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Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009 Today…. • Symbols and definitions reviewed • Understanding Z-scores • Using Z-scores to describe raw scores • Using Z-scores to describe sample means Symbols and Definitions Reviewed Definitions: Populations and Samples • Population : all possible members of the group of interest • Sample : a representative subset of the population Symbols 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 = X Symbols 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)2 Symbols 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 mean Symbols 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 = sx Normal Distribution and the Standard Deviation Mean=66.57 HEIGHT 14 Var=16.736 12 StdDev=4.091 10 8 6 Fre q u e n cy 4 2 0 51 61 56 71 66 58.38 HEIGHT 81 76 74.75 62.48 70.66 Normal Distribution and the Standard Deviation • IQ is normally distributed with a mean of 100 and standard deviation of 15 13% 70 13% 85 100 115 130 Understanding Z-Scores The 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 low Z-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 population Frequency Distribution of Attractiveness Scores Frequency Distribution of Attractiveness Scores Interpreting each score in relative terms: Slug: below mean, low frequency score, percentile low Binky: above mean, high frequency score, percentile medium Biff: above mean, low frequency score, percentile high To calculate these relative scores precisely, we use z-scores Z-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 book Z-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 mean Z-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 deviations Raw Score to Z-Score Formula • The formula for computing a z-score for a raw score in a sample is: XX z SX 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 = +0.5 XX z SX Z-Scores - Your Turn XX z SX • 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 inches Z-Scores - Your Turn XX z SX • Compute the z-scores for the following heights in the class. Mean = 66.57, StdDev=4.1 • • • • • 65 inches: (65 - 66.57) / 4.1 = -1.57 / 4.1 = -0.38 66.57 inches: (66.57 - 66.57) / 4.1 = 0 / 4.1 = 0 74 inches: (74 - 66.57) / 4.1 = 7.43 / 4.1 = 1.81 53 inches: (53 - 66.57) / 4.1 = -13.57 / 4.1 = -3.31 62 inches: (62 - 66.57) / 4.1 = -4.57 / 4.1 = -1.11 Z-Score to Raw Score Formula • When a z-score and the associated Sx and X are known, we can calculate the original raw score. The formula for this is: X ( z )( S X ) X Z-Score to Raw Score : Example X ( z )( S X ) X • Attractiveness scores. Mean = 60, StdDev=10 • What raw score corresponds to the following zscores? • +1 : X = (1)(10) + 60 = 10 + 60 = 70 • -4 : X = (-4)(10) + 60 = -40 + 60 = 20 • +2.5: X = (2.5)(10) + 60 = 25 + 60 = 85 Z-Score to Raw Score : Your Turn X ( z )( S X ) X • Height in class. Mean=66.57, StdDev=4.1 • What raw score corresponds to the following zscores? • • • • +2 -2 +3.5 -0.5 Z-Score to Raw Score : Your Turn X ( z )( S X ) X • Height in class. Mean=66.57, StdDev=4.1 • What raw score corresponds to the following zscores? • • • • +2: X = (2)(4.1) + 66.57 = 8.2 + 66.57 = 74.77 -2: X = (-2)(4.1) + 66.57 = -8.2 + 66.57 = 58.37 +3.5: X = (3.5)(4.1) + 66.57 = 14.35 + 66.57 = 80.92 -0.5: X = (-0.5)(4.1) + 66.57 = -2.05 + 66.57 = 64.52 Using Z-scores Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means Z-Distribution • A z-distribution is the distribution produced by transforming all raw scores in the data into z-scores • This will not change the shape of the distribution, just the scores on the x-axis • The advantage of looking at z-scores is the they directly communicate each score’s relative position • z-score = 0 • z-score = +1 Distribution of Attractiveness Scores Raw scores Z-Distribution of Attractiveness Scores Z-scores Z-Distribution of Attractiveness Scores Z-scores In a normal distribution, most scores lie between -3 and +3 Characteristics of the Z-Distribution • A z-distribution always has the same shape as the raw score distribution • The mean of any z-distribution always equals 0 • The standard deviation of any z-distribution always equals 1 Characteristics of the Z-Distribution • Because of these characteristics, all normal zdistributions are similar • A particular z-score will be at the same relative location on every distribution • • Attractiveness: z-score = +1 • Height: z-score = +1 You should interpret z-scores by imagining their location on the distribution Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means Using Z-Scores to compare variables • On your first Stats exam, you get a 21. On your first Abnormal Psych exam you get a 87. How can you compare these two scores? • The solution is to transform the scores into zscores, then they can be compared directly • z-scores are often called standard scores Using Z-Scores to compare variables • Stats exam, you got 21. Mean = 17, StdDev = 2 • Abnormal exam you got 87. Mean = 85, StdDev = 3 • Stats Z-score: (21-17)/2 = 4/2 = +2 • Abnormal Z-score: (87-85)/2 = 2/3 = +0.67 Comparison of two Z-Distributions Stats: X=30, Sx=5 English: X=40, Sx=10 Millie scored 20 Millie scored 30 Althea scored 38 Althea scored 45 Comparison of two Z-Distributions Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means Using Z-Scores to compute relative frequency • Remember your score on the first stats exam: • Stats z-score: (21-17)/2 = 4/2 = +2 • So, you scored 2 standard deviations above the mean • Can we compute how many scores were better and worse than 2 standard deviations above the mean? Proportions of Area under the Standard Normal Curve Relative Frequency • Relative frequency can be computed using the proportion of the total area under the curve. • The relative frequency of a particular z-score will be the same on all normal z-distributions. • The standard normal curve serves as a model for any approximately normal z-distribution Z-Scores XX z SX • z-scores for the following heights in the class. – Mean = 66.57, StdDev=4.1 • • • • • 65 inches: (65 - 66.57) / 4.1 = -1.57 / 4.1 = -0.38 66.57 inches: (66.57 - 66.57) / 4.1 = 0 / 4.1 = 0 74 inches: (74 - 66.57) / 4.1 = 7.43 / 4.1 = 1.81 53 inches: (53 - 66.57) / 4.1 = -13.57 / 4.1 = -3.31 62 inches: (62 - 66.57) / 4.1 = -4.57 / 4.1 = -1.11 Z-Scores XX z SX • z-scores for the following heights in the class. – Mean = 66.57, StdDev=4.1 • • • • • 65 inches: (65 - 66.57) / 4.1 = -1.57 / 4.1 = -0.38 66.57 inches: (66.57 - 66.57) / 4.1 = 0 / 4.1 = 0 74 inches: (74 - 66.57) / 4.1 = 7.43 / 4.1 = 1.81 53 inches: (53 - 66.57) / 4.1 = -13.57 / 4.1 = -3.31 62 inches: (62 - 66.57) / 4.1 = -4.57 / 4.1 = -1.11 • What are the relative frequencies of these heights? Z-Scores • How can we find the exact relative frequencies for these z-scores? • • • • • 65 inches: z = -0.38 66.57 inches: z = 0 74 inches: z = 1.81 53 inches: z = -3.31 62 inches: z = -1.11 Z-Scores • How can we find the exact relative frequencies for these z-scores? • • • • • 65 inches: z = -0.38 66.57 inches: z = 0 74 inches: z = 1.81 53 inches: z = -3.31 62 inches: z = -1.11 Proportions of Area under the Standard Normal Curve a th e T th e T the Proportions of Area under the Standard Normal Curve a Z = -0.38 a a Proportions of Area under the Standard Normal Curve a Z = -0.38 How many scores lie in this portion of the curve? a a Z-Scores • To find out the relative frequencies for a particular z-score, we use a set of standard tables – z-tables – They’re in the book Z-Scores • To find out the relative frequencies for a particular z-score, we use a set of standard tables – z-tables • 65 inches: z = -0.38 Z 0.38 area between mean & z 0.1480 area beyond z in tail 0.3520 Proportions of Area under the Standard Normal Curve a Z = -0.38 0.3520 of scores lie between this z-score and the tail a a Proportions of Area under the Standard Normal Curve a Z = -0.38 a a 0.1480 of scores lie between this z-score and the mean Z-Scores - Your turn • Find out what percentage of people are taller than the heights given below: – z-tables • • • • • 65 inches: z = -0.38 66.57 inches: z = 0 74 inches: z = 1.81 53 inches: z = -3.31 62 inches: z = -1.11 Z-Scores - Your turn • Find out what percentage of people are taller than the heights given below: – z-tables • • • • • 65 inches: z = -0.38 66.57 inches: z = 0 74 inches: z = 1.81 53 inches: z = -3.31 62 inches: z = -1.11 64.8% 50% 3.51% 99.95% 86.65% Using Z-scores to describe sample means Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means Sampling Distribution of Means • We can now describe the relative position of a particular score on a distribution • What if instead of a single score, we want to see how a particular sample of scores fit on the distribution? Sampling Distribution of Means • For example, we want to know if students who sit in the back score better or worse on exams than others • Now, we are no longer interested in a single score’s relative distribution, but a sample of scores • What is the best way to describe a sample? • So, we want to find the relative position of a sample mean Sampling Distribution of Means • To find the relative position of a sample mean, we need to compare it to a distribution of sample means • just like to find the relative position of a particular score, we needed to compare it to a distribution of scores • So first we need to create a new distribution, a distribution of sample means • How to do this? Sampling Distribution of Means • We want to compare the people in a sample to everyone else • To create a distribution of sample means, we can select 10 names at random from the population and calculate the mean of this sample • X1 = 3.1 • Do this over and over again, randomly selecting 10 people at a time • X2 = 3.3, X3 = 3.0, X4 = 2.9, X5 = 3.1, X6 = 3.2, etc etc Sampling Distribution of Means a a 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 Sampling Distribution of Means a a 2.3 2.5 2.7 2.9 3.1 3.3 3.5 Each score is not a raw score, but is instead a sample mean 3.7 3.9 Sampling Distribution of Means • In reality, we cannot infinitely draw samples from our population, but we know what the distribution would be like • The central limit theorem defines the shape, mean and standard deviation of the sampling distribution Central Limit Theorem • The central limit theorem allows us to envision the sampling distribution of means that would be created by exhaustive random sampling of any raw score distribution. Sampling Distribution of Means: Characteristics • A sampling distribution is approximately normal • The mean of the sampling distribution () is the same as the mean of the raw scores • The standard deviation of the sampling distribution (x) is related to the standard deviation of the raw scores Sampling Distribution of Means a a 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 Sampling Distribution of Means a Shape of distribution is normal a 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 Sampling Distribution of Means a Mean is the same as raw score mean a 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 Sampling Distribution of Means a SD related to raw score SD aa 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 Standard Error of the Mean • The standard deviation of the sampling distribution of means is called the standard error of the mean. The formula for the true standard error of the mean is: X X N Standard Error of the Mean - Example • Estimating Professor’s Age: • N = 197 • Standard deviation () = 4.39 X X N Standard Error of the Mean - Example • N = 197 • Standard deviation () = 4.39 • Standard error of the mean = 4.39 / √197 = 4.39 / 14.04 = 0.31 X X N Z-Score Formula for a Sample Mean • The formula for computing a z-score for a sample mean is: z X X Z-Score for a Sample Mean - Example • • • • Mean of population = 36 Mean of sample = 34 Standard error of the mean = 0.31 Z = (34 - 36) / 0.31 = -2 / 0.31 = -6.45 z X X Sampling Distribution of Means - Why? • We want to compare the people in sample to everyone else in population • Creating a sampling distribution gives us a normal distribution with all possible means • Once we have this, we can determine the relative standing of our sample • use z-scores to find the relative frequency Done for today • Read for next week. • Pick up quizzes at front of class.