The Mean

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The Mean

• A Balancing Act

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A little bit about seesaws
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A seesaw will balance if equal weights are placed
on the ends.
10 lbs.
10 lbs.
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10 lbs.
10 lbs.
This is called the fulcrum.
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Suppose that we had unequal weights on the seesaw.
20 lbs.
10 lbs.
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We could balance the seesaw by moving the fulcrum
to the left.
20 lbs.
10 lbs.
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Voila!
20 lbs.
10 lbs.
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Balance can always be achieved by moving the
fulcrum to a new location.
30 lbs.
10 lbs.
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30 lbs.
10 lbs.
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The Greek mathematician Archimedes discovered
this relationship between the weight of an object
and its distance from the fulcrum.
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Give me a spot where I can stand, and I shall
move the earth.
Archimedes discovered the law of levers.
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Can you figure out where the fulcrum should be
placed so that the seesaw is balanced?
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Lets make it more challenging. Where should the
fulcrum be placed to achieve balance?
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Try this one.
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The mean, or average, of a set of data is kind of
like a fulcrum for a seesaw.
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Consider the following
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Here are the heights of 5 people.
3 ft 4ft 5 ft 5
ft 6 ft
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Lets make a simple graph from this information.
Each smiley face represents one person.
3 4 5 6
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If this were a seesaw, where would the fulcrum go
to achieve balance?
3 4 5 6
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The mean is 4.6
3 4 5 6
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Lets try another one.
3ft 3ft 6 ft 6
ft 6 ft
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Where would the fulcrum go to achieve balance?
3 4 5 6
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The mean is 4.8
3 4 5 6
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Lets try another one.
3ft 6ft 6 ft 6
ft 6 ft
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Where would the fulcrum go to achieve balance?
3 4 5 6
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The mean is 5.4
3 4 5 6
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What is the mean for this set of data?
6ft 6ft 6 ft 6
ft 6 ft
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The mean is 6.
5 6 7
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Consider the following
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Can the mean of a set of data ever be more than
the greatest value?
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Lets look at a specific example. Here are the
ages of 6 people.
4 10 12
12 14 15
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Heres a graph of this data.
4 5 6 7 8 9 10
11 12 13 14 15
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Is it possible for the mean to be greater than 15?
4 5 6 7 8 9 10
11 12 13 14 15
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This is like asking, Can the fulcrum be placed
to the right of 15 to achieve balance?
4 5 6 7 8 9 10
11 12 13 14 15
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This is like asking, Can the fulcrum be placed
to the right of 15 to achieve balance?
4 5 6 7 8 9 10
11 12 13 14 15
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Obviously the seesaw will not balance if the
fulcrum is placed to the right of 15.
4 5 6 7 8 9 10
11 12 13 14 15
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The mean is 11 1/6
4 5 6 7 8 9 10
11 12 13 14 15
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The mean must be less than the greatest value in
the set.
4 5 6 7 8 9 10
11 12 13 14 15
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Likewise, the mean must greater than the smallest
value in the set.
4 5 6 7 8 9 10
11 12 13 14 15
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What is the mean age of the following 2 people?
10 years old 30 years old
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Here is a graph of this data.
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
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Because there are only two numbers, the mean will
fall directly in between 10 and 30.
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
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Because there are only two numbers, the mean will
fall directly in between 10 and 30.
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
The mean is 20.
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What is the mean age of the following 3 people?
10 years old 20 years old 30 years old
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Here is a graph of this data.
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
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Can you explain why the mean of these 3 numbers
is 20?
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
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The mean of 10 and 30 is 20.
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
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20 is the third number in this set of data.
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
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Because the 20 is the mean of 10 and 30,
including 20 in our set of data will not change
the mean.
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30
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What does the mean mean?
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The mean means equal distribution.
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Everyone gets the same amount.
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Suppose that there are three people with
different amounts of money.
1 2 6
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To find the mean, everyone should have the same
amount of money.
1 2 6
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1 2 6
59
1 3 5
60
2 3 4
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3 3 3
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The mean is 3.
3 3 3
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To find the mean, we had to divide all of the
money (9) into 3 equal groups.
1 2 6
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The total amount of money is 9.
1 2 6
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9 divided equally among 3 people means that each
person gets 3.
3 3 3
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Lets find the mean for this set of data.
1 2 6 8 8
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The total amount of money is 25
1 2 6 8 8 25
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There are 5 people.
1 2 6 8 8 25
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Therefore, the mean is 5.
5 5 5 5 5 25
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The mean is not always a whole number.
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Because you have to divide to find the mean, the
mean is usually a mixed number.
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Consider the following
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Suppose that Larry has 3 pets, and Moe has 2 pets.
Larry Moe
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To ensure that Larry and Moe have the same number
of pets, we must divide the extra dog in half.
Larry Moe
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To ensure that Larry and Moe have the same number
of pets, we must divide the extra dog in half.
Larry Moe
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We are not being cruel to animals.
Larry Moe
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Remember, the mean is just a number.
Larry Moe
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The mean of 2 and 3 is 2 ½ .
2 2 ½ 3
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2 3 55 ? 2 2 ½
2 2 ½ 3
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Practice Time!
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1) Why is 52 an unreasonable mean for this data?
Son
Dad
12 years old
40 years old
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1) Why is 52 an unreasonable mean for this data?
Son
Dad
12 years old
40 years old
52 is greater than 40 (max value).
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2) What is the mean age for this data?
Son
Dad
12 years old
40 years old
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2) What is the mean age for this data?
Son
Dad
12 years old
40 years old
(12 40) 2 26 years old
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3) If we include moms age in this set of data,
will the mean increase? Decrease? Stay the same?
Son
Dad
Mom
12 years old
40 years old
41 years old
Mean of Son Dad 26 years old
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3) If we include moms age in this set of data,
will the mean increase? Decrease? Stay the same?
Son
Dad
Mom
12 years old
40 years old
41 years old
Mean of Son Dad 26 years old
Mean of Son Dad Mom should be greater than 26
because Moms age is greater than 26.
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4) Find the mean age of Son, Dad, and Mom.
Son
Dad
Mom
12 years old
40 years old
41 years old
Mean of Son Dad 26 years old
Mean of Son Dad Mom
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4) Find the mean age of Son, Dad, and Mom.
Son
Dad
Mom
12 years old
40 years old
41 years old
Mean of Son Dad 26 years old
Mean of Son Dad Mom (12 40 41) 3 31
years old
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5) You have been hired by the MM/Mars company to
make sure that the machines that put the candy
into the bags are working properly. You sample
the contents of 5 bags of MMs. The machines are
supposed to put an average of 24.6 MMs per bag.
Based on these samples, are the machine working
properly?
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5) You have been hired by the MM/Mars company to
make sure that the machines that put the candy
into the bags are working properly. You sample
the contents of 5 bags of MMs. The machines are
supposed to put an average of 24.6 MMs per bag.
Based on these samples, are the machine working
properly?
24 21 28 27 23 123
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5) You have been hired by the MM/Mars company to
make sure that the machines that put the candy
into the bags are working properly. You sample
the contents of 5 bags of MMs. The machines are
supposed to put an average of 24.6 MMs per bag.
Based on these samples, are the machine working
properly?
24 21 28 27 23 123
123 5 24.6 Yes, the machines are working
properly.
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6) You have been hired by the General Mills
Company to make sure that their cereal boxes
contain a mean of 15.9 oz. of cereal. You
randomly sample 5 boxes of cereal to determine
whether their machines are putting the correct
amounts of cereal into the boxes. What is your
conclusion?
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6) You have been hired by the General Mills
Company to make sure that their cereal boxes
contain a mean of 15.9 oz. of cereal. You
randomly sample 5 boxes of cereal to determine
whether their machines are putting the correct
amounts of cereal into the boxes. What is your
conclusion?
16.2 15.8 15.5 16.3 15.8 79.6
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6) You have been hired by the General Mills
Company to make sure that their cereal boxes
contain a mean of 15.9 oz. of cereal. You
randomly sample 5 boxes of cereal to determine
whether their machines are putting the correct
amounts of cereal into the boxes. What is your
conclusion?
16.2 15.8 15.5 16.3 15.8 79.6
79.6 5 15.92 No, the machines are
slightly off.
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7) Mr. Dunlaps 3rd period class did an
experiment to find out how many licks it takes to
get to the center of a tootsie pop. These are
the results of 9 students.Find the mean number
of licks.
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7) Mr. Dunlaps 3rd period class did an
experiment to find out how many licks it takes to
get to the center of a tootsie pop. These are
the results of 9 students.Find the mean number
of licks.
58 70 65 92 84 79 81 96 68 693
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7) Mr. Dunlaps 3rd period class did an
experiment to find out how many licks it takes to
get to the center of a tootsie pop. These are
the results of 9 students.Find the mean number
of licks.
58 70 65 92 84 79 81 96 68 693
693 9 77 licks