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Probability and Statistics

Probability and Statistics

At a Glance - Mean/Average

When thinking about averages, we like to think about cookies. Okay, we always like to think about cookies...averages are only a convenient excuse.

Suppose Paul has 3 cookies and Mary has 5. That distribution doesn't seem quite fair. Let's redistribute so Paul and Mary each receive the same number of cookies. We have 3 + 5 = 8 cookies total, which means to divide the cookies fairly between Paul and Mary, we should give each person 4 cookies. We should do it quickly though, because Mary just caught our eye and we're fairly certain she's onto us.

Sample Problem

Anita got 2 cookies, Jonas got 3, and Ella got 2. If we were to redistribute the cookies fairly, how many would each person get?

We have 2 + 3 + 2 = 7 cookies total. If we divide these cookies evenly between the 3 people, we see that each person should receive:

Hope they're chewy rather than crunchy cookies, or cutting these things will be a nightmare.

The mean, or average, of a set of values is the size of a "fair" portion. To find the average of a set of values, we add up all the values and divide by the number of portions. "Average'' and ''mean'' refer to the same thing, and you may be asked to find either one. Their synonymousness doesn't extend outside the world of algebra though. Your mother won't care if you tell her a bully at school was being "average" to you.

Sample Problem

Louisa got 5 cookies, Danielle got 8 cookies, and Marcus got 2 cookies. What is the average number of cookies each person got?

The average number of cookies is the number each person would get if we divided the cookies fairly. We add up all the different numbers of cookies:

5 + 8 + 2 = 15

Then we divide by the number of portions, which is 3, since there are 3 people:

15 ÷ 3 = 5

The average number of cookies received by each person is 5. Again, we'd better take care of this situation with haste. Mary from our earlier problem caught wind of the cookie surplus and is on her way over.

Another phrase you might hear is the phrase "on average." This means roughly the same thing as "find the average of..." In the example above we could say that, on average, each person got 5 cookies. It's just another way of saying "5 is the average of the number of cookies each person got."

Word problems involving averages can do some interesting things. They can't bend their legs behind their heads...nothing that interesting, but still. Interesting.

Sample Problem

Linda bought four cookies that cost $0.25 each and two cookies that cost $0.50 each. On average, how much did she spend per cookie?

We want to make a list of all the cookie prices, and find the average of the numbers in that list. Since Linda bought four cookies for $0.25 each, the number $0.25 will show up four times in the list. Similarly, the number $0.50 will show up twice. The prices of the cookies were:

0.25, 0.25, 0.25, 0.25, 0.50, 0.50

If we add these numbers and divide by 6 (the total number of cookies purchased), we get


or about $0.33 per cookie. Maybe next time Linda will consider buying cookies by the box. Preferably in bulk from Costco.

Example 1

Juanita has $10, Stefan has $15, and Jorge has $11. If we redistribute the money so that each person gets the same amount, how much money will each person receive?

Example 2

The scores students received on the test were

60, 75, 84, 95, 96.

What was the mean of their scores?

Exercise 1

Find the mean for the following set of data: 10, 3, 5, 6. 

Exercise 2

Find the mean for the following set of data: 12, 14, 15, 15, 15, 17.

Exercise 3

Find the mean for the following set of data: 54, 66, 78, 80, 82, 84, 84, 90, 93.

Exercise 4

Find the mean for the following set of data: 0, 1, 1, 0.

Exercise 5

Find the mean for the following set of data: .

Exercise 6

Jen has $10, Connie has $15, and Elsa has $2. What is the average amount of money each person has?

Exercise 7

Xavier has 5 cats, Joann has 10 cats, Randy has 12 cats, and Betty has 1 cat. On average, how many cats does each person have? 

Exercise 8

Corinne bought three boxes of tea for $7 each and five boxes of tea for $5 each. What was the average price of one box of tea? 

Exercise 9

Tanya plans to paint her house using two different kinds of paint. She needs 10 gallons of paint costing $2 per gallon, and 5 gallons of paint costing $5 per gallon. How much will she spend, on average, per gallon of paint?

Exercise 10

Parker has four different piggy banks: one each for pennies, nickels, dimes, and quarters. If there are twenty coins in each of his four piggy banks, what is the average amount of money contained in each piggy bank? 

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