Weighted Averages and Indices Lesson

Adapted from Course materials (1.A Student Activity Sheet 6,7, &10) for AMDM developed under the leadership of the Charles A. Dana Center, in collaboration with the Texas Association of Supervisors of Mathematics and with funding from Greater Texas Foundation.

How do you determine your final grade in this course?

When a weighted average is applied to a set of numbers, more importance (weight) is placed on some components of the set. Your final average in this class is probably an example of a weighted average.

Consider two grading systems for determining your final class average. Each system is a weighted average of measures that include test grades, final exam grades, homework, and class participation.

In this case, you would probably prefer Grading System 2.

But what if you scored 10 points higher on the final exam, and got a 78 instead of a 68? Which system would you prefer then?

Answer: Under system 1, your average would be 85.1. Under system 2, your average would also be 85.1. In this case, either system gives you the same average.

What if you scored 6 points lower on the final exam and got a 62 instead of a 68? Which system would you prefer then?

Answer: System 1 would produce a final grade of 81.1. System 2 would produce a final grade of 82.7. You would probably prefer system 2.

You try it! Calculate a final course average using the two grading systems above.

You have four test grades: 80, 74, 82, and 88. Your final exam grade was an 84, homework was 90, and discussion was a 95. What is your average under each of the grading systems?

Answer: Test Average (80 + 74 + 82 + 88)/4 = 81
System 1: 81(0.40) + 84(0.25) + 90(0.25) + 95(0.10) = 85.4
System 2: 81(0.60) + 84(0.15) + 90(0.15) + 9 (0.10) = 84.2
You have a test average of 85, a homework average of 90, and a discussion average of 95. What grade do you need on the final exam to earn at least an 87 under each grading system?

Answer: System 1 requires a final exam grade of 84 and System 2 a final exam score of 87.

Index

An index is a numerical scale used to compare variables with one another or a reference number, to give information about general trends, and to help make comparisons and judgments. It is often calculated as a weighted sum of various factors resulting in a single summary number.

Fog Index

The Fog Index, which was developed by Robert Gunning in 1952, measures the readability of a piece of writing. Calculation of the index produces a number that indicates the years of education a person needs to have to easily understand the text. For example, a passage with a Fog Index of 8 has an eighth-grade reading level.

Writers can use the Fog Index to gauge whether or not their material is suitable for the targeted audience. It is not an index of how good your writings is, but instead how easy it is to understand. The ideal Fog Index number for a writing piece for the general population is 7 or 8. Technical material usually has a Fog Index of 10 to 15.

How to Use the Fog Index

Select a short passage and count the number of words.

For a lengthy document, select several different passages and average the Fog Index.

For example, this is an excerpt from the Supreme Court Decision, Mapp v. Ohio, 367 U.S. 643, 1961:

The ignoble shortcut to conviction left open to the State tends to destroy the entire system of constitutional restraints on which the liberties of the people rest. Having once recognized that the right to privacy embodied in the Fourth Amendment is enforceable against the States, and that the right to be secure against rude invasions of privacy by state officers is, therefore, constitutional in origin, we can no longer permit that right to remain an empty promise. Because it is enforceable in the same manner and to like effect as other basic rights secured by the Due Process Clause, we can no longer permit it to be revocable at the whim of any police officer who, in the name of law enforcement itself, chooses to suspend its enjoyment. Our decision, founded on reason and truth, gives to the individual no more than that which the Constitution guarantees him, to the police officer no less than that to which honest law enforcement is entitled, and, to the courts, that judicial integrity so necessary in the true administration of justice.

The judgment of the Supreme Court of Ohio is reversed and the cause remanded for further proceedings not inconsistent with this opinion.

Number of words = 200

Count the number of sentences within the passage.

Number of sentences = 5

Count the number of big words (3 or more syllables).

Exclude words in which "es" or "ed" form the third and final syllable, hyphenated words like "state-of-the-art", and compound words like "newspaper."

Number of "big words" = 23

Plug those numbers into the following equation

That's the Fog Index score.

Fog index score

The example passage requires a "21st" grade level of reading ability, in other words, the passage is at a post-graduate level of comprehension, something you would expect from the Supreme Court. Using technical terms or jargon tends to increase Fog Index scores. If these are words that the audience would be familiar with, you don't need to count them as big words.

Flesch/Flesch-Kincaid Readability Tests

Another widely used measure is the Flesch/Flesch-Kincaid Readability Tests, which consist of the Flesch Reading Ease (developed by Rudolph Flesch in 1948) and the Flesch-Kincaid Grade Level (developed by John P. Kincaid in 1976). The Flesch-Kincaid Grade Level Index converts the Flesch Reading Ease into a number that indicates the years of education a person needs to have to easily understand the text.

Calculate the Flesch-Kincaid Grade Level Index (FKGLI) using the following formula:

Formula for the Flesch-Kincaid Grade Level Index (FKGLI) showing the calculation: 0.39 times the number of words divided by the number of sentences, plus 11.8 times the number of syllables divided by the number of words, minus 15.59.

The following is an excerpt of the Michael Jordan Biography

Michael Jordan jumping and making a basket

Michael Jeffrey Jordan is an American former professional basketball player, entrepreneur, and majority owner and chairman of the Charlotte Bobcats. His biography on the National Basketball Association (NBA) website states:

By acclamation, Michael Jordan is the greatest basketball player of all time." Jordan was one of the most effectively marketed athletes of his generation and was considered instrumental in popularizing the NBA around the world in the 1980s and 1990s. Jordan is also noted for his product endorsements. He fueled the success of Nike's Air Jordan sneakers, which were introduced in 1985 and remain popular today.

Find the Fog Index and the Flesch-Kincaid Grade Level Index of the excerpt of the Michael Jordan Biography.
Check your answer for the Flesch-Kincaid Grade Level Index using MS Word. (Go to the Help section and look up Readability to find how to access this score.)
Find the Fog Index and the Flesch-Kincaid Grade Level Index for a passage of around 100 words from one of your favorite books or short stories.
Find the Fog Index and the Flesch-Kincaid Grade Level Index for a passage of around 100 words from your science textbook.
Write a proposal advertising a real or imagined product. The potential buyers of the product are high school students requiring a Fog Index and a Flesch-Kincaid Grade Level Index of between 10 and 12.

The following chart shows Michael Jordan's career statistics while playing professional basketball.

Calculate Jordan's career average points per game (PPG)?

Answer: 30.1

Notice that there is a year of play missing. What do you think happened to Michael Jordan in 1994?
Answer: He played minor league baseball

Baseball Slugging Average

One example of a weighted average in sports is a batter's slugging average (or percentage) in baseball. The slugging average (SLG) is calculated using the following equation:

SLG = (1x singles) + (2x doubles) + (3x triples )+ (4x home runs) / Total number of at bats

As you can see, each single has a weight of 1, each double a weight of 2, each triple a weight of 3, and each home run a weight of 4. An at-bat without a hit has a weight of 0.

In his first season with the New York Yankees, Babe Ruth set a record for slugging average that stood for more than 80 years. In 1920, Ruth pounded 172 hits in 458 at-bats. His hits consisted of 73 singles, 36 doubles, 9 triples, and 54 home runs, resulting in a total base count of (73 x 1) + (36 x 2) + (9 x 3) + (54 x 4) = 388. When his total number of bases (388) is divided by his total at-bats (458), the result is .847, his slugging percentage for the season. This record was broken in 2001 by Barry Bonds, who had 411 total bases in 476 at-bats for a slugging average of .863.

Is it possible to have a slugging average of more than 1?

Answer: Yes, unlike most percent averages, the slugging average can exceed 1 ( 100%), with a maximum possibly average of 4.

Think about this information and then answer the following questions

Theoretically, what is the highest possible value for the slugging average?
Answer: 4.0
Could a player ever achieve this value during a baseball season?
Answer: A 4.0 average would only be possible if a player hit a homerun at every at bat.
Give an example or explain why none exist.
Answer: Although a homerun at every at bat is theoretically possible and could even be achieved in a single game, it is extremely unlikely that a player could continue to hit a home run at every at bat over the course of several games

A slugging average of .500 or higher is considered a sign of an excellent player. Suppose a player had 4 triples and a batting average of .300 in 400 at-bats (batting average = hits/at-bats). Determine a combination of singles, doubles, and home runs that gives this player a slugging average higher than .500.

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