The Normal Curve Theoretical Symmetrical Known Areas For Each Standard Deviation or Z-score FOR EACH SIDE:  34.13% of scores in distribution are b/t the mean and 1 s from the mean  13.59% of scores are between 1 and 2 s’s from the mean  2.28% of scores are > 2 s’s from the mean

Z SCORE FORMULA Z = X i – X S X i = 120; X = 100; s=10 Z= 120 – 100 = The point is to convert your particular metric (e.g., height, IQ scores) into the metric of the normal curve (Z-scores). If all of your values were converted to Z- scores, the distribution will have a mean of zero and a standard deviation of one.

Normal Curve  Probability Important Link Normal curve is limited as most real world data is not “normally distributed” More important use has to do with probability theory and drawing samples from a population

Probability Basics What is the probability of picking a red marble out of a bowl with 2 red and 8 green? There are 2 outcomes that are red THERE ARE 10 POSSIBLE OUTCOMES p(red) = 2 divided by 10 p(red) =.20

Frequencies and Probability The probability of picking a color relates to the frequency of each color in the bowl 8 green marbles, 2 red marbles, 10 total p(Green) =.8 p(Red) =.2

Frequencies & Probability What is the probability of randomly selecting an individual who is extremely liberal from this sample? p(extremely liberal) = 32 =.024 (or 2.4%) 1,319

PROBABILITY & THE NORMAL DISTRIBUTION We can use the normal curve to estimate the probability of randomly selecting a case between 2 scores Probability distribution: Theoretical distribution of all events in a population of events, with the relative frequency of each event

PROBABILITY & THE NORMAL DISTRIBUTION The probability of a particular outcome is the proportion of times that outcome would occur in a long run of repeated observations. 68% of cases fall within +/- 1 standard deviation of the mean in the normal curve The odds (probability) over the long run of obtaining an outcome within a standard deviation of the mean is 68%

Probability & the Normal Distribution Suppose the mean score on a test is 80, with a standard deviation of 7. If we randomly sample one score from the population, what is the probability that it will be as high or higher than 89?  Z for 89 = 89-80/7 = 9/7 or 1.29  Area in tail for z of 1.29 =  P(X > 89) =.0985 or 9.85% ALL WE ARE DOING IS THINKING ABOUT “AREA UNDER CURVE” A BIT DIFFERENTLY (SAME MATH)

Probability & the Normal Distribution Bottom line: Normal distribution can also be thought of as probability distribution Probabilities always range from 0 – 1  0 = never happens  1 = always happens  In between = happens some percent of the time  This is where our interest lies

Inferential Statistics Inferential statistics are used to generalize from a sample to a population We seek knowledge about a whole class of similar individuals, objects or events (called a POPULATION) We observe some of these (called a SAMPLE) We extend (generalize) our findings to the entire class

WHY SAMPLE? Why sample? It’s often not possible to collect info. on all individuals you wish to study Even if possible, it might not be feasible (e.g., because of time, $, size of group)

WHY USE PROBABILITY SAMPLING? Representative sample One that, in the aggregate, closely approximates the population from which it is drawn

PROBABILITY SAMPLING Samples selected in accord with probability theory, typically involving some random selection mechanism  If everyone in the population has an equal chance of being selected, it is likely that those who are selected will be representative of the whole group  EPSEM – Equal Probability of SElection Method

PARAMETER & STATISTIC Population the total membership of a defined class of people, objects, or events Parameter the summary description of a given variable in a population Statistic the summary description of a variable in a sample (used to estimate a population parameter)

INFERENTIAL STATISTICS Samples are only estimates of the population Sample statistics will be slightly off from the true values of its population’s parameters Sampling error:  The difference between a sample statistic and a population parameter

μ = 4.5 (N=50) x=7 x=0 x=3 x=1 x=5 x=8 x=5 x=3 x=8 x=7 x=4 x=6 x=2 x=8 x=4 x=5 x=9 x=4 x=5 x=9 x=3 x=0 x=6 x=5 x=1 x=7 x=3 x=4 x=5 x=6 EXAMPLE OF HOW SAMPLE STATISTICS VARY FROM A POPULATION PARAMETER X=4.0 X=5.5 X=4.3 X=5.3X=4.7 CHILDREN’S AGE IN YEARS

By Contrast: Nonprobability Sampling Nonprobability sampling may be more appropriate and practical than probability sampling: When it is not feasible to include many cases in the sample (e.g., because of cost) In the early stages of investigating a problem (i.e., when conducting an exploratory study) It is the only viable means of case selection: If the population itself contains few cases If an adequate sampling frame doesn’t exist

Nonprobability Sampling: 2 Examples 1. CONVENIENCE SAMPLING When the researcher simply selects a requisite number of cases that are conveniently available 2. SNOWBALL SAMPLING Researcher asks interviewed subjects to suggest additional people for interviewing

Probability vs. Nonprobability Sampling: Research Situations For the following research situations, decide whether a probability or nonprobability sample would be more appropriate: 1. You plan to conduct research delving into the motivations of serial killers. 2. You want to estimate the level of support among adult Duluthians for an increase in city taxes to fund more snow plows. 3. You want to learn the prevalence of alcoholism among the homeless in Duluth.

( Back to Probability Sampling…) The “Catch-22” of Inferential Stats: When we collect a sample, we know nothing about the population’s distribution of scores We can calculate the mean (X) & standard deviation (s) of our sample, but  (population mean) and  (population standard deviation) are unknown The shape of the population distribution (normal?) is also unknown  Exceptions: IQ, height

PROBABILITY SAMPLING 2 Advantages of probability sampling: 1. Probability samples are typically more representative than other types of samples 2. Allow us to apply probability theory  This permits us to estimate the accuracy or representativeness of the sample

SAMPLING DISTRIBUTION From repeated random sampling, a mathematical description of all possible sampling event outcomes (and the probability of each one) Permits us to make the link between sample and population… & answer the question: “What is the probability that sample statistic is due to chance?” Based on probability theory

μ = 4.5 (N=50) x=7 x=0 x=3 x=1 x=5 x=8 x=5 x=3 x=8 x=7 x=4 x=6 x=2 x=8 x=4 x=5 x=9 x=4 x=5 x=9 x=3 x=0 x=6 x=5 x=1 x=7 x=3 x=4 x=5 x=6 Imagine if we did this an infinite amount of times… X=4.0 X=5.5 X=4.3 X=5.3X=4.7 CHILDREN’S AGE IN YEARS

What would happen… (Probability Theory) If we kept repeating the samples from the previous slide millions of times? What would be our most common sample mean? The population mean What would the distribution shape be? Normal This is the idea of a sampling distribution Sampling distribution of means

Relationship between Sample, Sampling Distribution & Population POPULATION SAMPLING DISTRIBUTION (Distribution of sample outcomes) SAMPLE Empirical (exists in reality) but unknown Nonempirical (theoretical or hypothetical) Laws of probability allow us to describe its characteristics (shape, central tendency, dispersion) Empirical & known (distribution shape, mean, standard deviation)

TERMINOLOGY FOR INFERENTIAL STATS Population  the universe of students at the local college Sample  200 students (a subset of the student body) Parameter  25% of students (p=.25) reported being Catholic; unknown, but inferred from sample statistic Statistic  Empirical & known: proportion of sample that is Catholic is 50/200 = p=.25 Random Sampling (a.k.a. “Probability”)  Ensures EPSEM & allows for use of sampling distribution to estimate pop. parameter (infer from sample to pop.) Representative  EPSEM gives best chance that the sample statistic will accurately estimate the pop. parameter