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NY Regents Exam - Geometry: Help and Review11 chapters | 135 lessons | 8 flashcard sets

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Lesson Transcript

Instructor:
*Emily Cadic*

Emily has a master's degree in engineering and currently teaches middle and high school science.

Learn about trend lines, including what they are, how to calculate them, and how to interpret them when they are used in a graph. You will see the concepts unfold in a series of examples that will prepare you for solving a variety of practice problems.

A **trend line**, often referred to as a line of best fit, is a line that is used to represent the behavior of a set of data to determine if there is a certain pattern. A trend line is an analytical tool used most often in conjunction with a scatter plot (a two-dimensional graph of ordered pairs) to see if there is a relationship between two variables.

Let's take a quick look at the main purposes of a trend line:

1. Determining if a set of points exhibits a positive trend, a negative trend, or no trend at all. Looking at the red trend lines in the examples illustrates various relationships with sets of data.

2. Predicting unknown or future data points.

This graph shows temperatures over the course of ten days. If you were attempting to predict the temperature on the 11th day based on this graph, a good estimate would be 70.5 degrees.

Determining the exact value of a trend line may not always be necessary. In some cases, an approximation is sufficient for gleaning the general behavior of the data. If the data set is linear, the trend line is simply a line running through each point. For all other data sets, there is a simple strategy for approximating the trend line: draw a line that is situated at a minimal distance from each point while trying to pass through as many as possible, so that the number of points falling above and below the line is roughly equal. Here is an illustration of this strategy:

In this graph, the trend line, though approximate, clearly indicates a positive relationship.

If we were asked to report an exact equation for the trend line that we approximated in the previous section, we would have to utilize the following formula:

This may look intimidating at first. It is much less so if you organize your computations in a table. In addition, there are easily accessible programs across the Internet that will automatically calculate trend lines that you can use to verify your work.

To test our new formula, we will use the data set featured in the previous section. To review, the ordered pairs were (1,2), (2,3), (3,6), (4,8), (5,10), and (6,12). As you calculate, you can create a table, like the one here, in which the formula has been split up into smaller groups of variables:

**Step 1: Complete each column of the table**

**Column 1:** the differences between each *x*-coordinate and the average of all of the *x*-coordinates

Row 1 Example:

1 - {(1+2+3+4+5+6) / 6} = 1 - 3.5 = -2.5

**Column 2:** the difference between each *y*-coordinate and the average of all of the *y*-coordinates

Row 1 Example:

2 - {(2+3+4+6+8+10) / 6} = 2 - 6.83 = -4.83

**Column 3:** multiply columns 1 and 2 = -2.5 * (-4.83) = 12.083

**Column 4:** multiply column 1 by itself = -2.5 * (-2.5) = 6.25

**Step 2: Calculate the slope ( m) of your trend line by dividing the total for Column 3 by the total for Column 4**

*m* = 36.5 / 17.5 = 2.0857

**Step 3: Calculate the y-intercept (b) of your trend line by using the average of the slope from Step 2 and the average of the x and y-coordinates**

*b* = 6.83 - (2.0857 * 3.5) = -0.46667

**Step 4: Report your equation and verify**

Here the trend line is equal to: *y* = 2.0857*x* - 0.4667

Up until this point, we have only spoken about trend lines, line being short for linear function. A first-order linear function may not be appropriate for all data sets, however. In some cases, a higher-order equation or a special function may provide the best match from which to draw conclusions about the behavior of the data and/or make predictions. Take, for instance,this graph:

There is a different type of curve that turns out to be more compatible with this data than the linear trend line; it connects or come closer to connecting to a greater number of points. As it turns out, like many problems dealing with monetary value, the data follows an exponential decay trend.

Please take note of a final word of caution about trend lines and when they may not be applicable. If you are dealing with an incomplete data set, it may be problematic to begin drawing conclusions from a trend line. You can't rely on a trend line that was generated using two points, for instance, as much as you could one generated using two hundred points.

In this lesson, we learned that a **trend line**, also referred to as a line of best fit, is a line that is used to represent the behavior of a data set to determine if there is a certain pattern. If the data set is linear, the trend line is simply a line running through each point. For all other data sets, draw a line that is situated at a minimal distance from each point while trying to pass through as many as possible, so that the number of points falling above and below the line is roughly equal.

To get an exact equation for a trend line, we use the following formula:

It is easiest to work with the formula if you create a table using the following steps:

**Step 1:** Complete each column of the table

**Step 2:** Calculate the slope (*m*) of your trend line by dividing the total for Column 3 by the total for Column 4

**Step 3:** Calculate the *y*-intercept (*b*) of your trend line using the average of the slope from Step 2 and the average of the *x-* and *y*-coordinates.

**Step 4:** Report your equation and verify.

Here, the trend line is equal to: *y* = 2.0857*x* - 0.4667

A trend line may not be appropriate for all data sets, however. If you are dealing with an incomplete data set, it may be problematic to begin drawing conclusions from a trend line. You can't rely on a trend line that was generated using two points. In some cases, a higher-order equation or a special function may provide the best match from which to draw conclusions about the behavior of the data and/or make predictions.

**Step 1:** Complete each column of the table: Column 1 = the differences between each *x*-coordinate and the average of all of the *x*-coordinates; Column 2= the difference between each *y*-coordinate and the average of all of the *y*-coordinates; Column 3 = multiply columns 1 and 2; Column 4 = multiply column 1 by itself.

**Step 2:** Calculate the slope (*m*) of your trend line by dividing the total for Column 3 by the total for Column 4.

**Step 3:** Calculate the *y*-intercept (*b*) of your trend line using the average of the slope from Step 2 and the average of the *x-* and *y*-coordinates.

**Step 4:** Report your equation and verify.

After studying all aspects of this video lesson, evaluate your ability to:

- Write the definition of a trend line
- Remember the strategy for approximating a trend line
- Calculate the trend line for a function using a table
- Note that trend lines may not be appropriate for all data sets

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NY Regents Exam - Geometry: Help and Review11 chapters | 135 lessons | 8 flashcard sets

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