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# Tweaks, Trends, and Correlation Help (page 2)

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## Correlation

Specialized graphs called scatter plots or scatter graphs can show the extent of correlation between the values of two variables when the values are obtained from a finite number of experimental samples.

If, as the value of one variable generally increases, the value of the other generally increases too, the correlation is considered positive. If the opposite is true – the value of one variable generally increases as the other generally decreases – the correlation is considered negative. If the points are randomly scattered all over the graph, then the correlation is considered to be 0.

Figure 1-18 shows five examples of scatter plots. At A the correlation is 0. At B and C, the correlation is positive. At D and E, the correlation is negative. When correlation exists, the points tend to be clustered along a well-defined path. In these examples the paths are straight lines, but in some situations they can be curves. The more nearly the points in a scatter plot lie along a straight line, the stronger the correlation.

Correlation is rated on a scale from a minimum of –1, through 0, up to a maximum of +1. When all the points in a scatter plot lie along a straight line that ramps downward as you go to the right, indicating that one variable decreases uniformly as the other variable increases, the correlation is –1. When all the points lie along a straight line that ramps upward as you go to the right, indicating that one variable increases uniformly as the other variable increases, the correlation is +1. None of the graphs in Fig. 1-18 show either of these extremes. The actual value of the correlation factor for a set of points is determined according to a rather complicated formula that is beyond the scope of this book.

## Tweaks, Trends, and Correlation Practice Problems

#### Practice 1

Suppose, as the value of the independent variable in a function changes, the value of the dependent variable does not change. This is called a constant function. Is its graph nonincreasing or nondecreasing?

#### Solution 1

According to our definitions, the graph of a constant function is both nonincreasing and nondecreasing. Its value never increases, and it never decreases.

#### Practice 2

Is there any type of function for which linear interpolation is perfectly accurate, that is, ''fills in the gap'' with zero error?

#### Solution 2

Yes. If the graph of a function is known to be a straight line, then linear interpolation can be used to ''fill in a gap'' and the result will be free of error. An example is the speed of a car that accelerates at a defined and constant rate. If its speed-versus-time graph appears as a perfectly straight line with a small gap, then linear interpolation can be used to determine the car's speed at points inside the gap, as shown in Fig. 1-19. In this graph, the heavy dashed line represents actual measured data, and the thinner solid line represents interpolated data.

Fig. 1-19. Illustration for Practice 2.

Practice problems for these concepts can be found at:

Background Math Practice Test

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