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Trend Lines

A trend line is a line added to a scatter plot that shows the generalization of all the data. Let's look at the scatter plot used in this explanation to show a trend line.

Example: Trend Line on Scatter Plot

We ask our friends their height and shoe size. The data is recorded below.

We're going to graph these points on a scatter plot.

This graph has a positive correlation. The $y$-value of each point generally increases as the $x$-value increases. We can add a trend line to this graph by adding a line that goes through the middle of the points.

Notice that the trend line has a positive slope. This matches the positive correlation of our scatter plot.

Line of Best Fit

The line of best fit is the trend line that fits the data the most closely. Take a look at this scatter plot with multiple trend lines:

The dotted green and dotted red lines are also trend lines. They generally go through the middle of our data set, and they are both positively sloping to match the correlation of the data. However, the solid purple line is the line of best fit as it fits our data the best.

It can be very difficult to calculate the equation for the line of best fit, but it can be made easier using programs such as Microsoft Excel or Google Sheets.

Interpolation and Extrapolation

The line of best fit can be used for estimation. Interpolation is when we use the trend line to estimate a value that is already inside of our data set. Extrapolation is when we use a trend line to estimate a value that is outside of the range of our data set.

Let's say we wanted to estimate the shoe size of someone that is $75 \text{ in}$ tall. $75 \text{ in}$ is within our data set, so we can follow our line of best fit to interpolate the shoe size. At $75$ on the $x$-axis, the $y$-value is about $14$, which means that we could estimate someone $75 \text{ in}$ tall would wear size $14$ shoes.

Investments

You may see these kinds of data points when you're looking at stock prices and other values when investing! Stocks can swing high and low over a time frame, and you may need to draw a line of best fit to figure out whether the price is steadily increasing or decreasing. One think you'll notice with this real-world example is that you may have a bunch of data points underneath the line or on top of the line. This is called support or resistance.

If a bunch of points are on top of the line, then the line acts as support. If there are a bunch of points under the line, then the line acts as resistance. With rapidly changing stock prices, you'll see trend line breaks, where the line suddenly changes direction.

Adding Trend Lines to Graphs

A trend line (also called the line of best fit) is a line we add to a graph to show the general direction in which points seem to be going.

Think of a "trend" as a pattern in math. Whatever shape you see on a graph or among a group of data points is a trend. The trend line is something we add to our graph to make the pattern even clearer.

So, now we can add a trend line to our graph to help make this trend even more visible.

So, how can we find the equation of the line of best fit (another name for a trend line)?

There's a pretty simple set of steps to follow:

Let's take a look at a sample scatterplot. It shows the number of bottles of water sold as the temperature outside varies:

First, let's find the $y$-intercept. This is where the data cross the $y$-axis. In other words, it's the number of bottles sold when the temperature is $0$° F.

This means that our $y$-intercept is $1$.

Let's add this to our formula for the equation of a line:

Now, let's work on Step 2: finding our slope.

For this, pick $2$ points from your data and use them to calculate slope. Recall that the formula for slope is:

From the graph, I'm going to pick $(60, 8)$ and $(90, 20)$ as the two points to calculate slope.

Let's add this into the equation of our line:

Now, all you need to do is graph the line.

We know it will start at $(0, 1)$, which is the $y$-intercept. Then, for each $5$ degrees we go, we increase by $2$ bottles. Here's what the line looks like:

(Video) Scatter Plots and Trend Lines

by Matt Pacione

This video by Matt Pacione covers trend lines on scatter plots and will cover how to draw trend lines.

$\large{\text{Summary}}$

Trend lines are lines used to approximate the general shape of a scatter plot. A positive trend line tells us the scatter plot has a positive correlation. A negative trend line tells us the scatter plot has a negative correlation.

The first example in the video is time studying and the grade you receive. You can expect that as you study more, your grades go up. This would be an example of a positive correlation.

You can see the trend that as time increases, so does grades. To create a trend line, draw a line on the graph that matches the slope of the data.

Here's the scatter plot with a trend line drawn in to approximate the data.

We can use the trend line to approximate values, such as a period of time. If you study $5 \text{ hours}$, you can look at the line and determine what grade you probably get. More scattered graphs will have a less accurate trend line.

The line of best fit is the trend line that is the closest approximation to all the data. The easiest way to find this is with programs like Excel or an online tool.

Here's the scatter plot with my guess of a trend line (dashed red). The blue line is the line of best fit. My guess was close.

Try drawing a scatter plot yourself using the data in the video. The $x$-axis is the hour on the internet per day, and the $y$-axis is the grade average in percentage. $$\begin{array}{|c|c|} \hline x&1&2&1&3&2&0&4&2&0&0&4&5\\ \hline y&90&87&95&80&83&96&79&86&93&90&70&50\\ \hline \end{array}$$

Let's see how you did!

The blue line is the line $y=-\frac{13}2x+96$. This is the best approximation compared to the others.