PSYCH 301 1st Edition Lecture 4Z SCORESOutline of Last Lecture I. Measure probability II. Learn why random sampling is a good thingIII. Calculate probability from a histogramOutline of Current Lecture I. Normal DistributionII. Z ScoresIII. QuestionsCurrent Lecture – How far is x from the mean?I. Knowing both the mean and standard deviation (or variance) allows us to compare one score to the populationa. Symbols: Sample vs. PopulationSamplestatisticsPopulationsscoresMean M μStandard Deviation SD σVarience SD² σ²Greek letters mean the data is talking about populationb. Normal Distribution is controlled by 2 parameters: i. Mean (location) and Standard Deviation (width)These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.-3 -2 -1 0 1 2 3II. Z Score: transforming a raw score into a standard score. (can only be used with bell-shaped data)a. Clarifies the relationship between one score and the rest of the data. (how far away from the mean is this score?)b. Difference from the mean/ variabilityc. There are 2 equal equations. The difference is who we are comparing the scores against. A sample mean or a population mean (Greek letters)i. The closer the score to the center (the mean) the smaller the z-scoreii. The further the score from the center (the mean) the larger the z-scored. Three steps for calculating z scores:i. Get mean and standard deviationii. Calculate deviation from the meaniii. Divide by standard deviatione. Relationship between numerator and denominator i. Z will be large when mean deviation is large and SD is smallii. Z will be small when mean deviation is small and SD is largeiii. Z will be small when mean deviation and SD are both largeiv. Z will be small when mean deviation and SD and both smallf. Z scores follow the normal distribution- μ=0, σ=1i. To determine how far away one z score is from the mean is equal to how ever many standard deviations away, above or below, the meang. Example problemsi. You take an IQ test and get a score of 130. The populations mean IQ is 100 and SD is 15. What is your IQ expressed as a z score?1. Plug in the numbers to the z score equationZ= (130-100)/152. Your IQ score of 130 has a z score of 2. This means that you are μ+2σ SDs above the mean score of the population.ii. What is the z score of scoring 6 runs when the bats scored 2, 2, 3, 3, 2, and 6 runs in their last 6 games?iii.n=6 M=3 SD= 1.412 - 3 = -1² = 12 - 3 = -1² = 12 - 3 = -1² = 13 - 3 = 0² = 03 - 3 = 0² = 06 - 3 = 3² = 9SS = 12h. Standardizing a Data set: converting an entire data set into Z scoresi. (raw scores → Z scores)Data Set : [ 2 4 6 8 ] n=4Mean: (2+4+6+8)/4 = 52-5= -3² = 94-5= -1² = 16-5= 1² = 18-5= 3² = 9SS= 20SD= √20/4 = √5 = 2.24SD = √12/6 = √2 = 1.414Z = 6 – 3 / 1.414 = 3 / 1.414 =Z = 2.12Z=2-5= -3/2.24 = -1.344-5= -1/2.24= -0.456-5= 1/ 2.24= 0.458-5= 3/2.24= 1.34i. Unstandardize data set (z scores → raw scores)i. Get x by itself: multiply each side by SD and add the mean1. X = ( z * σ ) + μj. Questions:i. What 2 pieces of information do you need from the population in order to calculate a Z score?1. Mean or SDii. Explain why knowing only the mean doesn’t tell you everything about how the scores relate to one another.1. The mean doesn’t tell you anything about how the scores vary from each other. You need to know the SD to figure out the relation.iii. What happens to a z score if the mean difference stays the same but the SD is increased?1. You are increasing the number in the denominator, so the z score will be smalleriv. What happens to a z score if the mean difference stays the same but the SD is decreased?1. You are decreasing the number in the denominator, so the z score will be