Psych 230 Psychological Measurement and StatisticsToday….Symbols and Definitions ReviewedDefinitions: Populations and SamplesSymbols and Definitions: MeanSymbols and DefinitionsSymbols and Definitions: VariabilitySlide 8Normal Distribution and the Standard DeviationNormal Distribution and the Standard DeviationUnderstanding Z-ScoresThe Next StepSlide 13Z-ScoresFrequency Distribution of Attractiveness ScoresSlide 16Slide 17Slide 18Slide 19Slide 20Raw Score to Z-Score FormulaZ-Scores - ExampleZ-Scores - Your TurnSlide 24Z-Score to Raw Score FormulaZ-Score to Raw Score : ExampleZ-Score to Raw Score : Your TurnSlide 28Using Z-scoresUses of Z-ScoresSlide 31Z-DistributionDistribution of Attractiveness ScoresZ-Distribution of Attractiveness ScoresSlide 35Characteristics of the Z-DistributionSlide 37Slide 38Using Z-Scores to compare variablesSlide 40Comparison of two Z-DistributionsSlide 42Slide 43Using Z-Scores to compute relative frequencyProportions of Area under the Standard Normal CurveRelative FrequencySlide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Z-Scores - Your turnSlide 61Using Z-scores to describe sample meansSlide 63Sampling Distribution of MeansSlide 65Slide 66Slide 67Slide 68Slide 69Slide 70Central Limit TheoremSampling Distribution of Means: CharacteristicsSlide 73Slide 74Slide 75Slide 76Standard Error of the MeanStandard Error of the Mean - ExampleSlide 79Z-Score Formula for a Sample MeanZ-Score for a Sample Mean - ExampleSampling Distribution of Means - Why?Done for todayPsych 230Psychological Measurement and StatisticsPedro WolfSeptember 16, 2009Today….•Symbols and definitions reviewed•Understanding Z-scores•Using Z-scores to describe raw scores•Using Z-scores to describe sample meansSymbols and Definitions ReviewedDefinitions: Populations and Samples•Population : all possible members of the group of interest•Sample : a representative subset of the populationSymbols and Definitions: Mean•Mean–the most representative score in the distribution–our best guess at how a random person scored•Population Mean = x•Sample Mean = XSymbols and Definitions•Number of Scores or Observations = N•Sum of Scores = ∑X•Sum of Deviations from the Mean = ∑(X-X)•Sum of Squared Deviations from Mean = ∑(X-X)2•Sum of Squared Scores = ∑X2•Sum of Scores Squared = (∑X)2Symbols and Definitions: Variability•Variance and Standard Deviation–how spread out are the scores in a distribution –how far the is average score from the mean•Standard Deviation (S) is the square root of the Variance (S2)•In a normal distribution:–68.26% of the scores lie within 1 std dev. of the mean–95.44% of the scores lie within 2 std dev. of the meanSymbols and Definitions: Variability•Population Variance = 2X•Population Standard Deviation = X•Sample Variance = S2x•Sample Standard Deviation = Sx•Estimate of Population Variance = s2x•Estimate of Population Standard Deviation = sxNormal Distribution and the Standard Deviation Mean=66.57Var=16.736StdDev=4.091HEIGHT81767166615651HEIGHTFrequency1412108642062.48 70.6658.38 74.75Normal Distribution and the Standard Deviation•IQ is normally distributed with a mean of 100 and standard deviation of 15 70 85 100 115 13013% 13%Understanding Z-ScoresThe Next Step•We now know enough to be able to accurately describe a set of scores–measurement scale–shape of distribution–central tendency (mean)–variability (standard deviation)•How does any one score compare to others in the distribution?The Next Step•You score 82 on the first exam - is this good or bad?•You paid $14 for your haircut - is this more or less than most people?•You watch 12 hours of tv per week - is this more or less than most?•To answer questions like these, we will learn to transform scores into z-scores –necessary because we usually do not know whether a score is good or bad, high or lowZ-Scores•Using z-scores will allow us to describe the relative standing of the score–how the score compares to others in the sample or populationFrequency Distribution of Attractiveness ScoresFrequency Distribution of Attractiveness ScoresInterpreting each score in relative terms:Slug: below mean, low frequency score, percentile lowBinky: above mean, high frequency score, percentile mediumBif: above mean, low frequency score, percentile highTo calculate these relative scores precisely, we use z-scoresZ-Scores•We could figure out the percentiles exactly for every single distribution–e ≈ 2.7183, π≈ 3.1415•But, this would be incredibly tedious•Instead, mathematicians have figured out the percentiles for a distribution with a mean of 0 and a standard deviation of 1–A z-distribution•What happens if our data doesn’t have a mean of 0 and standard deviation of 1?–Our scores really don’t have an intrinsic meaning–We make them up•We convert our scores to this scale - create z-scores•Now, we can use the z-distribution tables in the bookZ-Scores•First, compare the score to an “average” score•Measure distance from the mean–the deviation, X - X–Biff: 90 - 60 = +30–Biff: z = 30/10 = 3–Biff is 3 standard deviations above the mean.Z-Scores•Therefore, the z-score simply describes the distance from the score to the mean, measured in standard deviation units•There are two components to a z-score:–positive or negative, corresponding to the score being above or below the mean–value of the z-score, corresponding to how far the score is from the meanZ-Scores•Like any score, a z-score is a location on the distribution. A z-score also automatically communicates its distance from the mean•A z-score describes a raw score’s location in terms of how far above or below the mean it is when measured in standard deviations–therefore, the units that a z-score is measured in is standard deviationsRaw Score to Z-Score Formula•The formula for computing a z-score for a raw score in a sample is:XSXXzZ-Scores - Example•Compute the z-scores for Slug and Binky•Slug scored 35. Mean = 60, StdDev=10•Slug: = (35 - 60) / 10 = -25 / 10 = -2.5•Binky scored 65. Mean = 60, StdDev=10•Binky: = (65 - 60) / 10 = 5 / 10 = +0.5XSXXzZ-Scores - Your Turn•Compute the z-scores for the following heights in the class. Mean = 66.57, StdDev=4.1•65 inches •66.57 inches •74 inches •53 inches •62 inchesXSXXzZ-Scores -