency (N)10 Sum (Σ) Variance (S 2 ) Standard Deviation (S)

Computations cont.  Pooled variance  Standard Error = =

Computations cont.  Compute t calc  Decision: Reject H 0. Girls scored statistically significantly higher on the reading test than boys did. =

Confidence Intervals 24

Statistical Significance vs. Importance of Effect  Does finding that p <.05 mean the finding is relevant to the real world?  Not necessarily…   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 25

Cohen’s d  Equation:  Guidelines:  d =.2 = small  d =.5 = moderate  d =.8 = large  Not only is our effect statistically significant, but the effect size is large. =