Download Mean Absolute Deviation

MEAN ABSOLUTE
DEVIATION
Warm-Up Problems:
Find the mean, median, mode and range of the
following data set:
{9, 7, 7, 4, 5, 7}
Objective:
By the end of this lesson, you will be able to:


find the standard deviation of a data
set
find the mean absolute deviation of a
data set
In statistics, the measure of central tendency gives a
single value that represents the whole value.
But the central tendency cannot describe the
observation fully.
The measure of dispersion helps us to study the
variability of the items.
Two formulas which find the dispersion of data about
the mean are:
standard deviation – squares each
difference from the mean to
eliminate the negative differences.
mean absolute deviation – uses
absolute value of each difference
from the mean to eliminate the
negative differences.
Standard Deviation
Standard deviation is a number used to tell how
measurements for a group are spread out from the
average (mean), or expected value.
A low standard deviation means that most of the
numbers are very close to the average. A high
standard deviation means that the numbers are
spread out.
Standard deviation is used in a variety of fields and professions.
Often, the standard deviation is used as a measurement of how
accurate or reliable a set of data might be.
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

In a scientific study, the standard deviation of the data can indicate
whether the data was consistent and therefore can help scientists
consider how solid any scientific conclusion based on their data
might be.
In meteorology, the standard deviation of weather data can help an
observer understand how reliable or predictable a certain weather
forecast will be.
In predicting the movement of the stock market and in determining
how profitably or reliably an investment can be made to function. If
there is a large amount of disparity between a stock's value or profit
at different times, that could indicate that the stock is volatile; if
there is a low standard deviation, that could indicate that a stock is
solid and would make a safe investment.
Example of Standard Deviation
The owner of the Chez Tahoe restaurant is interested in how much
people spend at the restaurant. He examines 10 randomly selected
receipts for parties of four and writes down the following data.
44, 50, 38, 96, 42, 47, 40, 39, 46, 50
He calculated the mean by adding and dividing by 10 to
get
x = 49.2
Continued…
Below is the table for getting the standard deviation:
x
x - 49.2
(x - 49.2 )2
44
-5.2
27.04
50
0.8
0.64
38
11.2
125.44
96
46.8
2190.24
42
-7.2
51.84
47
-2.2
4.84
40
-9.2
84.64
39
-10.2
104.04
46
-3.2
10.24
50
0.8
0.64
Total
2600.4
Continued…
We first find the variance:
2600.4 = 288.7
10 – 1
The standard deviation is the square root of 289 = 17.
Since the standard deviation can be thought of measuring how far the
data values lie from the mean, we take the mean and move one
standard deviation in either direction.
The mean for this example was about 49.2 and the standard deviation
was 17.
We have:
49.2 - 17 = 32.2
and
49.2 + 17 = 66.2
What this means is that most of the patrons probably spend between
$32.20 and $66.20.
Mean Absolute Deviation
Mean Absolute Deviation, referred to as MAD, is a
better measure of dispersion than the standard
deviation when there are outliers in the data.
An outlier is a data point which is far removed in
value from the others in the data set. It is an
unusually large or an unusually small value
compared to the others.
Mean Absolute Deviation
mean absolute deviation (MAD)

The mean absolute deviation (MAD) of a data set is
the average of the absolute values of all deviations
from the mean in that set.
The Mean Absolute Deviation is calculated in five
simple steps.
1) Determine the Mean: Add all numbers and
divide by the count
2) Determine deviation of each variable from the
Mean: Subtract the mean from each number
3) Make the deviation 'absolute' by squaring and
determining the roots (i.e. eliminate the negative
aspect)
4) Find the sum of the absolute values
5) Divide the sum by the number of
data items
Mean Absolute Deviation
For example, in the data set: {1, 2, 3, 4, 5},
the mean is 3.
The deviations from the mean are {-2, -1, 0, 1, 2}.
i.e. 1 – 3, 2 – 3, 3 – 3, etc.
The absolute deviations from the mean are
{|-2|, |-1|, |0|, |1|, |2|}.
The MAD is (2 + 1 + 0 + 1 + 2) / 5 = 1.2.
The MAD is a measure of, on average, how far the
values in a data set are from the mean.
Mean Absolute Deviation

What does a low mean deviation means?
A low standard deviation would mean that there is not
much variation from the mean value.
Find the mean absolute deviation
Test scores for 6 students were :
85, 92, 88, 80, 91 and 20.
1. Find the mean:
(85+92+88+80+91+20)/6=76
2. Find the deviation from the mean:
85-76=9 92-76=16 88-76=12
80-76=4 91-76=15 20-76=-56
3. Find the absolute value of each
deviation from the mean:
85  76  9 92  76  16 88  76  12
80  76  4 91  76  15 20  76  56
4. Find the sum of the absolute values:
9 + 16 + 12 + 4 + 15 + 56 = 112
5. Divide the sum by the number of
data items:
112/6 = 18.7
The mean absolute deviation is 18.7.
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