Transcript Lecture 18 - Standard Deviation

Standard Deviation
Lecture 18
Sec. 5.3.4
Tue, Oct 4, 2005
Deviations from the Mean


Each unit of a sample or population deviates
from the mean by a certain amount.
Define the deviation of x to be (x –x).
0
1
2
3
4
x = 3.5
5
6
7
8
Deviations from the Mean

Each unit of a sample or population deviates
from the mean by a certain amount.
deviation = -3.5
0
1
2
3
4
x = 3.5
5
6
7
8
Deviations from the Mean

Each unit of a sample or population deviates
from the mean by a certain amount.
dev = -1.5
0
1
2
3
4
x = 3.5
5
6
7
8
Deviations from the Mean

Each unit of a sample or population deviates
from the mean by a certain amount.
dev = +1.5
0
1
2
3
4
x = 3.5
5
6
7
8
Deviations from the Mean

Each unit of a sample or population deviates
from the mean by a certain amount.
deviation = +3.5
0
1
2
3
4
x = 3.5
5
6
7
8
Deviations from the Mean



How do we obtain one number that is
representative of the set of individual
deviations?
If we add them up to get the average, the
positive deviations will cancel with the negative
deviations, leaving a total of 0.
That’s no good.
Sum of Squared Deviations



We will square them all first. That way, there
will be no canceling.
So we compute the sum of the squared
deviations, called SSX.
Procedure
Find the deviations
 Square them all
 Add them up

Sum of Squared Deviations

SSX = sum of squared deviations
SSX  x  x 
2

For example, if the sample is {0, 2, 5, 7}, then
SSX = (0 – 3.5)2 + (2 – 3.5)2 + (5 – 3.5)2 + (7 – 3.5)2
= (-3.5)2 + (-1.5)2 + (1.5)2 + (3.5)2
= 12.25 + 2.25 + 2.25 + 12.25
= 29.
The Population Variance


Variance of the population – The average
squared deviation for the population.
The population variance is denoted by 2.
 x   
SSX
2
 

N
N
2
The Population Standard Deviation

The population standard deviation is the square
root of the population variance.


SSX

N
2


x

x

N
We will interpret this as being representative of
deviations in the population (hence the name
“standard”).
The Sample Variance


Variance of a sample – The average squared
deviation for the sample, except that we divide by
n – 1 instead of n.
The sample variance is denoted by s2.
 x  x 
SSX
2
s 

n 1
n 1
2

This formula for s2 makes a better estimator of
2 than if we had divided by n.
Example


In the example, SSX = 29.
Therefore,
s2 = 29/3 = 9.667.
The Sample Standard Deviation

The sample standard deviation is the square root
of the sample variance.
s

SSX

n 1
 x  x 
2
n 1
We will interpret this as being representative of
deviations in the sample.
Example


In our example, we found that s2 = 9.667.
Therefore, s = 9.667 = 3.109.
Example

Use Excel to compute the mean and standard
deviation of the sample {0, 2, 5, 7}.
Do it once using basic operations.
 Do it again using special functions.


Then compute the mean and standard deviation
for the on-time arrival data.

OnTimeArrivals.xls.
Alternate Formula for the
Standard Deviation

An alternate way to compute SSX is to compute

x 

2
SSX  x


2
n
Note that only the second term is divided by n.
Then, as before
SSX
s 
n 1
2
Example



Let the sample be {0, 2, 5, 7}.
Then  x = 14 and
 x2 = 0 + 4 + 25 + 49 = 78.
So
SSX = 78 – (14)2/4
= 78 – 49
= 29,
as before.
TI-83 – Standard Deviations


Follow the procedure for computing the mean.
The display shows Sx and x.
Sx is the sample standard deviation.
 x is the population standard deviation.


Using the data of the previous example, we have
Sx = 3.109126351.
 x = 2.692582404.

Interpreting the Standard
Deviation



Both the standard deviation and the variance are
measures of variation in a sample or population.
The standard deviation is measured in the same
units as the measurements in the sample.
Therefore, the standard deviation is directly
comparable to actual deviations.
Interpreting the Standard
Deviation


The variance is not comparable to deviations.
The most basic interpretation of the standard
deviation is that it is roughly the average
deviation.
Interpreting the Standard
Deviation


Observations that deviate fromx by much
more than s are unusually far from the mean.
Observations that deviate fromx by much less
than s are unusually close to the mean.
Interpreting the Standard
Deviation
x
Interpreting the Standard
Deviation
s
s
x
Interpreting the Standard
Deviation
s
x – s
s
x
x + s
Interpreting the Standard
Deviation
A little closer than normal tox
but not unusual
x – s
x
x + s
Interpreting the Standard
Deviation
Unusually close tox
x – s
x
x + s
Interpreting the Standard
Deviation
A little farther than normal fromx
but not unusual
x – 2s
x – s
x
x + s
x + 2s
Interpreting the Standard
Deviation
Unusually far fromx
x – 2s
x – s
x
x + s
x + 2s
Let’s Do It!


Let’s Do It! 5.13, p. 329 – Increasing Spread.
Example 5.10, p. 329 – There Are Many
Measures of Variability.