Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009

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=66.57Var=16.736StdDev=4.091

HEIGHT

8176

7166

6156

51

HEIGHTFr

eque

ncy14

12

10

8

6

4

2

0

62.48 70.6658.38 74.75

Normal Distribution and the Standard Deviation

• IQ is normally distributed with a mean of 100 and standard deviation of 15

70 85 100 115 130

13% 13%

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 lowBinky: above mean, high frequency score, percentile mediumBiff: 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:

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 = +0.5

XSXXz

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 inches

XSXXz

Z-Scores - Your Turn

• 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

XSXXz

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:

XSzX X ))((

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 = 10 + 60 = 70• -4 : X = (-4)(10) + 60 = -40 + 60 = 20• +2.5: X = (2.5)(10) + 60 = 25 + 60 = 85

XSzX X ))((

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• -2• +3.5• -0.5

XSzX X ))((

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) + 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

XSzX X ))((

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, Sx=5Millie scored 20Althea scored 38

English: X=40, Sx=10Millie scored 30Althea 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=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

XSXXz

Z-Scores

• 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?

XSXXz

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

T the

the

T the

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 = -0.38

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 = -0.38

Z area between mean & z area beyond z in tail

0.38 0.1480 0.3520

Proportions of Area under the Standard Normal Curve

a

a

Z = -0.380.3520 of scores lie between this z-score and the tail

a

Proportions of Area under the Standard Normal Curve

a

a

Z = -0.38 0.1480 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 = -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 64.8%• 66.57 inches: z = 0 50%• 74 inches: z = 1.81 3.51%• 53 inches: z = -3.31 99.95%• 62 inches: z = -1.11 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 3.7 3.9

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

2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9

Sampling Distribution of Means

a

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

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

a

SD related to raw score SD

a

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:

NX

X

Standard Error of the Mean - Example

• Estimating Professor’s Age:

• N = 197• Standard deviation () = 4.39

NX

X

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

NX

X

Z-Score Formula for a Sample Mean

• The formula for computing a z-score for a sample mean is:

X

Xz

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

X

Xz

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.