# Scatter Plots

A Scatter (XY) Plot has points that show the relationship between two sets of data.

In this example, each dot shows one person's weight versus their height.

(The data is plotted on the graph as "Cartesian (x,y) Coordinates")

## Example:

The local ice cream shop keeps track of how much ice cream they sell versus the noon temperature on that day. Here are their figures for the last 12 days:

Temperature °C Ice Cream Sales Ice Cream Sales vs Temperature 14.2° \$215 16.4° \$325 11.9° \$185 15.2° \$332 18.5° \$406 22.1° \$522 19.4° \$412 25.1° \$614 23.4° \$544 18.1° \$421 22.6° \$445 17.2° \$408

And here is the same data as a Scatter Plot:

It is now easy to see that warmer weather leads to more sales, but the relationship is not perfect.

## Line of Best Fit

We can also draw a "Line of Best Fit" (also called a "Trend Line") on our scatter plot:

Try to have the line as close as possible to all points, and as many points above the line as below.

But for better accuracy we can calculate the line using Least Squares Regression and the Least Squares Calculator.

## Example: Sea Level Rise

 A Scatter Plot of Sea Level Rise: And here I have drawn on a "Line of Best Fit".

## Interpolation and Extrapolation

Interpolation is where we find a value inside our set of data points.

Here we use linear interpolation to estimate the sales at 21 °C.

Extrapolation is where we find a value outside our set of data points.

Here we use linear extrapolation to estimate the sales at 29 °C (which is higher than any value we have).

Careful: Extrapolation can give misleading results because we are in "uncharted territory".

As well as using a graph (like above) we can create a formula to help us.

### Example: Straight Line Equation

We can estimate a straight line equation from two points from the graph above

Let's estimate two points on the line near actual values: (12°, \$180) and (25°, \$610)

First, find the slope:

 slope "m" = change in y change in x = \$610 − \$180 25° − 12° =  \$430 13° = 33 (rounded)

Now put the slope and the point (12°, \$180) into the "point-slope" formula:

y − y1 = m(x − x1)

y − 180 = 33(x − 12)

y = 33(x − 12) + 180

y = 33x 396 + 180

y = 33x 216

#### INTERpolating

Now we can use that equation to interpolate a sales value at 21°:

y = 33×21° 216 = \$477

#### EXTRApolating

And to extrapolate a sales value at 29°:

y = 33×29° 216 = \$741

The values are close to what we got on the graph. But that doesn't mean they are more (or less) accurate. They are all just estimates.

Don't use extrapolation too far! What sales would you expect at 0° ?

y = 33×0° 216 = \$216

Hmmm... Minus \$216? We extrapolated too far!

Note: we used linear (based on a line) interpolation and extrapolation, but there are many other types, for example we could use polynomials to make curvy lines, etc.

## Correlation

When the two sets of data are strongly linked together we say they have a High Correlation.

The word Correlation is made of Co- (meaning "together"), and Relation

• Correlation is Positive when the values increase together, and
• Correlation is Negative when one value decreases as the other increases

Like this:

## Negative Correlation

Correlations can be negative, which means there is a correlation but one value goes down as the other value increases.

### Example : Birth Rate vs Income

The birth rate tends to be lower in richer countries.

Below is a scatter plot for about 100 different countries.

 Country Yearly Production per Person Birth Rate Madagascar \$800 5.70 India \$3,100 2.85 Mexico \$9,600 2.49 Taiwan \$25,300 1.57 Norway \$40,000 1.78

It has a negative correlation (the line slopes down)

Note: I tried to fit a straight line to the data, but maybe a curve would work better, what do you think?