Some Key Ingredients for Inferential Statistics Chapter 4 Some Key Ingredients for Inferential Statistics The Normal Curve, Probability, and Population Versus Sample Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Inferential Statistics Methods used by social and behavioral scientists to go from results of research studies to conclusions about theories or applied procedures What most of statistics entails Beyond mere descriptive statistics Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

The Normal Curve Bell-shaped Unimodal Symmetrical Exactly half of the scores above the mean Exactly half of the scores below the mean Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

The Normal Curve There are known percentages of scores above or below any given point on a normal curve 34% of scores between the mean and 1 SD above or below the mean An additional 14% of scores between 1 and 2 SDs above or below the mean Thus, about 96% of all scores are within 2 SDs of the mean (34% + 34% + 14% + 14% = 96%) Note: 34% and 14% figures can be useful to remember Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Normal Curve Table Normal curve table gives the precise percentage of scores between the mean (Z score of 0) and any other Z score. Can be used to determine Proportion of scores above or below a particular Z score Proportion of scores between the mean and a particular Z score Proportion of scores between two Z scores Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Normal Curve Table Continued By converting raw scores to Z scores, can be used in the same way for raw scores. Can also use it in the opposite way Determine a Z score for a particular proportion of scores under the normal curve Table lists positive Z scores Can work for negatives too Why? Because curve is symmetrical Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Steps for Figuring Percentage Above or Below a Z Score Convert raw score to Z score, if necessary Draw a normal curve Indicate where Z score falls Shade area you’re trying to find Make rough estimate of shaded area’s percentage Find exact percentage with normal curve table Check to verify that it’s close to your estimate Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Steps for Figuring a Z Score or Raw Score From a Percentage Draw normal curve, shading approximate area for the percentage desired Make a rough estimate of the Z score where the shaded area starts Find the exact Z score using normal curve table Check to verify that it’s close to your estimate Convert Z score to raw score, if desired Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Probability Abbreviated as p, as in “p < .05” Expected relative frequency of a particular outcome Probability = possible successful outcomes divided by all possible outcomes Represented as Proportion (number between 0 and 1) Percentage (between 0% and 100%) Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Probability and Frequency Distributions For any frequency distribution the percentage of scores in a particular region corresponds to the probability of selecting a score from that region. For example, the normal curve Histogram to the right 10 out of 50 people scored 7 or higher. Thus, the probability of randomly selecting a person with a score of 7 or higher is 10/50, or .20 Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Sample vs. Population Sample Population Relatively small number of instances that are studied in order to make inferences about a larger group from which they were drawn Population The larger group from which a sample is drawn Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Sample vs. Population Examples a. pot of beans b. larger circle c. histogram Sample a. spoonful b. smaller circle c. shaded scores Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Why Study Samples? Often not practical to study an entire population Instead, researchers attempt to make samples representative of populations Random selection Each member of population has an equal chance of being sampled Good but difficult Haphazard selection Take steps to ensure samples do not differ from the population in systematic ways Not as good but much more practical Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall