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Math Lecture Notes
What is Average?

In-Class Activity link to here link to index

Typical of a Group

A green triangle pattern block weighs 1.5 grams.

green weighs one and one-half grams

1. What is the typical weight of a block in this group?

a group of pattern blocks

Hm. What do we mean by typical? Discuss this with your classmates and develop an answer that you are prepared to defend before the class, using the pattern blocks as props if that helps.

1a. Is it the block that appears most often? That would be the blue block, because we have six of those.

1b. Perhaps we do not care about counting which block that appears the most, but when looking at the blocks do get a sense that a block type that only appears once or twice does not appear often enough to be considered representative of the group. We would still pick the blue block as typical, because there are not enough green or red blocks to matter compared to the bunch of blues.

1c. Is it about the average weight of all the blocks?

Finding an Average

In other math classes you have probably learned than an average is when we first sum the values and then divide by how many values there were.

The total weight is (1.5 × 2 greens) + (3 × 6 blues) + (4.5 × 1 red) = 25.5 total grams.

Then we divide by how many blocks there are to find an average weight of 25.5 total grams ÷ 9 blocks ≈ 2.8 grams.

The average weight of 2.8 grams is slightly less than a blue block. Should we say that the blue block is typical? Or is the blue block a bit more than typical?

There is no right answer. The definition of the word typical is an English issue, not a math issue.

However, notice how all three ideas in the answer above about the word typical—and any new ideas you or your classmates had—will all agree that if we have to pick one block as typical the best answer is the blue block.

2. Someone added two big blocks! Now what is the typical weight of a block in the group?

another group of pattern blocks

2a. Is it the block that appears most often? That would still be the blue block, because we have six of those.

2b. Perhaps we do not care about counting which block that appears the most, but when looking at the blocks do get a sense that a block type that only appears once or twice does not appear often enough to be considered representative of the group. We would still pick the blue block as typical, because there are not enough green, red, or yellow blocks to matter compared to the bunch of blues.

2c. Is it about the average weight of all the blocks?

Finding Another Average

In other math classes you have probably learned than an average is when we first sum the values and then divide by how many values there were.

The total weight is (1.5 × 2 greens) + (3 × 6 blues) + (4.5 × 1 red) + (9 × 2 yellows) = 34.5 total grams.

Then we divide by how many blocks there are to find an average weight of 34.5 total grams ÷ 11 blocks ≈ 3.1 grams.

The average weight of 3.1 grams is slightly more than a blue block. Should we say that the blue block is typical? Or is the blue block a bit less than typical?

Once again there is no right answer. We can disagree about definition of the word typical. Yet it again seems that no matter how you define typical the best answer is the blue block.

It was perhaps Benjamin Disraeli who first said, "There are three kinds of lies: lies, damned lies, and statistics." Surely our uncertainly about how to use the word typical demonstrates that people can...exaggerate a bit...without really lying...when trying to be convincing.

Let's see what mathematicians do to remove the uncertainly and ambiguity. We want some math terms to help us recognize when someone might be misleading us with statistics.

Facts link to here link to index

Histograms

Below is a chart that shows midterm scores for a Math 25 class back in Winter term 2016.

math class test histogram

This type of chart is called a histogram. Each bar shows a different category of things. (In this case, students with a certain test score.) The height of the bar counts how many things are in each category.

3. How many students scored between 80% and 90% (inclusive) on the histogram above?

3. The word "inclusive" tells us to include both the 80% bar and the 90% bar. So we are being asked to total the heights of all the yellow bars.

1 + 1 + 2 + 3 + 2 + 1 = 10 students

4. How many students took that midterm?

4. We are being asked to total the heights of all the bars. Let's group them by color just to help avoid making a careless mistake.

(1 + 1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 2 + 3 + 2 + 1) + (3 + 2 + 1) = 23 students

Many histograms are like the one above and use categories with numeric values. The categories have a natural order because they are numbers for ages, weights, heights, test scores, etc.

Not all histograms have numeric categories. Common examples include populations for different countries, costs of living in different cities, popularity of different foods, etc.

The histogram above was also color-coded to group the categories into D, C, B, and A grades. But color-coding is not normally a part of histograms.

Bell Curves

One Bell Curve

The histogram above had a big "clump" of B students with yellow bars, a smaller clump of "A" students with green bars, and a few students that scored below 80% without much clumping

Some histogams form one very symmetric clump. These are called bell curves because they are shaped like a bell.

Bell curves can happen when an average result is the most common, being higher or lower than average is equally likely, and being way higher or lower could happen but is rare.

one bell curve broken into three parts

Random events such as the sum of two dice will create a bell curve.

But to make a bell curve happen we need a histogram that counts a very large number of things. If we find the sum of two dice only a dozen times then wacky luck might produce some crazy histograms. But if we find the sum of two dice two hundred times we will always get a bell curve.

Many physical characteristics of people such as heights and weights will form a bell curve if a large enough group of people are measured.

One Bell Curve

Consider a pair of mostly similar bell curves.

two bell curves

Let's say these bell curves are both histograms counting people.

Because the two bell curves have the same area, they both count the same number of people.

Because the two bell curves have the same middle value, the average is the same for both sets of people.

The pink bell curve is taller. More of its people are average or nearly average, and fewer are notably higher or lower. Because of how bell curves work, this effect is exaggerated near the middle and extremes. The pink bell curve has a lot more average people, and a lot fewer extremely high or low people.

The blue bell curve is shorter. More of its people are funky. Most are still average or nearly average. But comparatively more are notably higher or lower. Because of how bell curves work, this effect is exaggerated near the middle and extremes. The blue bell curve has a lot fewer average people, and a lot more extremely high or low people.

Both bell curves have mostly average people. The "tails" of both bell curves remain above zero: both curves have extremely high or low people.

Imagine these curves measured height. The pink bell curve is measuring a group of people that strongly tend to be about average height—not too many are much taller or shorter, and the group has very, very few giants and dwarves. The blue bell curve is measuring a group of people with less tendancy to be average height—many more are taller or shorter than average, and although the group still has very few giants and dwarves, if you happened to see a giant or dwarf it would probably be from the blue group.

Formulas link to here link to index

Mean and Median

Mean

The average you are most used to in other math classes is formally called the mean.

The Mean

To find the mean of a group of numbers, first add up all numbers and then divide by how many numbers are in the group.

When people say "average" it should be safe to assume they are talking about the mean, unless they say otherwise. In most situations the mean is a very realistic average that provides useful information about the size of a typical value in that set of numbers.

5. In a certain preschool classroom, the students' weights are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41 pounds. What is the average weight of the twelve students in the room?

weights of preschoolers

5. First we add up all the numbers.

34 + 36 + 36 + 36 + 37 + 37 + 38 + 39 + 40 + 40 + 41 = 453 pounds

Then we divide that total by 12, because there are 12 students.

453 pounds ÷ 12 = 37.75 ≈ 38 pounds

This answer feels right. Some of the preschoolers weigh more than 38 pounds, and some weight less.

The mean is also where a histogram balances.

Notice that when you look at the histogram of preschoolers' weights it looks like the balance point is slightly below 38 pounds, matching our arithmetical answer.

mean on a balance scale

6. Double check that this picture of a balance scale is accurate, and the mean really is where the balance point is drawn at 5.

6. First we add up all the numbers.

2 + 2 + 6 + 10 = 20

Then we divide that total by 4, because there are 4 blocks.

20 ÷ 4 = 5

(We can also see this illustration in a different way. The two blocks on the left side are each 3 spots from the center, for a total left hand weight of 6. The two blocks on the right side are 1 and 5 spots from the center, for a total right hand weight of 6. The balance spot is accurate because the left and right hand weights both total six.)

Median

The median is a different kind of average. It picks the middle number of a sorted list.

The Median

To find the median of a group of numbers, first sort the list of numbers in order and then pick the middle number in that sorted list.

If the list has an even number of values then there will be no middle value. Instead we find the mean of the two values most in the middle.

7. In a certain preschool classroom, the students' weights are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41 pounds. Three preschoolers insist their favorite stuffed animals, with weights 1, 1, and 2 pounds, also be included. Find the mean of all fifteen "friends" in the room.

7. First we add up all the numbers.

34 + 36 + 36 + 36 + 37 + 37 + 38 + 39 + 40 + 40 + 41 + 1 + 1 + 2 = 457 pounds

Then we divide that total by 15, because there are 15 "friends".

457 pounds ÷ 15 ≈ 30 pounds

This answer feels wrong. All of the preschoolers weigh more than 30 pounds. The number 30 does not really represent anyone in the room. It feels wrong that the stuffed animals have such a big influence.

8. In a certain preschool classroom, the students' weights are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41 pounds. Three preschoolers insist their favorite stuffed animals, with weights 1, 1, and 2 pounds, also be included. Find the median of all fifteen "friends" in the room.

8. First we sort the list.: 1, 1, 2, 34, 36, 36, 36, 37, 37, 38, 39, 40, 40, 41

The two middle values are 36 and 37. Their mean is (36 + 37) ÷ 2 = 36.5 pounds

This answer feels right. There are preschoolers who weigh around 36.5 pounds. It is less than the mean when we did not include any stuffed animals, but it feels okay that the stuffed animals have a measurable but not dramatic influence.

mean versus median

The process of finding the median "throws out" any atypical smallest or largest values.

You can think of the median as ignoring the most extreme values on a histogram and then finding where the rest of them balance. That is not quite how the median works, especially if the group of numbers is small. But in the real life situations where the median is used it will ideally behave that way.

The median is appropriate and often used for company salaries, state incomes, house values, and other situations where the lowest and highest values are really should not be thought of as representative of the group.

Mode

Just for the sake of completeness, know there is a third kind of average called the mode.

The Mode

The mode of a group of numbers is the number that appears most ofen. If there is a tie, all ties are modes.

(On a histogram, the modes are the tallest bars.)

9. In a certain preschool classroom, the students' weights are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41 pounds. Which weight is the mode?

9. The only number to appear three times is 36. So the mode is 36 pounds.

The mode is so very rarely used in real life that it is only taught in math classes because of tradition. It will not appear in homework or a test.

Homework link to here link to index

More Practice

three distribution curves

28. The home values on a certain street, in thousands of dollars, are: 384, 364, 342, 346, 360, 356, 265, 417, and 530. What is the mean and median of these numbers? Which type of average best communicates the "typical" home on that street?

29. A shipping company needs to transport seven freight containers. Their weights are 10, 16, 16, 18, 20, 60, and 77 tons. What is the mean and median of these numbers? Which type of average best communicates the "typical" weight of the containers?

30. Two company clerks receive a report that only contains the mean and median weights, and number of containers, from the previous problem. The first clerk tries to find the total weight by multiplying the mean by the number of containers. The second clerk tries to find the total weight by multiplying the median by the number of containers. Which clerk is correct? Why? How much error does the other clerk have?

28. For the home values:

The sum is $3,364,000. There are nine homes. Divide to find the mean, which is $374,000.

The sorted list is 265, 342, 346, 356, 360, 364, 384, 417, and 530. So the median is the middle value of 360, corresponding to $360,000.

(By the way, the median is more traditional for home values so one atypical neighbor does not affect the local values as much. But someone selling a house on the street might use the mean so the neighboring property values sound higher.)

29. For the shipping containers:

The sum is 217 tons. There are seven containers. Divide to find the mean, which is 31 tons.

The list is already sorted, and the middle value is 18 tons. In this particular case this is most "typical" for five of the seven shipping containers.

30. The first clerk gets 31 × 7 = 217 tons, which is correct. He "worked backwards" from the mean to the total. The second clerk gets 18 × 7 = 126 tons, which is too light by 217 − 126 = 91 tons. The median neglected the two much heavier containers which made his estimate is too small.

B4. During the 2007 strike of the Writer's Guild of America, two different news reports painted very different pictures of these screen and television writers.

• According to CNBC, there were 4,434 guild writers who worked full-time in 2006, and their average salary was $204,000. (CNBC headline, October 11, 2007)

• According to the Los Angeles Times, the median income of the writers from their guild-covered employment is $5,000 a year. (Howard A. Rodman, October 17, 2007)

Which was true? Were Hollywood's writers very wealthy and going to strike even though they earned much more than most Americans? Or were they poor and going on strike to defend the few thousand dollars they could earn from their writing? The truth was that a very few writers earn millions of dollars (what did that do to the mean?) and almost half of the guild's writers don't write anything in a given year (what does having half the list values zero do to the median?) How could averages been used better to explain and report that situation?

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Homework

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