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 = ∑X 2 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 (S 2 ) 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 =  2 X Population Standard Deviation =  X Sample Variance = S 2 x Sample Standard Deviation = S x Estimate of Population Variance = s 2 x Estimate of Population Standard Deviation = s x

Normal Distribution and the Standard Deviation Mean=66.57 Var= StdDev=

Normal Distribution and the Standard Deviation IQ is normally distributed with a mean of 100 and standard deviation of %

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

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 ≈ , π≈ 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: = +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:

Z-Scores - Example Compute the z-scores for Slug and Binky Slug scored 35. Mean = 60, StdDev=10 Slug: = ( ) / 10 = -25 / 10 = -2.5 Binky scored 65. Mean = 60, StdDev=10 Binky: = ( ) / 10 = 5 / 10 = +0.5

Z-Scores - Your Turn Compute the z-scores for the following heights in the class. Mean = 66.57, StdDev= inches inches 74 inches 53 inches 62 inches

Z-Scores - Your Turn Compute the z-scores for the following heights in the class. Mean = 66.57, StdDev= inches: ( ) / 4.1 = / 4.1 = inches: ( ) / 4.1 = 0 / 4.1 = 0 74 inches: ( ) / 4.1 = 7.43 / 4.1 = inches: ( ) / 4.1 = / 4.1 = inches: ( ) / 4.1 = / 4.1 = -1.11

Z-Score to Raw Score Formula When a z-score and the associated S x and X are known, we can calculate the original raw score. The formula for this is:

Z-Score to Raw Score : Example Attractiveness scores. Mean = 60, StdDev=10 What raw score corresponds to the following z- scores? +1 : X = (1)(10) + 60 = = : X = (-4)(10) + 60 = = : X = (2.5)(10) + 60 = = 85

Z-Score to Raw Score : Your Turn Height in class. Mean=66.57, StdDev=4.1 What raw score corresponds to the following z- scores?

Z-Score to Raw Score : Your Turn Height in class. Mean=66.57, StdDev=4.1 What raw score corresponds to the following z- scores? +2: X = (2)(4.1) = = : X = (-2)(4.1) = = : X = (3.5)(4.1) = = : X = (-0.5)(4.1) = = 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 z- distributions 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 z- scores, 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, S x =5 Millie scored 20 Althea scored 38 English: X=40, S x =10 Millie scored 30 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 z-scores for the following heights in the class. – Mean = 66.57, StdDev= inches: ( ) / 4.1 = / 4.1 = inches: ( ) / 4.1 = 0 / 4.1 = 0 74 inches: ( ) / 4.1 = 7.43 / 4.1 = inches: ( ) / 4.1 = / 4.1 = inches: ( ) / 4.1 = / 4.1 = -1.11

Z-Scores z-scores for the following heights in the class. – Mean = 66.57, StdDev= inches: ( ) / 4.1 = / 4.1 = inches: ( ) / 4.1 = 0 / 4.1 = 0 74 inches: ( ) / 4.1 = 7.43 / 4.1 = inches: ( ) / 4.1 = / 4.1 = inches: ( ) / 4.1 = / 4.1 = 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 = inches: z = 0 74 inches: z = inches: z = inches: z = -1.11

Z-Scores How can we find the exact relative frequencies for these z-scores? 65 inches: z = inches: z = 0 74 inches: z = inches: z = inches: z = -1.11

Proportions of Area under the Standard Normal Curve a T the th e T th e

Proportions of Area under the Standard Normal Curve a a a Z = -0.38

Proportions of Area under the Standard Normal Curve a a Z = How many scores lie in this portion of the curve? 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 = Zarea between mean & z area beyond z in tail

Proportions of Area under the Standard Normal Curve a a Z = of scores lie between this z-score and the tail a

Proportions of Area under the Standard Normal Curve a a Z = of scores lie between this z-score and the mean a

Z-Scores - Your turn Find out what percentage of people are taller than the heights given below: – z-tables 65 inches: z = inches: z = 0 74 inches: z = inches: z = 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 = % inches: z = 0 50% 74 inches: z = % 53 inches: z = % 62 inches: z = %

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 X 1 = 3.1 Do this over and over again, randomly selecting 10 people at a time X 2 = 3.3, X 3 = 3.0, X 4 = 2.9, X 5 = 3.1, X 6 = 3.2, etc etc

Sampling Distribution of Means a a

Sampling Distribution of Means a a Each score is not a raw score, but is instead a sample mean a

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 aa

Sampling Distribution of Means a a Shape of distribution is normal a

Sampling Distribution of Means a a Mean is the same as raw score mean a

Sampling Distribution of Means a a SD related to raw score SD a

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:

Standard Error of the Mean - Example Estimating Professor’s Age: N = 197 Standard deviation (  ) = 4.39

Standard Error of the Mean - Example N = 197 Standard deviation (  ) = 4.39 Standard error of the mean = 4.39 / √197 = 4.39 / = 0.31

Z-Score Formula for a Sample Mean The formula for computing a z-score for a sample mean is:

Z-Score for a Sample Mean - Example Mean of population = 36 Mean of sample = 34 Standard error of the mean = 0.31 Z = ( ) / 0.31 = -2 / 0.31 = -6.45

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.