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Advanced Psychological Statistics
Standard Deviation - The distance of a value in a population (or sample) from the mean value of the population (or sample). Probability - The mathematical Language of uncertainty. A measure of the likelihood of an event. It is the ratio of the number of ways a certain event can occur to the number of possible outcomes. A possibility of genetic combinations for the next generation from test crosses. Axioms of Probability - All probabilities greater than or equal to 0; P (Omega) = 1; additive rule (probability of A is the sum of the probabilities of a partition's parts) Subjective Probability - Uses a probability value based on an educated guess or estimate, employing opinions and inexact information. Frequentist probability - a true frequentist only relies on past data. Mutual exclusivity (disjoint) - two events are mutually exclusive if they can not both be true. Independent events - the probability of one event occurring in no way affects the probability of the other event occurring
Can an event be both mutually exclusive and independent? - Yes. if the probability of one event is equal to zero. See multiplication rule of independent events. Bayes Theorem - describes the probability of an event occurring based on conditions that might be related to the event. Instead of our usual P(D/H) Bayes theorem uses P(H/D). Getting the prior probability (D), in this case is extremely important, and rather contentious. Limits - We're not so interested in how the function behaves at a particular point, but more of how the function behaves as you approach a particular point Derivative - the slope of a tangent line at a given point. The rate of change at a given point. In regards to driving- speed. Derivative of a derivative - How much the change is changing- in a car, acceleration Integration - area under a curve. Statistical Inference - P(D/Ho) Process of deducing properties of an underlying distribution by analysis of data. The population is assumed to be larger than the observed data set; in other words, the observed data is assumed to be sampled from a larger population. Makes presumptions about a population, using data from the population with some sort of sampling. Given a hypothesis about a population, for which we wish to draw inferences. Consists of; 1. selecting a model of the process that generates the data. 2. deducing propositions from the model.
A variable in which the units are in the whole numbers, or "discrete" units (for example, number of children, number of defects). continuous variable - A variable (such as age, test score, or height) that can take on a wide or infinite number of values. nominal scale - most basic, does not provide measurable info, merely classifies names, labels, or identifies by group, has NO TRUE ZERO point and DOES NOT INDICATE ORDER ordinal scale - A scale of measurement using ranks rather than actual numbers. Interval scale - Scale of measurement with the property of order (magnitude) and equal intervals (equal distances between points or values), but without a true zero. Can add & subtract. Can add to calculate a mean or standard deviation. Score of 0 (z-score = 0) is arbitrary, does not imply the absence of the trait being measured (0 F). (ex. scores on most standardized tests) Use mode (Mo) or Median (Md) or (preferred) Mean (M) to measure central tendency of data on this scale of measurement. Think IQ Ratio Scale - What scale has a true zero point, graded into equal increments, and also orders them? random variable - A variable whose value is determined by the outcomes of a probability experiment statistical significance - a statistical statement of how likely it is that an obtained result occurred by chance a statistical criterion for rejecting the assumption of no differences in a particular study
Scientists have decided that 5% is the cutoff for statistically significant results. This means that in an experiment design, there must be less than a 5% chance that the results occurred by chance. Estimation - You're going to use a sample to talk about populations. Pretty much all psychological experiments. the beautiful triangle - Losing a degree of freedom, not losing a degree of freedom in a population because everything is set. Standard Deviation - The square root of variance Variance - A measure of spread within a distribution (the square of the standard deviation). skewness - a measure of the shape of a data distribution. Data skewed to the left result in negative skewness; a symmetric data distribution results in zero skewness; and data skewed to the right result in positive skewness kurtosis - the peakedness of a frequency distribution Z test - a statistical test for means and proportions of a population, used when the population is normally distributed and the population standard deviation is known Z= x-u/ (population variance, standard error)
student's t-test - any statistical hypothesis test in which the test statistic follows a Student's t-distribution if the null hypothesis is supported. It can be used to determine if two sets of data are significantly different from each other, and is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a Student's t distribution. Central Limit Theorem - If a random variable y has a population mean μ and population variance σ2, then the sample mean, y, based on n observations, has an approximate normal distribution with mean μ and variance σ2, for sufficiently large n. (p. 64) n More generally, the theorem pertains to the limiting form of the cumulative distribution function (cdf) of a normal random variable (Casella and Berger, 2002, p. 236). Asymptotically, the distribution of a normal random variable converges to that of a normal distribution as n approaches infinity. You have to have at least n=30 for it to work z score - a measure of how many standard deviations you are away from the mean t score - on a t distribution, number of standard deviations from the mean effect size - in studies involving means of one or two groups, measure of difference between populations Statistical Power - The probability that the null hypothesis will be rejected if it is false. A higher statistical power means that we can be more certain that the null hypothesis was not rejected incorrectly or achieve a lower type 1 error alpha level -
The ____________ is the probability of making a type I error; it is designated at the end of the tail in a distribution. Is alpha equal to power? - no
- α ≠ p value or power α is a criteria we set arbitrarily and is the probability of making a Type I error. The p value is an actual value that is obtained based on the distribution of interest that we can compare to the α value. The p value states the actual probability (based on the integral calculation until the distribution curve) of obtaining the data statistic of interest or greater. 1 ¬- confidence interval (if not in %) = α CI and α are values applied to the Ho distribution Power comes into play when we have an alternative distribution and is the probability of rejecting the null hypothesis given that the null is false. Power = 1 - β ≠ α β is associated with the H1 distribution, α with Ho β is the probability of making a Type II error Picture the overlapping distributions The vertical line he put that crossed both distributions is the criteria we set (α) The entire area to the right of that line, including the portion of the Ho distribution on the right side of the criteria line (because that is also still part of the H1 distribution) is the power, 1 - β. The small portion of the Ho distribution on the right side of the criteria line is α for the Ho distribution. The portion of the H1 distribution on the left side of the line is β. p value - An inferential statistic that indicates if the data from an experiment are statistically significant. In order for the results to be statistically significant, this must be =.05.
