Cost per No. of Applications

Correlation

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Getting Started	General Instructions \| Introduction to Your Study
Descriptive Statistics	Histograms \| Scatter Plots \| Central Tendency \| Standard Deviation \| Confidence Intervals
Relating Variables	Correlation
Important Concepts	The Normal Distribution \| Z Scores \| Probability Distributions
Levels	You are currently on Correlation at level 2. Level 1 \| Level 2 \| Level 3
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Explanation

Calculating Correlation
The correlation between two variables may be calculated in more than one way. We will tell you about a measure known as Pearson's Product Moment Coefficient, which is commonly used.

Pearson's measure is denoted by the letter r. In order to understand what the calculation for r is doing, we need to think for a moment about what correlation really means. We saw in level 1 that the correlation coefficient tells you two things about the linear relationship between two variables:

The direction of the relationship (positive or negative)
The strength of the relationship - that is, how consistently linear it is

It is easiest to think of all these things with reference to a scatter plot of the data. If there is a linear relationship between the two variables, they will tend to form a straight line on the scatter plot. If the points on the scatter plot all line up perfectly, then correlation is 1 (or -1 if the slope goes down). If the points cluster about a line, forming a thin cloud, then correlation will be less. The thicker the cloud, the weaker the correlation.

The section below steps you through a visual explanation of how the formula for Pearson's Correlation Coefficient is derived. Use the Next and Previous links to step through it.

Step 1 of 8
Consider the diagram to the right. It shows a scatterplot of two variables, X and Y. You can see that they are related, but not perfectly.
Next

Exploration

Here is the formula for calculating Pearson's Correlation Coefficient.

The top part calculates the contribution of the distances from the mean, as explained above, and the bottom part divides by the number of data points minus 1 so that the value is comparable whatever the size of data, and divides by the product of the standard deviations of the two variables to remove the effect of wider variation in the data.

Application

To calculate the correlation coefficient of your data, you will need to know the following:

The mean of cost (denoted as Cost) is 517.53 and the mean of no. apps (denoted as No. Apps) is 56.73
The standard deviation of cost is 298.47
The standard deviation of no. apps is 30.64
You have 30 data points

Your data are shown below. The first five rows have spaces for you to fill in the calculations, but the rest have been completed for you.

Subtract the mean of cost from each cost value and enter the result in the first column (keep the minus sign if there is one).
Do the same for each value of no. apps and its mean and enter its value in the second column.

Finally, multiply the two differences together to give you their product and enter that in the final column.
Enter answers to 2 decimal places.

Cost	No. Apps	Cost - Cost	No. Apps - No. Apps	Product
71	5
663	62
980	110
138	12
861	83
145	41	-372.53	-15.73	5859.9
493	46	-24.53	-10.73	263.21
548	75	30.47	18.27	556.69
251	23	-266.53	-33.73	8990.06
1024	100	506.47	43.27	21914.96
435	41	-82.53	-15.73	1298.2
772	75	254.47	18.27	4649.17
71	10	-446.53	-46.73	20866.35
663	59	145.47	2.27	330.22
381	62	-136.53	5.27	-719.51
138	15	-379.53	-41.73	15837.79
861	78	343.47	21.27	7305.61
145	59	-372.53	2.27	-845.64
493	56	-24.53	-0.73	17.91
548	56	30.47	-0.73	-22.24
251	27	-266.53	-29.73	7923.94
1009	110	491.47	53.27	26180.61
435	46	-82.53	-10.73	885.55
772	110	254.47	53.27	13555.62
493	46	-24.53	-10.73	263.21
548	72	30.47	15.27	465.28
251	23	-266.53	-33.73	8990.06
988	100	470.47	43.27	20357.24
435	41	-82.53	-15.73	1298.2
663	59	145.47	2.27	330.22

Calculator
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Now add up all the values in the product column and enter the total here:
You have completed the calculation for the top of the formula: ∑ (x - x)(y - y).
Now multiply (n-1) by the standard deviation for cost multiplied by the standard deviation for no. apps:
This is (n-1)S_xS_y, the bottom of the formula.
Finally, divide result from step 4 by the result from step 5 to get the correlation coefficient:

Samples and Populations | The Normal Distribution