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Lesson 7: MAD – Mean Absolute Deviaton

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Lesson 7:MAD – Mean Absolute Deviaton

Have rulers and pennies ready.

Part 1 has 3 examples…

are you ready?

Think of your ruler as a number line…

Jacob is super interested, for some reason, in how long it takes for students to get to school…

So he asks two kids how long it took them to get to school today.

One kid said it took him 1 minute, the other said it took him 11.

Lets measure that data…

6

- Where is the mean on the number line?

Right in between the points!

What is the mean of this data?

- Now try putting 2 MORE pennies on the 4 and the 8 of your ruler... Can you still balance the ruler in the same place?

Now try balancing some pennies to the 1 and the 11... Can you balance the ruler on the 6 with your fingers?

1 + 4 + 8 + 11

4

= 6

- What if you moved the penny on number 4 to the 6? What can you do to the penny on 8 to keep it balanced?

What is the mean of 1, 4, 8, and 11?

Move the one on the 8 to the 6 too!

- Now… is the mean still 6?

1 + 6 + 6 + 11

4

= 6

YES!!

- So what does this tell us about Mean? “what is the meaning of the ‘mean’?”

Move the one on the 8 to the 6 too!

- We call this distance from the mean the “Deviation from the mean”.

- Deviation is a measurement that tells us how far away a value is from the mean.

Lets go back to “considering” Jacob and his obsession with people getting to school… where the times were just 1 and 11…

- Numbers to the right are said to have a positive deviation while the left ones have a negative deviation.

- The 11 has a deviation of +5, and the 1 has a deviation of -5.

- We get this by subtracting the mean from the value: 1 – 6 = -5, and 11 – 6 = +5.

- Write this: The deviation is the value subtracted by the mean.

4 – 6 = -2

8 – 6 = +2

1 – 6 = -5

11 – 6 = +5

So what are all the deviation values of these pennies?

-5

So whenever I add the deviations together, my answer should ALWAYS be…

-2

= 0

+2

+

+5

If I were to add all of these up…. What number would I get?

Example 2 !!!

- What is their mean?

2 + 9 + 7

3

= 6

- What are their deviations?

2 – 6 = -4

7 – 6 = +1

9 – 6 = +3

Lets go back to Jacob… He found three more people!… one took 9 minutes to get to to school, one took 7 minutes, and the other took 2

Example 2 continued…

So if I hold my fingers at the mean on this ruler, will it be balanced?

Draw a number line with these points and show their Deviations…

-5

-3

-3

-1

+

+

+4

+2

= -6

= 0

- Is the mean of this data set 6?

NOPE! If it were the sum would be zero!

- Is the mean of this data set 4?

Yes!!

1 – 4 = -3

3 – 4 = -1

8 – 4 = +4

Use your Calculators!

Find the mean yearly temperature of both cities…

New York City: 63 degrees

San Francisco: 64 degrees

Mean Temperature for

New York City: 63 degrees

Mean Temperature for

San Francisco: 64 degrees

Do you think the Mean is a good way to compare these cities?

This is a result of Variability

For which of these would the mean be a better indicator?

Write down: San Francisco! Because the data is much closer to the mean and has less Variability

- Part 2 is next week… where we finally figure out what MAD is!
- In the Meantime, here’s your homework
- Worksheet
- Homework Packets Due
- BRING BACK YOUR MATH TEXTBOOK!

- … where we finally figure out what MAD is!

Data from “City G”

(the mean is 63)

1) Which of these cities has the greatest variability?… the least?

(don’t copy all the graphs)

2) Finish this chart for City G!

(what should the sum of your deviations be?)

Example 1

Here are the answers to your problems…

Data from “City G”

(the mean is 63)

Awesome! You’ve got a head start!

Go ahead and add “Lesson 7 Part 2” to your notes…

Here’s another way you can show the deviations…

ON A LINE PLOT!

(please don’t copy)

This is a great way to see the variation of the data. But we never did find a way to give a “measurement” to the variation last time…

Don’t copy… just listen…

What would be the total sum of just the negative deviations?

Is there another way to write the distance so that the distance is positive? (where have we measured distance before?)

-22

The Absolute Value! (it looks like this: |-22|)

What would be the total sum of just the positive deviations?

+22

Example 1 (now back to work)

Data from “City G”

(the mean is 63)

Write down the absolute value of the deviations…

Abs. Value of Dev.

|-10| =

10

6

3

3

1

Now, do you think the sum would still be zero?!…

1

1

1

NOPE!

…so what is it?

1

5

5

7

44!

44

Example 1 (now back to work)

Data from “City G”

(the mean is 63)

So now we have the distance of each value from the mean, which shows us the variability.

Abs. Value of Dev.

|-10| =

10

6

3

3

What would be a good way to describe ALL of the deviations and variability with just ONE number?

1

1

1

1

1

5

How about the mean of the deviations?

5

7

44

Example 1 (now back to work)

Data from “City G”

(the mean is 63)

So now we have the distance of each value from the mean, which shows us the variability.

Abs. Value of Dev.

|-10| =

10

6

3

3

What would be a good way to describe ALL of the deviations and variability with just ONE number?

1

1

1

1

1

5

How about the mean of the deviations?

5

7

44

44

__ 2 3.66 or 3

3

Example 1 (now back to work)

Data from “City G”

(the mean is 63)

How about the mean of the deviations?

Abs. Value of Dev.

|-10| =

10

6

3

4412

3

=

1

1

1

THIS…

Is what we call the Mean Absolute Deviation or the MAD

1

1

5

5

7

(Write that down NOW!)

44

44

Mean Absolute Deviation (MAD):

The average distance of all values from the mean

This is the BEST way to describe the variation of the data.

Write this all down and you’re DONE!

Mean Absolute Deviation (MAD):

The average distance of all values from the mean

- So here is your proceedure:
- Find the mean of the data
- Use the mean to find the deviations
- (subtracting the number by the mean)

- Find the absolute value of ALL of the deviations
- Find the mean one more time
- (of the deviations)

- Part 2 is next week… where we finally figure out what MAD is!
- Lets practice this in your homework together…
- “Lesson 9” Worksheet