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A Look at High School Dropout Rates

Average Rates of Change and Trend Lines

In this module, we calculate and interpret the meaning of various

average rates of change and look at trend lines in the context of high school dropout rates.

Prepared for SSAC byFrank C. Wilson, Chandler-Gilbert Community College

© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. *2006*

SSAC2006:LC142.FCW1.1

Core Quantitative IssueData analysis: Time series: Rate of change

Supporting Quantitative IssuesRates: Average rate of changeVisualization: Graphs: XY scatter plot; trend lineAnalysis: Function; independent, dependent variablesNumber sense: Unit analysis

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Slide 3 states the problem.

Slides 4 - 9 calculate and interpret average rates of change of status dropout rates.

Slides 10 – 12 draw a scatter plot.

Slides 12 - 13 find the trend line and interpret its meaning.

Slide 14 compares the slope of the trend line to the average rate of change.

Slides 15 - 16 ask you to analyze data on event dropout rates.

Overview of Module

High school dropout rates are used by policy makers as an indicator of the effectiveness of individual schools as well as the educational system as a whole. Unfortunately, a lot of confusion exists among politicians and the public due to the fact that there is not a common standard for measuring the dropout rate. Two common measures used by the U.S. Census Bureau are:

(1) Status dropout rate – the percentage of 18 – 24-year olds who never graduated and are not currently enrolled in a high school completion program;

(2) Event dropout rates – the percentage of 10th – 12th grade students who drop out.

In this module, we will investigate both measures of high school dropout rates.

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What percentage of the 18 – 24-year olds in the country this year did not complete high school and are not enrolled in a high school completion program. That percentage is this year’s status dropout rate. Status dropout rates for 1975 to 2003 are listed in this table.

Dropout rates are socially significant because they are often looked to as a measure of the effectiveness of an educational system and, as such, can have direct impact on the allocation of federal and state funding.

Problem

What conclusions can you draw from the data? Justify your conclusions using the concepts of average rate of change and trend line.

Year

Status Dropout Rate(percentage of 18 - 24 year

olds who didn't complete high school and aren't enrolled)

1975 15.6

1980 15.6

1985 13.9

1990 14.4

1995 13.9

1997 13.0

1998 13.9

1999 13.1

2000 12.4

2001 13.0

2002 12.3

2003 11.8

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What is a Rate of Change?

The term rate of change refers to the rate at which a quantity is changing. A car’s speed (in mph) is the rate at which its distance is changing. A car whose distance from a junction is changing at a constant rate of 60 mph will be 60 miles further away from the junction after traveling for one hour.

Note the y-axes. How is one related to the other?

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What is an Average Rate of Change?

Rates of change are often variable. For example, the speed of a certain car on a freeway may fluctuate between 47 mph and 73 mph. The concept of average rate of change effectively smoothes out these fluctuations. If the car with the variable speed travels 60 miles in a one-hour period, we say that its average rate of change in distance is 60 mph. This means that a hypothetical car traveling at a constant rate of 60 mph would end up in the same position as the variable-rate car after one hour.

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There are multiple ways to calculate an average rate of change. One way is to calculate the rate of change between each pair of consecutive data points and then average the rates of change. This method works only if the input values of the data set are equally spaced.

Calculating an Average Rate of Change: Method #1

Quick Example:The average rate of change in the dropout rate between 2000 and 2003 is

YearStatus

Dropout Rate(percent)

Annual Rate of Change

(percentage points

per year)

2000 12.4

2001 13.0 0.6

2002 12.3 0.7

2003 11.8 0.5

0.6 0.7 0.5

30.6

percentage point per year30.2 percentage point per year

Method #1 works in this example because the consecutive years of the data set are equally spaced one year apart.

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Another way to calculate the average rate of change of a function over an interval [a, b] is to divide the difference in the two outputs (dependent variable) by the difference in the corresponding inputs (independent variable). That is,

is the average rate of change in the function f between a and b.

Calculating an Average Rate of Change: Method #2

Quick Example:The average rate of change in the dropout rate between 2000 and 2003 is

net change in output

net change in input

f b f a

b a

YearStatus Dropout Rate

(percent)

2000 12.4

2001 13.0

2002 12.3

2003 11.8

11.8 12.4 percentage points

2003 2000 year

0.6 percentage point per year

30.2 percentage point per year

Method #2 works even if consecutive points in the data set are not equally spaced. Note that this result is the same as what we got with Method #1.

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Keeping track of the units of an average rate of change is key to interpreting its meaning. The units of an average rate of change are units of output (dependent variable) per unit of input (independent variable) of the original data set.

A positive average rate of change indicates increasing behavior.A negative average rate of change indicates decreasing behavior.

Interpreting an Average Rate of Change

Quick Example #1:The average rate of change in the dropout rate between 2000 and 2003 is -0.2 percentage point per year. This means that, although the dropout rate varied between 2000 and 2003, the average rate of decrease in the dropout rate was 0.2 percentage point per year.

Quick Example #2:The average rate of change in the distance of a car from a junction is 40 miles per hour between the first and third hour of a trip. This means that, although the speed of the car may have varied between the first and third hour, the distance from the junction increased at an average rate of 40 miles per hour.

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Calculating Average Rates of Change in Excel

1. Create two columns of data with the left-hand column representing the year and the right-hand column representing the dropout rate.

2. Label a third column “average rate of change.”

3. In the Cell D4, type in an equation to calculate the average rate of change between 1975 and 1980.

4. Copy the formula by grabbing the handle on the lower right hand corner of the Cell D4 and dragging it down to the Cell D14.

5. Calculate the average rate of change in the dropout rate between 1975 and 2003 using methods #1 and #2, respectively. Which result is correct and why?

= cell with a number in it

= cell with a formula in it

B C D

2 YearStatus Dropout

Rate (%)

Average Rate of Change

(percentage points per yr)

3 1975 15.64 1980 15.6 0.05 1985 13.9 -0.36 1990 14.4 0.17 1995 13.9 -0.18 1997 13.0 -0.59 1998 13.9 0.910 1999 13.1 -0.811 2000 12.4 -0.712 2001 13.0 0.613 2002 12.3 -0.714 2003 11.8 -0.51516 Method #1 -0.1817 Method#2 -0.14

Note: A rate of change expressed as % per

year means something else. Why?

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Drawing a Scatter Plot

One way to visually analyze a data set is through the use of a scatter plot. For a scatter plot, the input values of the table and their corresponding output values are plotted on a rectangular coordinate system with the horizontal axis representing the input (independent variable) and the vertical axis representing the output (dependent variable).

1. Select the two columns of the data set.

2. Click on the Chart Wizard icon on the tool bar to open up the Chart Wizard dialog box.

3. Select the XY(Scatter) chart type

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Drawing a Scatter Plot (continued)

4. Click Finish to display the scatter plot.

5. Does the status dropout rate appear to be decreasing or increasing? Explain.

Status Dropout Rate

0

2

4

6

8

10

12

14

16

18

1970 1975 1980 1985 1990 1995 2000 2005

Pe

r c

en

t

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Graphing a Trend Line

A trend line (or line-of-best-fit) is the line that best models a data set. Depending on how well the line fits the data, it may or may not represent the data well.

1. Right-click on a data point on the graph of the scatter plot to open the Format Data Series drop-down menu.

2. Select Add Trendline from the drop-down menu to open the Add Trendline dialog box.

3. Select Linear and click OK to draw the trend line.

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Graphing a Trend Line (continued)

4. Does the trend line seem to fit the data well? Explain.

5. Estimate the slope of the trend line and compare/contrast it with the average rate of change of the data set.

Status Dropout Rate

0

2

4

6

8

10

12

14

16

18

1970 1975 1980 1985 1990 1995 2000 2005

Per

cen

t

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Comparing the Slope of a Trend Line and an Average Rate of Change

If a trend line fits a particular data set well, then the slope of the trend line and the average rate of change of the data set will be close in value. (If the trend line fits the data set perfectly then they will be equal in value.)

If a trend line does not fit a particular data set well, then the slope of the trend line and the average rate of change might not be close in value.

slope of trend line = 1.7

average rate of change = 1.75

slope of trend line = 0.1

average rate of change = 0.5

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Show You Know Activities (End-of-Module Assignments)

Another way to measure the dropout rate is to determine the percentage of 10th 12th graders who drop out of school in a given year. This measure gives the results shown in the table.

1. Explain what an average rate of change is in the context of the event dropout data.

2. Referring to the event dropout data, give a time interval when the percentage of dropouts is increasing on average and a time interval when the percentage of dropouts is decreasing on average.

3. Calculate the average rate of change in the event dropout rate between 1975 and 2003 for this new set of dropout data.

Year

Event Dropout Rate(percentage of 10th – 12th grade

students who dropped out)

1975 5.8

1980 6.0

1985 5.2

1990 4.5

1995 5.4

1997 4.3

1998 4.4

1999 4.7

2000 4.5

2001 4.7

2002 3.3

2003 3.8

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Show You Know Activities (continued)

4. Create a scatter plot of the event dropout rate.

5. Does your calculation of the average rate of change in question #3 appear to be supported by the scatter plot? Explain.

6. Draw a trend line for the scatter plot.

7. Compare and contrast the slope of the trend line with the average rate of change found in question #3 of this activity.

8. Do you think that the event dropout rate or the status dropout rate better represents the high school dropout issue? Explain your reasoning.

Year

Event Dropout Rate(percentage of 10th – 12th grade

students who dropped out)

1975 5.8

1980 6.0

1985 5.2

1990 4.5

1995 5.4

1997 4.3

1998 4.4

1999 4.7

2000 4.5

2001 4.7

2002 3.3

2003 3.8