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VCE Further Mathematics: Exam Prep & Study Guide21 chapters | 136 lessons | 16 flashcard sets

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Lesson Transcript

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.

Seasonal indices can be used to deseasonalize and, thereby, smooth time plot data. That means seasonal fluctuations or patterns can be removed from the data, and forecasts can be made with regard to future data values.

Let's suppose that a fictional company sells widgets. This company recorded the number of sales of these widgets on a quarterly basis through two years, or eight times, which you can see in the table:

What will be the company's projected sales in quarter 9 (the first quarter of 2018) or quarter 10 (the second quarter of 2018)?

This is where we can implement seasonal indices to **deseasonalize** and, thereby, smooth data to allow for forecasting of trends. A **seasonal index** is a measure of how a particular season through some cycle compares with the average season of that cycle. By deseasonalizing data, we're removing seasonal fluctuations, or patterns in the data, to predict or approximate future data values.

Our fictional company wishes to project sales of widgets into 2018. Let's help them do this using:

- Seasonal indices
- Forecasting or trending
- Deseasonalized linear regression (remember that
**linear regression**involves determining a straightforward relationship between an independent and dependent variable)

Before we begin, let's slightly reformat the table given our example to represent the number of sales in millions of widgets over eight quarters through 2016 and 2017. To avoid confusion, note that this table contains the same data as the prior table; it's just represented differently.

Furthermore, let's make a time-plot of that data:

We can use the time plot to make a visual note of the general shape and behavior of our sales through time.

To calculate seasonal indices, we first take the yearly average, or mean, of the quarterly sales, which you can see on the table:

Secondly, we divide each quarterly sales figure by its respective yearly mean, which gives us the following indices in the table:

Note how these index values rise and fall with their respective sales values. We then take the mean of these indices for each quarter to get our seasonal (or quarterly) index values:

Note how the index values at the bottom add up to 4, or the number of quarters. Also notice how the index values rise and fall with their respective quarterly sales.

We can use these seasonal (or quarterly) indices to deseasonalize and, thereby, smooth our sales data over the eight quarters. As shown, we divide each original sales figure by its respective quarterly index:

If we were to plot this deseasonalized sales data through eight quarters and superimpose it onto the original time plot for sales, it would look like this graph:

Note in this graph how this deseasonalized time plot could very well be helpful in forecasting future trends, as sharp seasonal peaks and troughs are smoothed, providing more basic visual aids for trending.

For instance, if we were to draw a straight trend line (by eye) through our deseasonalized sales data, we might see the following, noting that our trend line is shown here with a dotted line:

We also might use this line to visually predict upcoming sales for quarters in 2018.

Let's take a look at an example of predicting further sales numbers using linear regression. In this example, a linear or least-squares regression for the deseasonalized sales of widgets will be given in the form of *y* = *mx* + *b*. While linear regression can be determined by hand, we'll derive a linear regression from the deseasonalized sales data above via a scientific calculator (in this case, a TI-83 or TI-84). We'll then use this linear regression equation to more accurately forecast sales trend figures of widgets into 2018.

Given our deseasonalized sales figures (as listed in the STAT Edit Table on the TI-calculator) through eight quarters of 2016 and 2017:

We can use a linear regression function under STAT CALC on our TI-calculator:

We'll be given the following linear regression to fit our deseasonalized data:

Note that the line of the graph (as displayed on the calculator screen) of this linear regression equation *y* = -0.912*x* + 42.6 is almost identical to that of the deseasonalized widget sales trend line in the prior time plot given in orange.

Therefore, if we want to make a prediction as to how many widgets (in millions) will be sold in the ninth quarter (or the first quarter of 2018), we would merely plug 9 into *y* = -0.912*x* + 1.42.

*y* = -0.912(9) + 42.6 = 34.4

This figure, 34.4, is our projected widget sales figure (in millions) for the first quarter of 2018! Similarly, if we would like to make a prediction as to how many will be sold in the second quarter of 2018, we would plug 10 in for *x*.

*y* = -0.912(10) + 42.6 = 33.5

There would be 33.5 million widgets sold! Like our time plot, this equation shows sales declining in 2018.

Let's take a few moments to review what we've learned about seasonal indices. Seasonal indices can provide a means of smoothing time plot data and allow us to more easily spot **trends** in it. In short, a **seasonal index** is a measure of how a particular season through some cycle compares with the average season of that cycle. By deseasonalizing data, we're removing seasonal fluctuations, or patterns in the data, to predict or approximate future data values.

To illustrate, we can first determine indices for the data set. Second, we can **deseasonalize** and smooth our data. We can then draw a trend line or, better yet, develop a **linear regression**, which involves determining a straightforward relationship between an independent and dependent variable. When developing a linear regression, we're fitting the deseasonalized data and making more precise approximate numerical predictions of the time plot data, which is always preferable to more general predictions.

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VCE Further Mathematics: Exam Prep & Study Guide21 chapters | 136 lessons | 16 flashcard sets

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