# Graphical Smoothing of Time Series Data

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Time plots serve to provide a visual representation of data and can aid in visualizing the behavior of a data set. We address trending in a time plot, smoothing time plot data to better see trends that we might not see otherwise.

## Smoothing a Time Plot

Let's say that Frank is in the business of selling a fictional item or widget. What will the time plot look like if he gives us a list of the number of widgets sold per year from 2000 to 2008? Once Frank obtains the time plot of his data, how can he smooth his plot to better spot trends, using a 3-mean smooth or a 5-mean smooth? To help Frank with this problem, let's start with his numbers. So first, we look at the table of data for his number of widget sales per year.

We have a list of widget sales from 2000 to 2008. If we plot this data set as points on a line graph, where our horizontal axis (or x-axis) is represented by the year and our vertical axis (or y-axis) is represented by the number of sales, we have a graph that looks like this:

## Graphical Smoothing

Notice how the time plot has some sharp peaks and valleys, much like those seen in stock market indices. Upon looking at this data, we might have trouble seeing any trend regarding the increase or decrease of widget sales over time. If we perform a 3-mean smooth or a 5-mean smooth on this time plot, perhaps we might more easily see a possible overall trend.

### 3-Mean Smooth

Looking at the operations in the 3-mean smooth column on the table, we see that a 3-mean smooth is performed on the number of sales by merely taking the mean or the average of groups of 3. We see that the first mean of the first 3 numbers 5, 11 and 5 is 7.00, represented to the right of 11 in the sales column. We get this mean by taking the sum of 5 + 11 + 5 = 21 and dividing this sum by three: 21 / 3 = 7.00. We then group the next 3 numbers, 11, 5 and 4 and find the mean, which is 6.67. The next group of 5 + 4 + 9 has a mean of 6.00 and so on. We repeat this process until we run out of groups of 3 to mean.

### 5-Mean Smooth

With the 5-mean smooth, we do the same operation as in finding the mean for the 3-mean smooth. However, as opposed to groups of 3, we are calculating means for groups of 5. Looking at the column representing the 5-mean smooth of sales, we take the sum of the first group of 5 numbers, 5 + 11 + 5 + 4 + 9, and we divide it by 5. We get 6.80, as seen to the right of 5 in the sales column. We repeat this process until, as with the 3-mean smooth, we run out of groups of 5 to find the mean. We graph all three sets of data:

1. the original time plot of sales vs. time (1st series in blue)
2. the 3-mean smooth plot of sales vs. time (2nd series in orange)
3. the 5-mean smooth plot of sales vs. time (3rd series in grey)

We get a graph that looks like this:

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