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Basic Statistics Correlation

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**Relationships Associations**

Var Relationships Associations Var Var Var Var

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**Independent variables**

In Research Information Dependent variable X1 ? Y X2 COvary X3 Independent variables

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**The Concept of Correlation**

Association or relationship between two variables Co-relate? r relation X Y Covary---Go together

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**Patterns of Covariation**

Zero or no correlation X Y Correlation Covary Go together X Y X Y Negative correlation Positive correlation

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**Scatter plots allow us to visualize the relationships**

The chief purpose of the scatter diagram is to study the nature of the relationship between two variables Linear/curvilinear relationship Direction of relationship Magnitude (size) of relationship Scatter Plots

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**Scatter Plot A Variable Y Variable X high low low high**

Represents both the X and Y scores Variable Y Exact value low low high Variable X An illustration of a perfect positive correlation

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**An illustration of a positive correlation**

Scatter Plot B high Variable Y Estimated Y value low low high Variable X An illustration of a positive correlation

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**Scatter Plot C Variable Y Variable X high low low high**

Exact value low low high Variable X An illustration of a perfect negative correlation

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**An illustration of a negative correlation**

Scatter Plot D high Variable Y Estimated Y value low low high Variable X An illustration of a negative correlation

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**Scatter Plot E Variable Y Variable X high low low high**

An illustration of a zero correlation

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**An illustration of a curvilinear relationship**

Scatter Plot F high Variable Y low low high Variable X An illustration of a curvilinear relationship

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**The Measurement of Correlation**

The Correlation Coefficient The degree of correlation between two variables can be described by such terms as “strong,” ”low,” ”positive,” or “moderate,” but these terms are not very precise. If a correlation coefficient is computed between two sets of scores, the relationship can be described more accurately. A statistical summary of the degree and direction of relationship or association between two variables can be computed

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**Pearson’s Product-Moment Correlation Coefficient r**

No Relationship Negative correlation Positive correlation Direction of relationship: Sign (+ or –) Magnitude: 0 through +1 or 0 through -1

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**The Pearson Product-Moment Correlation Coefficient**

Recall that the formula for a variance is: If we replaced the second X that was squared with a second variable, Y, it would be: This is called a co-variance and is an index of the relationship between X and Y.

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**Conceptual Formula for Pearson r**

This formula may be rewritten to reflect the actual method of calculation

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**Calculation of Pearson r**

You should notice that this formula is merely the sum of squares for covariance divided by the square root of the product of the sum of squares for X and Y

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**Formulae for Sums of Squares**

Therefore, the formula for calculating r may be rewritten as:

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**Calculation of r Using Sums of Squares**

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An Example Suppose that a college statistics professor is interested in how the number of hours that a student spends studying is related to how many errors students make on the mid-term examination. To determine the relationship the professor collects the following data:

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**The Stats Professor’s Data**

Student Hours Studied (X) Errors (Y) X2 Y2 XY 1 4 15 16 225 60 2 12 144 48 3 5 9 25 81 45 6 10 36 100 7 8 49 64 56 28 42 18 Total X = 70 Y = 73 X2 =546 Y2=695 XY=429

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**The Data Needed to Calculate the Sum of Squares**

X Y X2 Y2 XY Total X = 70 Y = 73 X2 =546 Y2=695 XY=429 = /10 = = 56 = /10 = = 162.1 = 429 – (70)(73)/10 = 429 – 511 = -82

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**Calculating the Correlation Coefficient**

= -82 / √(56)(162.1) = Thus, the correlation between hours studied and errors made on the mid-term examination is -0.86; indicating that more time spend studying is related to fewer errors on the mid-term examination. Hopefully an obvious, but now a statistical conclusion!

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**Pearson Product-Moment Correlation Coefficient r**

perfect negative correlation Zero correlation Perfect positive correlation -1 +1 Negative correlation Positive correlation

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Numerical values .73 - .35 Negative correlation Zero correlation Positive correlation Perfect Strong Moderate

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**The Pearson r and Marginal Distribution**

The marginal distribution of X is simply the distribution of the X’s; the marginal distribution of Y is the frequency distribution of the Y’s. Y Bivariate relationship Bivariate Normal Distribution X

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**Marginal distribution of X and Y are precisely the same shape.**

Y variable X variable

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**Interpreting r, the Correlation Coefficient**

Recall that r includes two types of information: The direction of the relationship (+ or -) The magnitude of the relationship (0 to 1) However, there is a more precise way to use the correlation coefficient, r, to interpret the magnitude of a relationship. That is, the square of the correlation coefficient or r2. The square of r tells us what proportion of the variance of Y can be explained by X or vice versa.

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**How does correlation explain variance?**

Suppose you wish to estimate Y for a given value of X. high How does correlation explain variance? Explained Variable Y Free to Vary 49% of variance is explained Explained low low high Variable X An illustration of how the squared correlation accounts for variance in X, r = .7, r2 = .49

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**Now, let’s look at some correlation coefficients and their corresponding scatter plots.**

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**What is your estimate of r?**

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Y X What is your estimate of r? r = -1.00 r2 = 1.00 = 100%

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Y X What is your estimate of r? r = +1.00 r2 = 1.00 = 100%

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**What is your estimate of r?**

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**What is your estimate of r?**

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