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### Course: 6th grade (Illustrative Mathematics) > Unit 8

Lesson 10: Lesson 11: Deviation from the mean# Mean absolute deviation example

This lesson teaches how to calculate the mean and mean absolute deviation (MAD) using a bar graph. The graph shows the number of bubbles blown by four people. The mean is found by adding all the values and dividing by the number of data points. The MAD is calculated by finding the average of the absolute differences from the mean.

## Want to join the conversation?

- Is there a way for the MAD to be negative other than if the data values are negative?(31 votes)
- Actually, regardless of whether data values are zero, positive, or negative, the MAD can
**never**be negative. This is because the MAD is calculated by finding**absolute values**of the deviations (or differences) from the mean, and then taking the average (or mean) of these absolute values. Note that the absolute value of a quantity is never negative.(55 votes)

- I'm still confused after the step after calculating the mean. Can someone help me?(3 votes)
- After we calculate the mean, we need to subtract it from every data point and take the absolute value of each result. Adding all that together and dividing by the number of values you have will give you the MAD. Here's an example:

Let's say you have set 1, 2, 3, 4, 5 with a mean of 3. To solve for MAD, you would do the following:

|1 - 3| + |2 - 3| + |3 - 3| + |4 - 3| + |5 - 3| / 5

= |-2| + |-1| + |0| + |1| + |2| / 5

Taking the absolute value eliminates all negative signs.

= 2 + 1 + 0 + 1 + 2 / 5

= 6 / 5

= 1.2

Hope this helps!(15 votes)

- I understand this video but when i try to do it on the acual questions i dont understand because when i do the absoulute deviantion i divid and then i get the same number in the begining(4 votes)
- So you found the mean of the numbers, divided the mean by how many numbers are in the data set then subtracted the answer from the division problem. Maybe you did the math wrong(3 votes)

- at2:03, what does Sal mean by deviate?(3 votes)
- Just in case you were wondering, deviate also has a more general application in everyday language, meaning how far you are from the 'original' point. For example, if you're walking on a path in a nature reserve, and you see something far off to your right and start walking off the path and into the bushes, you could say you 'deviated from the path'. Or if you have to do an unprepared speech about horses and start off talking about horses, but end up doing most of your speech about how high kangaroos can jump, then you have 'deviated from the topic'. So basically, it's kinda like if you 'stray' from the topic (or path) or other things. :)(7 votes)

- If all the numbers equal the mean will the MAD be 0?(5 votes)
- if all the numbers are equal to mean then there would be no deviation at all and hence mean absolute deviation would be zero(7 votes)

- All right,

I got a problem that I'm confused on. I will try to solve it right now, narrating all the steps. (If I do something wrong or forget a step, please correct me!)

We have a number line. One dot on 1, one dot on 4, one dot on 5, one dot on 7, and one dot on 8. We know (because the problem tells us) that the mean for the data on the line plot is 5.

The question asks:

What is the absolute deviation for the data point at 7?

When you find the absolute deviation you find the mean of a data set. 1+4+5+7+8=25. 25 divided by 5 is 5.

That is the mean. Then, we get multiple number out of it, which is the step I don't really get.

(I was trying to solve a problem with 47, 45, 44, 41, and 48. When you add them up, you get 225, and divide it by 5. You get 45. And then, all of a sudden, we have the numbers 2, 0, 1, 4, and 3. Where did we get those numbers? How did we go from 45 to 2, 0, 1, 4, and 3?!)

I'm so confused!

Please explain both problems if you can!(2 votes)- Hi!

In the first problem, they are asking for the absolute difference between 7 and the mean. Because the mean is 5...

|7 - 5| =**2**

The || (absolute value) guarantees that any operation performed inside it will become positive or stay positive.

To solve for the MAD in the second problem, we need to find the mean of the absolute values of (each data point minus the mean of the original data set). Here is the equation:

For a data set with 47, 45, 44, 41, 48 and a mean of 45,

|47-45| + |45-45| + |44-45| + |41-45| + |48-45| / 5*5 is the amount of numbers in the "new data set" that makes up the numerator*

2 + 0 + |-1| + |-4| + 3 / 5

2 + 0 + 1 + 4 + 3 / 5

10 / 5 =**2**

Hope this helps!😄(7 votes)

- find the MAD by

1. finding the mean(average) of the set of numbers

2. find the distance of all the numbers from the mean.

3. Find the mean of those numbers.(5 votes) - Can you have a MAD less than 1?(4 votes)
- Don't know why I'm answering a 4 year old comment, but yes, because decimals and fractions can be positive and less than one(2 votes)

- How to solve mean absolute deviation

Take each number in the data set, subtract the mean, and take the absolute value. Then take the sum of the absolute values. Now compute the mean absolute deviation by dividing the sum above by the total number of values in the data set. Finally, round to the nearest tenth.(4 votes) - Which MAD shows the most consistency(4 votes)
- it depends on the set of numbers used and the typ of number you want.(1 vote)

## Video transcript

- This bar graph here tells us bubbles blown by each gum-chewer. We have four different gum-chewers, and they tell us how many
bubbles each of them blew. What I wanna do is, I
wanna figure out first the mean of the number of bubbles blown, and then also figure out
how dispersed is the data, how much do these vary from the mean. I'm gonna do that by calculating the mean absolute deviation. Pause this video now. Try to calculate the mean of
the number of bubbles blown. And then, after you do that, see if you can calculate the mean absolute deviation. Step one, let's figure out the mean. The mean is just going to be the sum of the number of bubbles blown divided by the number of datapoints. Manueala blew four bubbles. She blew four bubbles. Sophia blew five bubbles. Jada blew six bubbles. Tara blew one bubble. We have one, two, three, four datapoints. So let's divided by four. And so, this is going to be
equal to four plus five is nine, plus six is 15, plus one is 16. So it's equal to 16 over four, which is 16 divided by
four is equal to four. The mean number of bubbles blown is four. Lemme actually do this with
a bold line right over here. This is the mean number of bubbles blown. Now what I wanna do is I wanna figure out the mean absolute deviation. Mean. MAD: Mean Absolute Deviation. What we wanna do is we wanna
take the mean of how much do each of these datapoints
deviate from the mean. I know I just used the word
mean twice in a sentence, so it might be a little confusing, but as we work through it, hopefully, it'll make a little bit of sense. How much does Manueala's, the
number of bubbles she blew, how much does that deviate from the mean? Well, Manueala actually blew four bubbles, and four is the mean. So her deviation, her absolute deviation
from the mean is zero. Is zero. Actually, lemme just write this over here. Absolute deviation, that's AD, absolute deviation from the mean. Manueala didn't deviate
at all from the mean. Now let's think about Sophia. Sophia deviates by one from the mean. We see that right there, she's one above. Now, we would say one whether
it's one above or below, 'cause we're saying absolute deviation. Sophia deviates by one. Her absolute deviation is one. And then, we have Jada. How much does she deviate from the mean? We see it right over here. She deviates by two. She is two more than the mean. And then, how much does
Tara deviate from the mean? She is at one, so that
is three below the mean. That is three below the mean. Once again, this is two, this is three. She deviates. Her absolute deviation is three. And then we wanna take the
mean of the absolute deviation. That's the M in MAD, in
Mean Absolute Deviation. This is Manueala's absolute deviation, Sophia's absolute deviation,
Jada's absolute deviation, Tara's absolute deviation. We want the mean of those, so we divide by the number of datapoints, and we get zero plus one,
plus two, plus three, is six over four. Six over four, which is
the same thing as 1 1/2. Or, lemme just write it
in all the different ways. We could write it as three
halves, or 1 1/2, or 1.5. Which gives us a measure of how much do these datapoints vary
from the mean of four. I know what some of you are thinking. "Wait, I thought there was a formula "associated with the
mean absolute deviation. "It seems really complex. "It has all of these absolute-value signs "and whatever else." That's all we did. When we write all those
absolute-value signs, that's just a fancy way of
looking at each datapoint, and thinking about how much
does it deviate from the mean, whether it's above or below. That's what the absolute value does. It doesn't matter, if it's
three below, we just say three. If it's two above, we just say two. We don't put a positive or negative on. Just so you're comfortable seeing how this is the exact same thing you
would've done with the formula, let's do it that way, as well. So the mean absolute deviation
is going to be equal to. Well, we'll start with Manueala. How many bubbles did she blow? She blew four. From that you subtract the mean of four, take the absolute value. That's her absolute deviation. Of course, this does evaluate
to this zero, to zero here. Then you take the absolute value. Sophia blew five bubbles,
and the mean is four. Then you do that for Jada. Jada blew six bubbles; the mean is four. And then you do it for Tara. Tara blew one bubble,
and the mean is four. Then you divide it by the
number of datapoints you have. Lemme make it very clear. This right over here,
this four, is the mean. This four is the mean. You're taking each of the datapoints, and you're seeing how far
it is away from the mean. You're taking the absolute
value 'cause you just wanna figure out the absolute distance. Now you see, or maybe you see. Four minus four, this is. Different color. Four minus four, that is a zero. That is that zero right over there. Five minus four, absolute value of that? That's going to be. Lemme do this in a new color. This is just going to be one. This thing is the same
thing as that over there. We were able to see
that just by inspecting this graph, or this chart. And then, six minus four,
absolute value of that, that's just going to be two. That two is that two right over here, which is the same thing as
this two right over there. And then, finally, our one
minus four, this negative three, but the absolute value of
that is just positive three, which is this positive
three right over there, which is this distance right over here. You divide it by four, you get 1.5 again. Hopefully you found this mean
absolute deviation example as interesting as I did.