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## 6th grade (Eureka Math/EngageNY)

### Unit 6: Lesson 2

Topic B: Summarizing a distribution that is approximately symmetric using the mean and mean absolute deviation- Statistics intro: Mean, median, & mode
- Mean, median, & mode example
- Calculating the mean
- Calculating the mean
- Calculating the mean: data displays
- Missing value given the mean
- Mean as the balancing point
- Missing value given the mean
- Median & range puzzlers
- Median & range puzzlers
- Impact on median & mean: removing an outlier
- Impact on median & mean: increasing an outlier
- Effects of shifting, adding, & removing a data point
- Mean absolute deviation (MAD)
- Mean absolute deviation example
- Mean absolute deviation (MAD)

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# Mean absolute deviation (MAD)

Mean absolute deviation (MAD) of a data set is the average distance between each data value and the mean. Mean absolute deviation is a way to describe variation in a data set. Mean absolute deviation helps us get a sense of how "spread out" the values in a data set are.

## Want to join the conversation?

- I still don't get how to find the MAD, can anyone pls help me(26 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.(96 votes)

- Wait, so we have to find the mean and then the absolute value right?(23 votes)
- Is there an easier way to calculate MAD? So much writing!(7 votes)
- Well, we can solve the writing problem by doing mental math, but we can't solve the easy way part.(1 vote)

- but how do you do these things and not get them wrong:{(1 vote)
- There are a lot of calculations and it's easy to get one wrong.

Be patient, take your time, and never assume you got it right on your first try.(18 votes)

- Is this different from standard deviation? I find that I get different answers from both, but they seem like the same concept. Can you please explain the difference and purpose of each?(8 votes)
- There was a distinction made between a sample variance/standard deviation and a population variance/standard deviation. The population variance is calculated by taking the sum of the squared deviations from each data point to the population mean, and then dividing by the number of data points in the population. On the other hand, the sample variance goes through the same process as above, except it's with respect to the sample mean, and you should also divide by one less than the number of data points in your sample, to correct the bias (Bessel's Correction). I'm wondering if a similar notion exists for the Mean Absolute Deviation (MAD)? In other words, whether it's a sample or population we're dealing with, is there any significant difference in the way that the MAD is calculated for either of them?(6 votes)
- what how to do it(2 votes)
- Mean Absolute Deviation (MAD) is a way to measure how spread out a set of data is.

The first step is to calculate the mean (average) of the set of data. If we have the set of data [-1,2,3,7,9,12,17], the mean would be [-1+2+3+7+9+12+17] / 7, so 7 is the mean.

The second step is to measure how far each point of data is from the mean, so [7-(-1)] + [7-2] + [7-3] + [7-7] + [7-9] + [7-12] + [7-17]. If we have a number that is bigger than the mean, like 9,12, and 17 in this case, we take the absolute value, so usually 7-9 = -2 (but with absolute value) 2.

The third step is to add all the values above, and divide them by

the number of data points, so 7. Eventually, we get : 34/7, or [4 6/7]

Hope this helps.(8 votes)

- Ummm. Do the numbers in a data set need to be ordered? Thanks...................(3 votes)
- It is not necessary in this case, the statistic that is easiest to do by ordering is the median (middle number). To find the mean, order does not matter because addition is commutative, and order may help to find the mode (number with the most results), but it does not matter for MAD either. On the other hand, it never hurts to order the data for convenience and possibly easier calculations.(3 votes)

- I still don't under stand how you come up with the two different data sets do I split my data in half?(0 votes)
- Sal uses two completely different data sets to show how MAD describes the variability of a single data set.

2,2,4,4 - number of donuts I ate each of the last four days

1,1,6,4 - number of times I scored in my last four soccer games

Both data sets have a mean of 3. On average, I eat 3 donuts a day, and score 3 goals per game [I wish].

The MAD of the donut data is 1, showing that I am pretty consistent on eating donuts. The average day is 2 to 4 donuts (1 donut more or less than 3).

However, The MAD of the soccer data is 2, showing that there is more variability in my goal scoring. An average game is 1 to 5 goals (2 goals more or less than 3).(12 votes)

- what does the ratio difference in means/mean absolute deviation tell you about how much visual overlap there is between two distributions with similar variations?(3 votes)

## Video transcript

- [Voiceover] Let's say that I've got two different data sets. The first data set, I have two, another two, a four, and a four. And then, in the other
data set, I have a one. We'll do this on the
right side of the screen. A one, a one, a six, and a four. Now, the first thing I
wanna think about is, "Well, how do I ... "Is there a number that can give me "a measure of center of
each of these data sets?" And one of the ways that
we know how to do that is by finding the mean. So let's figure out the mean
of each of these data sets. This first data set, the mean ... Well, we just need to sum
up all of the numbers. That's gonna be two plus two plus four plus four. And then we're gonna divide by the number of numbers that we have. So we have one, two, three, four numbers. That's that four right over there. And this is going to be,
two plus two is four, plus four is eight, plus four is 12. This is gonna be 12 over four, which is equal to three. Actually, let's see if
we can visualize this a little bit on a number line. Actually I'll do kind of a ... I'll do a little bit of a dot plot here so we can see all of the values. If this is zero, one, two, three, four, and five. We have two twos. Why don't I just do ... So for each of these twos ... Actually, I'll just do it in yellow. So I have one two, then
I have another two. I'm just gonna do a dot plot here. Then I have two fours. So, one four and another
four, right over there. And we calculated that the mean is three. The mean is three. A measure of central
tendency, it is three. So I'll just put three right over here. I'll just mark it with that dotted line. That's where the mean is. All right. Well, we've
visualized that a little bit. That does look like it's the center. It's a pretty ... It makes sense. So now let's look at this
other data set right over here. The mean, the mean over here is going to be equal to one plus one plus six plus four, all of that over, we still
have four data points. And this is two plus six is eight, plus four is 12, 12 divided by four ... This is also three. So this also has the same mean. We have different numbers, but we have the same mean. But there's something about this data set that feels a little bit
different about this. And let's visualize it, to see if we can see a difference. Let's see if we can visualize it. I have to go all the way up to six. Let's say this is zero, one, two, three, four, five, six, and I'll go one more, seven. So we have a one. We have a one, we have another one. We have a six. And then we have a four. And we calculated that the mean is three. So we calculated that the mean is three. So the mean is three. When we measure it by the mean, the central point, or
measure of that central point which we use as the mean, well, it looks the same, but
the data sets look different. How do they look different? Well, we've talked about notions of variability or variation. And it looks like this data
set is more spread out. It looks like the data
points are on average further away from the mean than these data points are. That's an interesting question that we ask ourselves in statistics. We just don't want a measure
of center, like the mean. We might also want a
measure of variability. And one of the more straightforward ways to think about variability is, well, on average, how far
are each of the data points from the mean? That might sound a little complicated, but we're gonna figure out what that means in a second, (chortles) not
to overuse the word "mean." So we wanna figure out, on average, how far each of these
data points from the mean. And what we're about to calculate, this is called Mean Absolute Deviation. Absolute Deviation. Mean Absolute Deviation, or if you just use the acronym, MAD, mad, for Mean Absolute Deviation. And all we're talking about, we're gonna figure out how
much do each of these points, their distance, so absolute deviation. How much do the deviate from the mean, but the absolute of it? So each of these points at two, they are one away from the mean. Doesn't matter if they're less or more. They're one away from the mean. And then we find the mean
of all of the deviations. So what does that mean? (chuckles) I'm using the word "mean," using it a little bit too much. So let's figure out the
Mean Absolute Deviation of this first data set. We've been able to figure
out what the mean is. The mean is three. So we take each of the data points and we figure out, what's its absolute
deviation from the mean? So we take the first two. So we say, two minus the mean. Two minus the mean, and we
take the absolute value. So that's its absolute deviation. Then we have another two, so we find that absolute
deviation from three. Remember, if we're just
taking two minus three, taking the absolute value, that's just saying its absolute deviation. How far is it from three? It's fairly easy to
calculate in this case. Then we have a four and another four. Let me write that. Then we have the absolute deviation of four from three, from the mean. Then plus, we have another four. We have this other four right up here. Four minus three. We take the absolute value, because once again,
it's absolute deviation. And then we divide it, and then we divide it by the
number of data points we have. So what is this going to be? Two minus three is negative one, but we take the absolute value. It's just going to be one. Two minus three is negative one. We take the absolute value. It's just gonna be one. And you see that here visually. This point is just one away. It's just one away from three. This point is just one away from three. Four minus three is one. Absolute value of that is one. This point is just one away from three. Four minus three, absolute value. That's another one. So you see in this case, every data point was exactly one away from the mean. And we took the absolute value so that we don't have negative ones here. We just care how far it
is in absolute terms. So you have four data points. Each of their absolute
deviations is four away. So the mean of the absolute deviations are one plus one plus one plus one, which is four, over four. So it's equal to one. One way to think about it is saying, on average, the mean of the
distances of these points away from the actual mean is one. And that makes sense because all of these are exactly one away from the mean. Now, let's see how, what results we get for this
data set right over here. And I'll do it ... Let me actually get some space over here. At any point, if you get inspired, I encourage you to calculate the Mean Absolute Deviation on your own. So let's calculate it. The Mean Absolute Deviation here, I'll write MAD, is going to be equal to ... Well, let's figure out
the absolute deviation of each of these points from the mean. It's the absolute value
of one minus three, that's this first one, plus the absolute deviation, so one minus three, that's the second one, then plus the absolute
value of six minus three, that's the six, then we have the four, plus the absolute value
of four minus three. Then we have four points. So one minus three is negative two. Absolute value is two. And we see that here. This is two away from three. We just care about absolute deviation. We don't care if it's to
the left or to the right. Then we have another one
minus three is negative two. It's absolute value, so this is two. That's this. This is
two away from the mean. Then we have six minus three. Absolute value of that
is going to be three. And that's this right over here. We see this six is three to the right of the mean. We don't care whether it's
to the right or the left. And then four minus three. Four minus three is one,
absolute value is one. And we see that. It is one to the right of three. And so what do we have? We have two plus two is four, plus three is seven, plus one is eight, over four, which is equal to two. So the Mean Absolute Deviation ... Let me write it down. It fell off over here. Here, for this data set, the Mean Absolute
Deviation is equal to two, while for this data set, the Mean Absolute
Deviation is equal to one. And that makes sense. They have the exact same means. They both have a mean of three. But this one is more spread out. The one on the right is more spread out because, on average, each of these points are two away from three, while on average, each of these points are one away from three. The means of the absolute
deviations on this one is one. The means of the absolute
deviations on this one is two. So the green one is more
spread out from the mean.