z-score and correlationr

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z-Score and Correlation

FOUNDATIONS of INFERENTIAL STATISTICS

z-Score:Location of Scores and

Standardized Distribution

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PREVIEW In the preceding parts, we

concentrated on method for describing entire distributions using the basic parameters of central tendency and variability

Now, we will examine a procedure for standardizing distributions and for describing specific locations within a distribution

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PREVIEW In particular, we will convert each

individual score into a new, standardize score, so that the standardized score provides a meaningful description of its exact location within the distribution

We will use the mean as a reference point to determine whether the individual is above or below average

The standard deviation will serve as yardstick for measuring how much an individual differ from the group average

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EXAMPLE Suppose you received a score of X =

76 on a statistics exam. How did you do?

It should be clear that you need more information to predict your grade

Your score could be one of the best score in class, or it might be the lowest score in the distribution

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X = 76, the best score or the lowest score?

To find the location of your score, you must have information about the other score in the distribution

If the mean were μ = 70 you would be in better position than the mean were μ = 86

Obviously, your position relative to the rest of the class depends on mean

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X = 76 and μ = 70 However, the mean by itself is not sufficient

to tell you the exact location of your score At this point, you know that your score is six

points above the mean Six points may be a relatively big distance

and you may have one of the highest score in class, or

Six points may be a relatively small distance and you are only slightly above the average

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Two Distribution of Exam Scores

70 73

X=76

70 82

X=76

σ = 3 σ=12

• For both distributions μ = 70, but for blue distribution σ = 3 and for the red distribution σ = 12

• The position of X = 76 is very different for these two distributions

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THE z-SCORE FORMULA

z =X - μ

σ

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z-Score and Correlation z-Score and Location In a

Distribution One of the primary purpose of a z-

Score is to describe the exact location of a score within a distribution

The z-Score accomplishes this goal by transforming each X value into a signed number (+ or -), so that:○The sign tells whether the score is located

above (+) or below (-) the mean, and○The number tells the distance between

the score and the mean in term of the number of standard deviation

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For a population with μ = 50 and σ =10, find the z-score for each of the following scores○ X = 65, X = 50, X = 40

For a population with μ = 50 and σ =10, find the X value corresponding to each of the following z-scores○z = +1,4; z = -0,8; z = +2,5

In a population with a mean μ = 65, a score of X = 59 corresponding to z = -2,0. What is the standard deviation for the population?

LEARNING CHECK

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For a population with μ = 100 and σ = 10, ○Find the z-Score that corresponds to X

= 106 and X = 87○Find the raw score (X) that

corresponds to z = 1.20 and z = -0.80 A population of scores has μ = 85 and

σ = 20. Find the raw score (X value) corresponds to z = 0.60 and z = -2.30

LEARNING CHECK

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If every X value is transformed into a z-score, then the distribution of z-score will have the following properties: Shape of the z-score distribution will be the same

as the original distribution of raw scores. Each individual has exactly the same relative position in the X distribution and the z-score distribution

The Mean will always have a mean of zero. The subject with score same as the mean is transformed into z = 0

The Standard Deviation will always have a standard deviation of 1. The subject with score same as the +1S from the mean is transformed into z = +1

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z-Score and Correlation Using z-Scores for Making

Comparison When two scores come from different

distribution, it is impossible to make any direct comparison between them

Suppose, for example, Bob received a score of X = 60 on a Statistics exam and a score of X = 56 for Biology test

For which course should Bob expect the better grade?

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z-Score and Correlation Bob score: X = 60 on a

Statistics and a score of X = 56 for Biology Without additional information, it is even

impossible to determine whether Bob is above or below the mean in either distribution

Before you can begin to make comparison, you must know the values for the mean and standard deviation for each distribution

THEN….. compare the z-Score for each subject

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For a population with μ = 50 and σ = 44, corresponds to z = -1.50. what is the standard deviation for this distribution?

For a population with σ = 20, a raw score of X = 110 correspond to z = 1.50. What is the mean for this distribution?

On a statistics exam, you have a score of X = 73. If the mean for this exam is μ = 65, would you prefer a standard deviation of σ = 8 or σ = 16?

LEARNING CHECK

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The State College requires applicants to submit test scores from either the College Placement Exam (CPE) or the College Board Test (CBT). Score on the CPE have a mean of μ = 200 with σ = 50, and scores on the CBT average μ = 500 with σ = 100.

Tom’s application includes a CPE score of X = 235, and Bill’s application reports a CBT score of X = 540.

Base on these scores, which students is more likely to be admired?

LEARNING CHECK

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z-Score and Correlation Ratios of the Total Range to the Standard

Deviation in a Distribution for Different Values of N

Rough check for a computed SD○ The actual percentage of a case between +1 SD

and -1 SD deviates 68 percents○ In very large sample (N = 500 or more) the SD

as about one-sixth of the total range

N Range/S

N Range/S

N Range/S

5 2.3 40 4.3 400 5.910

3.1 50 4.5 500 6.1

15

3.5 100

5.0 700 6.3

20

3.7 200

5.5 1000

6.5

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No single statistical procedure has opened so many new avenues of discovery in

psychology, and possibly in the behavioral science in

general, as that ..…

This is understandable when we remember that scientific progress depends upon finding out what things are co-related and what things are not

CORRELATION

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KKI’s 2006 Height and WeightSUB

J.H W

A 168

56

B 168

68

C 165

58

D 159

42

E 164

48

F 155

46

G 169

65

H 157

68

SUBJ.

H W

I 179

72

J 165

60

K 168

50

L 166

54

M 168

52

N 148

42

O 175

85

P 157

47

SUBJ.

H W

Q 158

47

R 168

45

S 175

56

T 154

41

U 174

55

V 167

55

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kg

cm

40 45 50 55 60 65 70 75 80 85

180

175

170

165

160

155

150

145

0

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kg

cm

40 45 50 55 60 65 70 75 80 85

180

175

170

165

160

155

150

145

0

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CORRELATION … is a statistical technique that used

to measure and describe a relationship between two variables

Usually the two variables are simply observed as they exist naturally in environment, there is no attempt to control or manipulate the variables

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z-Score and Correlation A correlation measures three characteristics of the

relationship:1. The DIRECTION of the

relationship2. The FORM of the relationship3. The DEGREE of the relationship

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The DIRECTION of the relationship

Correlation can be classified into two basic categories, positive and negative○In a positive correlation, the two

variables tend to move in the same direction

○In a negative correlation, the two variables tend to go in opposite direction

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kg

cm

40 45 50 55 60 65 70 75 80 85

180

175

170

165

160

155

150

145

0

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The FORM of the relationship

?

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z-Score and Correlation The DEGREE of the

relationship A correlation measures how well the data fit

the specific form being considered The DEGREE of the relationship is measured

by the numerical value of the correlation A perfect correlation always is identified by a

correlation of 1.00 and indicated a perfect fit At the other extreme, a correlation of 0

indicates not fit at all The numerical value of the correlation also

reflects the degree to which there is a consistent, predictable relationship

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WHERE and WHYCORRELATION ARE USED

1. Prediction2. Validity3. Reliability4. Theory Verification

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CORRELATION ARE USED…1. Prediction

○ If two variables are known to related in some systematic way, it is possible to use one of the variables to make accurate prediction about the other

○ Example: Tes Potensi Akademik - IPK

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CORRELATION ARE USED…2. Validity

Suppose that you are develops a new test for measuring intelligence, how could you show that this test is measuring what it claims (valid)?

One common technique for demonstrating validity is to use correlation

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CORRELATION ARE USED…3. Reliability

○ The consistency of scores obtained by the same persons when they are examined with the same test on different occasions, or

○ With different sets of equivalent items, or

○ Under other variable examining conditions

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CORRELATION ARE USED…4. Theory Verification

○ For example, a theory (hypothesis) may predict a relation between parents’ heights and child’s heights; anxiety and exam score

○ In each case, the prediction of theory (hypothesis) could be tested by determining the correlation between the two variables

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THE PEARSON CORRELATION

rXY =∑zXzY

n

•The product of the z-score determines the strength and direction of the correlation

•z-scores are considered to be the best way to describe a location within a distribution

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THE PEARSON CORRELATION

rXY =∑xy

NSxSy

rXY =∑xy

(∑x2)(∑y2)√

rxy =N∑XY – (∑X) (∑Y)

[N∑X2 – (∑X)2] [N∑Y2 – (∑Y)2] √

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KKI’s 2006 Height and WeightSUB

J.H W

A 168

56

B 168

68

C 165

58

D 159

42

E 164

48

F 155

46

G 169

65

H 157

68

SUBJ.

H W

I 179

72

J 165

60

K 168

50

L 166

54

M 168

52

N 148

42

O 175

85

P 157

47

SUBJ.

H W

Q 158

47

R 168

45

S 175

56

T 154

41

U 174

55

V 167

55

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UNDERSTANDING and INTREPRETING The PEARSON CORRELATION (1)

Correlation simply describes a relationship between two variables. It does not explain why the two variables are related. Specifically, a correlation should not and cannot be interpreted as proof of a cause-effect relationship

The value of a correlation can be affected greatly by the range of scores represented in data

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UNDERSTANDING and INTREPRETING The PEARSON CORRELATION (2)

One or two extreme data points (outliers) can have dramatic effect on the value of a correlation

When judging how good a relationship is, it is tempting to focus on the numerical value of the correlation

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CORRELATION and CAUSATION Correlation simply describes a relationship

between two variables. It does not explain why the two variables are related. Specifically, a correlation should not and cannot be interpreted as proof of a cause-effect relationship

One of the most common error in interpreting correlations is to assume that a correlation necessarily implies a cause-and-effect relationship between two variables

Cigarette smoking is related to heart disease, alcohol consumption is related to birth defect, etc

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z-Score and Correlation CORRELATION and RESTRICTED

RANGE The value of a correlation can be

affected greatly by the range of scores represented in data

Whenever a correlation is computed from scores that do not represent the full range of possible values, you should be cautious in interpreting the correlation

To be safe, you should not generalize any correlation beyond the range of data represented in the sample

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OUTLIERS An outlier is an individual with X

and/or Y values that are substantially different (larger or smaller) from the values obtained for the other individual in the data set

One or two extreme data points (outliers) can have dramatic effect on the value of a correlation

Look at figure 16.8 page 518

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CORRELATION and THE STRENGTH of THE RELATIONSHIP

When judging how good a relationship is, it is tempting to focus on the numerical value of the correlation

A correlation measure the degree of relationship between two variables on a scale from 0 to 1.00

One of the common uses of the correlation is for prediction. If two variables are correlated, you can use the value of one variable to predict the other

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COEFFICIENT of DETERMINATION The value r2 is called the coefficient of

determination because it measures the proportion of variability in one variable that can be determined from the relationship of variability with the other variable

A correlation of r = 0.80 (or -0.80) means that r2 = 0.64 (or 64%) of the variability in the Y (or X) scores can be predicted from the relationship with X (or Y)

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HYPOTHESIS TESTS WITH THE r A sample correlation often is used to

answer question about the general population

For example, we would like to know whether there is a relationship between IQ and creativity

To answer the question, a sample would be selected, and the sample data would be used to compute the correlation value

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The HYPOTHESIS The basic question for this hypothesis

test is whether or not a correlation exist in the population ○ H0 ρ = 0 there is no population

correlation○ H1 ρ ≠ 0 there is a real correlation

The correlation from the sample data (r) will be used to evaluate these hypotheses

There will be some discrepancy (sampling error) between a sample statistic and the corresponding parameter.

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A high school counselor would like to know if there is a relationship between mathematical skill and verbal skill. A sample of n = 25 students is selected, and the counselor record achievement test scores in mathematics and English for each student.

The Pearson correlation for this sample is r=+0.50. Do these data provide sufficient evidence for a real relationship in the population? Test at the .05 level, two tail.

z-score and correlationr

Documents

extreme data

standard deviation

learning check

kkis 2006

dramatic effect

affected greatly

statistics

sample data