Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.
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Transcript of Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.
![Page 1: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/1.jpg)
Psych 230
Psychological Measurement and Statistics
Pedro Wolf
September 16, 2009
![Page 2: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/2.jpg)
Today….
• Symbols and definitions reviewed
• Understanding Z-scores
• Using Z-scores to describe raw scores
• Using Z-scores to describe sample means
![Page 3: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/3.jpg)
Symbols and Definitions Reviewed
![Page 4: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/4.jpg)
Definitions: Populations and Samples
• Population : all possible members of the group of interest
• Sample : a representative subset of the population
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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
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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
![Page 7: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/7.jpg)
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
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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
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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
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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%
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Understanding Z-Scores
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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?
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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
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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
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Frequency Distribution of Attractiveness Scores
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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
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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
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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.
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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
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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
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Raw Score to Z-Score Formula
• The formula for computing a z-score for a raw score in a sample is:
XSXXz
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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
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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
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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
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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 ))((
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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 ))((
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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 ))((
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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 ))((
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Using Z-scores
![Page 30: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/30.jpg)
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
![Page 31: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/31.jpg)
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
![Page 32: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/32.jpg)
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
![Page 33: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/33.jpg)
Distribution of Attractiveness Scores
Raw scores
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Z-Distribution of Attractiveness Scores
Z-scores
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Z-Distribution of Attractiveness Scores
Z-scores
In a normal distribution, most scores lie between -3 and +3
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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
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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
![Page 38: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/38.jpg)
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
![Page 39: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/39.jpg)
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
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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
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Comparison of two Z-Distributions
Stats: X=30, Sx=5Millie scored 20Althea scored 38
English: X=40, Sx=10Millie scored 30Althea scored 45
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Comparison of two Z-Distributions
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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
![Page 44: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/44.jpg)
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?
![Page 45: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/45.jpg)
Proportions of Area under the Standard Normal Curve
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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
![Page 47: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/47.jpg)
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
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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
![Page 49: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/49.jpg)
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
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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
![Page 51: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/51.jpg)
Proportions of Area under the Standard Normal Curve
a
T the
the
T the
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Proportions of Area under the Standard Normal Curve
a
a
a
Z = -0.38
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Proportions of Area under the Standard Normal Curve
a
a
Z = -0.38
How many scores lie in this portion of the curve?
a
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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
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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
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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
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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
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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
![Page 61: Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.](https://reader036.fdocuments.in/reader036/viewer/2022062504/5a4d1b567f8b9ab0599a9934/html5/thumbnails/61.jpg)
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%
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Using Z-scores to describe sample means
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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
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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?
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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
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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?
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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
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Sampling Distribution of Means
a
a
2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
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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
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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
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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.
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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
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Sampling Distribution of Means
a
aa
2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
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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
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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
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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
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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
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Standard Error of the Mean - Example
• Estimating Professor’s Age:
• N = 197• Standard deviation () = 4.39
NX
X
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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
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Z-Score Formula for a Sample Mean
• The formula for computing a z-score for a sample mean is:
X
Xz
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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
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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
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Done for today
• Read for next week.• Pick up quizzes at front of class.