Download Standard Deviation

Objectives
The student will be able to:
•
describe the variation of the
data.
•
find the mean absolute deviation
of a data set.
Two formulas which find the dispersion of
data about the mean:
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.
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.
Outlier
Test scores for 6 students were :
85, 92, 88, 80, 91 and 20.
The score of 20 would be an outlier.
The standard deviation is greatly
changed when the outlier is included
with the data.
The mean absolute deviation would
be a better choice for measuring the
dispersion of this data.
Mean Absolute Deviation
1. Find the mean of the data.
2. Subtract the mean from each value –
the result is called the deviation from
the mean.
3. Take the absolute value of each
deviation from the mean.
4. Find the sum of the absolute values.
5. Divide the total by the number of items.
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
Find the mean absolute
deviation
3. Find the absolute value of each
deviation from the mean
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.
Standard Deviation
Standard Deviation shows the
variation in data. If the data is close
together, the standard deviation will
be small. If the data is spread out, the
standard deviation will be large.
Standard Deviation is often denoted
by the lowercase Greek letter sigma,
.
The bell curve which represents a
normal distribution of data shows
what standard deviation represents.
One standard deviation away from the mean (  ) in
either direction on the horizontal axis accounts for
around 68 percent of the data. Two standard
deviations away from the mean accounts for roughly
95 percent of the data with three standard deviations
representing about 99 percent of the data.
Analyzing the data:
The standard deviation of quiz scores for
Class A is about 1.2. Describe the quiz
scores that are within one standard
deviation of the mean.
Quiz Scores: 9, 8, 6, 7, 8, 9, 9, 10, 7, 10,
8, 8
Answer Now
Analyzing the data:
Step 1: Find the mean
Mean = 8.25
Step 2: Find the range of values that are
within one standard deviation of the mean.
8.25 - 1.2 = 7.05
8.25 + 1.2 = 9.45
Quiz scores that are between 7.05 and 9.
45 points are within one standard deviation
of the mean.
Summary:
As we have seen, standard deviation
measures the dispersion of data.
The greater the value of the
standard deviation, the further the
data tend to be dispersed from the
mean.