In some research scenarios, we might turn our attention towards co-existing properties of a set of phenomena where the properties are quantitative in nature. As in any association study, we are interested in finding if phenomena having particular values with regard to one of the co-existing properties (X) also have particular values with regard to the other co-existing property (Y). In a statistical study of such a suspected association, we ask if an observed pattern of differences in property X are related to observed patterns of differences in property Y for a particular set of phenomena.

Some illustrative examples would include:

• HOURS OF SLEEP and GPA measured for a group of college students;
• UNEMPLOYMENT RATE and PRESIDENTIAL APPROVAL RATING;
• TOTAL US PER CAPITA MILITARY CAUALTIES and PRESIDENTIAL APPROVAL RATING; and
• SOCIAL MEDIA EXPOSURE (in hours per day) and CIVICS KNOWLEDGE TEST SCORES.

For quantitative properties, differences in observed values are based on comparisons of "larger" or "smaller" as typically assessed by subtraction.

With regard to patterns of co-occurrences, we might find the following:

• We might find larger values of one property consistently co-occurring with larger values of the other property, and smaller values of the first property consistently co-occurring with smaller values of the second property. This is said to be a "direct" relationship.

• We might find larger values of one property consistently co-occurring with smaller values of the other property, and smaller values of the first property consistently co-occurring with larger values of the second property. This is said to be an "inverse" relationship.

• We might find larger values of one property co-occurring with both lower and larger values of the other property, and smaller values of the first property co-occurring with both smaller and larger values of the second property. Since such a pattern of co-occurrence is neither a direct relationship nor an inverse relationship, it is said to be a "null" or "non-relationship." From the standpoint of probability theory, such a "null" relationship is consistent with the two properties being "stochastically independent."

One method by which we can mathematically assess the pattern of co-occurrences is to assess what is said to be the co-variation of the two properties among the observations.

From a statistical perspective regarding a set of observations, the most direct method of defining "larger" and "smaller" is to define "larger" values of the relevant property as being above the mean observed value for that property (mean x) and "smaller" values as being below the mean. Mathematically, this is assessed by subtraction. That is for each observed co-occurrence (x, y), we have:

x - (mean x) = x and

y - (mean y) = y.

• If x is greater than the mean x, x will be positive.
• If x is less than the mean x, x will be negative.
• If y is greater than the mean y, y will be positive.
• If y is less than the mean y, y will be negative.

Secondly, a standard mathematical method of representing the co-occurrence of values for a phenomenon is to multiply the values together:

x * y.

Representing x as x and y as y, we have:

x * y.

• If x and y are both positive, their co-occurrence will be represented as positive.
• If x and y are both negative, their co-occurrence will be represented as positive. This is consistent with a direct association.
• If x is positive and y is negative, their co-occurrence will be represented as negative.
• If x is negative and y is positive, their co-occurrence will be represented as negative. positive. This is consistent with an inverse association.

For a set of observations, then, we can represent their combined co-occurrences as their mean:

( (x * y)) n, where n is the number of observations.

This is said to be their Covariance, denoted as Cov (X,Y), and the values will be some positive number, some negative number, or zero:

• If most of the co-occurrences are positive, as found in a direct association, the mean will be positive. This is said to be a direct association.
• If most of the co-occurrences are negative, as found in an inverse association, the mean will be negative. This is said to be an inverse association.
• If the co-occurrences are a mix of positive and negative values, representing a non-association, the mean will be near zero. This is said to be a null on non-association.

Your assignment is to calculate the Covariance of the observations presented below:

The following observations have been made regarding Daily Internet Use (Hours) and Score (10= "low" and 50 = "high") on a Civics Test for a set of 50 High School Seniors:

Internet Hours	Score = 0	Score = 10	Score = 20	Score = 30	Score = 40	Score = 50	Total (Internet)
0	0	0	0	0	0	3	3
1	0	0	0	0	5	2	7
2	0	0	0	5	4	0	9
3	0	0	5	4	1	0	10
4	0	5	4	1	0	0	10
5	1	4	1	0	0	0	6
6	4	1	0	0	0	0	5
Total (Score)	5	10	10	10	10	5	50

First, calculate the mean Internet Use (MeanH):

Internet Hours	Frequency	Contribution Hours * Frequency
0	3
1	7
2	9
3	10
4	10
5	6
6	5
Total	50
Mean		Total Contribution Total Frequency =

Second, calculate the mean Score (MeanS):

Score	Frequency	Contribution Hours * Frequency
0	5
10	10
20	10
30	10
40	10
50	5
Total	50
Mean		Total Contribution Total Frequency =

Third, calculate the Covariance:

Internet Hours	Hours - MeanH = (Diff H)	Score	Score - MeanS = (Diff S)	(Diff H) * (Diff S)	Frequency	Contribution = (Diff H * Diff S) * Frequency
0		0			0
0		10			0
0		20			0
0		30			0
0		40			0
0		50			3
1		0			0
1		10			0
1		20			0
1		30			0
1		40			5
1		50			2
2		0			0
2		10			0
2		20			0
2		30			5
2		40			4
2		50			0
3		0			0
3		10			0
3		20			5
3		30			4
3		40			1
3		50			0
4		0			0
4		10			5
4		20			4
4		30			1
4		40			0
4		50			0
5		0			1
5		10			4
5		20			1
5		30			0
5		40			0
5		50			0
6		0			4
6		10			1
6		20			0
6		30			0
6		40			0
6		50			0
TOTAL	XXXXXXX	XXXXX	XXXXXXX	XXXXXXX	50

The Covariance is then the mean of the Contributions:

Total of the "Contributions"
Total number of observations	50
Covariance = Total Contributions / Total Observations
Does this suggest a direct, inverse, or non-association?

While assessing the Covariance is useful in identifying direct, inverse, or non-associations, a more useful version of the Covariance is obtained by dividing the Covariance by the combined standard deviations of X (SDX) and Y (SDY):

Cov (X,Y) / (SDX * SDY).

That is, the Covariation in the two properties is compared as a percentage of the overall combined variation in the two properties. This is said to be the Correlation Coefficient. It is denoted as r, it is a number between 1.00 and +1.00, and it is interpreted in the following way:

• A value of 1.00 to 0.60 is considered a strong, inverse association.
• A value of 0.59 to 0.01 is considered a weak, inverse association.
• A value of 0.01 to + 0.01 is considered a non-association.
• A value of +0.01 to +0.59 is considered a weak, direct association.
• A value of +0.60 to +1.00 is considered a strong, direct association.

For the observations presented, we have the following:

• Standard deviation for Internet Hours (SDX) is 1.69

• Standard deviation for Score (SDY) is 15

Correlation Coefficient = Covariance / (SDX * SDY) =
Is the relationship strong or weak?
Is the relationship direct, inverse, or null?
What do these assessments suggest regarding Internet Use and Civics Knowledge?