Z-scorespecifies the precise location of each X value within a distribution. The sign of the z-score (+ or -) signifies whether the score is above or below the mean. The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and μ.z-score serves two purposes:1. It tells us exactly where a score is compared to the rest of the distribution
2. It standardizes the distribution, allowing us to compare scores across tests (allows us to compare student A to student B).Z Score ShapeThe shape of a z-score distribution will be the same as the original distribution of raw scores.Z Score MeanThe mean of a z-score distribution is always equal to zero.Z Score Standard DeviationThe standard deviation of a z-score distribution is always equal to one.Standardized distributioncomposed of scores that have been transformed to create predetermined values for μ and σ. Standardized distributions are used to make dissimilar distributions comparable.probabilitydefined as a fraction or a proportion of all the possible outcomes.Random Samplingeach individual in the population has an equal chance of being selected. The probabilities of being selected must remain constant from one selection to the next if more than one individual is selected.sampling with replacementwhen you pick an individual from a population, you return that individual back to the population before picking again.Sampling errorthe natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.ΣSummationsSample standard deviationXRaw scoreZZ scoreσ2Population varianceMSample meannSample number of scoresμPopulation means2Sample varianceSSSum of squaresNPopulation number of scoresσPopulation standard deviationDistribution of sample meanscollection of sample means for all possible random samples of a particular size (n) that can be obtained from a population.Sampling distributiona distribution of statistics obtained by selecting all the possible samples of a specific size (n) from a population.Characteristics of the distribution of sample means1. The sample means should pile up around the population mean.
2. The pile of sample means should tend to form a normal distribution.
3. The larger the sample size, the closer the sample means should be to the population mean.The distribution of sample means is almost perfectly normal if1. The population the sample means are from is normal.
2. The number of scores (sample size n) in each sample is around 30 or more.expected value of MThe mean of the distribution of sample means is equal to the mean of the population scores, μstandard error of MThe standard deviation of a distribution of sample meanstwo purposes for the standard error as the standard deviation1. The standard deviation describes the distribution of scores.
2. The standard error measures how well an individual sample mean represents the entire distribution.Standard ErrorHow much distance we expect, on average, between a sample mean and the population mean.Law of Large NumbersThe larger the sample size (n), the more probable it is that the sample mean will be close to the population mean.
