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Car depreciation once again:

Year	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001	2002

Car's Worth	15000	13000	11400	10150	8700	7300	6200	4500	3800	3200	2800

Give the linear correlation coefficient r.
Find "best fit" linear regression line y = mx + b.

Solution:

Use formulas or use the Excel spreadsheet:

We get r= -0.9887 (very good negative correlation.)

m= -1295 b= 15480

Best regression line is Y = -1295X + 15480, where X is the number of years from X=0
and Y is the Car Price.

What is the equation of the "best" straight line to indicate this change of prices:

The x-variable is 1977 prices.
The y-variable is 1984 prices.

Straight line is y = mx + b where slope m and y-intercept b are given by formulas:

m= 0.281 (About an average increase in prices of 28% in 7 years.)
b= -0.111

Straight line is given by: y = 0.281x - 0.111
We could put in a food price for 1977(x-value) and predict it's price in 1984(y-value)

Clearly there is a linear correlation.

There is a straight-line trend.

A simple graph with only 2 variables.

Is there a trend? Is there positive correlation?

There are very affirmative indications that the two variables are very positively correlated.

Two variables are analyzed together in this chapter.

There may be "cause and effect" although this is not proven by the mathematics alone.

Recent Populations (in millions) of Canada and the U.S.

Year	1951	1961	1971	1981	1991	2001

Canadian	14.0	18.2	21.6	24.3	27.3	30.0

U.S.	151.3	179.3	203.3	226.5	248.7	276.2

Does x-variable, the Canadian population, correlate positively with y-variable, the U.S. population?

Yes, it does, and it is easier to show this on an XY-scatterplot.

Another example:

x-variable	1	2	3	4	5	6	7	8	9	10
y-variable	46	67	43	89	49	19	48	7	12	87

The analyst is looking for patterns in the above data.

Maybe there is an upward trend and then a downward trend? Linear correlation is quite strict--it wants only ONE trend.

This looks like a very generic example. That's what it is.

The two "no-name" variables are poorly correlated.

When x-variable increases , then y-variable sometimes increases and sometimes decreases.

Automobile depreciation

A new car depreciates in value every year. To approximate this we use the data

Year	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001	2002

Car's Worth	15000	13000	11400	10150	8700	7300	6200	4500	3800	3200	2800

There is clearly a trend. A mathe-matician would indicate it is not a perfect linear trend.

However we call this a negative correlation -- a good negative correlation.

As one variable, the year, increases the car's worth decreases. (negative correlation.)

Fundamentals of correlation:

Scatterplots are not sufficient to analyze correlation. The best judge of whether you have a good correlation or not is the calculation for r, as seen below. The r is the correlation coefficient.

The calculation for r is:

Look again at Canada vs US populations

They should be positively correlated. Calculate r.

(Let Microsoft Excel do the work)
We get r= 0.9976.

So it's true. We have r very close to 1.000 and this indicates strong positive linear correlation.

Year	1951	1961	1971	1981	1991	2001
Canadian	14.0	18.2	21.6	24.3	27.3	30.0
U.S.	151.3	179.3	203.3	226.5	248.7	276.2

Generic example (re-visited):

x-variable	1	2	3	4	5	6	7	8	9	10
y-variable	46	67	43	89	49	19	48	7	12	87

Remember this is the obvious application which should indicate poor correlation.

Calculate value for r.

We get r= -0.22

This value is far removed from 1.000 or from -1.000 indicating poor correlation.

Food Prices are often in the news because they seem
to affect everyone.

Food Prices	A	B	C	D	E	F	G	H	I	J
Price 1977	0.95	0.95	0.69	0.98	0.52	0.76	0.45	1.18	0.71	2.09
Price 1984	1.89	1.87	0.8	1.19	0.82	1.33	0.83	2.85	0.89	3.49

Put this into a scatterplot and get.