Transcript Lecture 18 - Standard Deviation

Standard Deviation
Lecture 18
Sec. 5.3.4
Mon, Feb 18, 2007
Variability
Our ability to estimate a parameter
accurately depends on the variability of the
population.
 What do we mean by variability in the
population?
 How do we measure it?

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).

1
2
3
4
5
6
7
8
9
10
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).

1
2
3
4
5
x = 6
6
7
8
9
10
Deviations from the Mean

Each unit of a sample or population
deviates from the mean by a certain
amount.
deviation = –5
1
2
3
4
5
x = 4
6
7
8
9
10
Deviations from the Mean

Each unit of a sample or population
deviates from the mean by a certain
amount.
dev = –2
1
2
3
4
5
x = 4
6
7
8
9
10
Deviations from the Mean

Each unit of a sample or population
deviates from the mean by a certain
amount.
dev = +1
1
2
3
4
5
x = 4
6
7
8
9
10
Deviations from the Mean

Each unit of a sample or population
deviates from the mean by a certain
amount.
dev = +2
1
2
3
4
5
x = 4
6
7
8
9
10
Deviations from the Mean

Each unit of a sample or population
deviates from the mean by a certain
amount.
deviation = +4
1
2
3
4
5
x = 4
6
7
8
9
10
Deviations from the Mean
How do we obtain one number that is
representative of the whole set of
individual deviations?
 Normally we use an average to summarize
a set of numbers.
 Why will the average not work in this
case?

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.

Sum of Squared Deviations

Procedure
 Find
the average
 Find the deviations from the average
 Square the deviations
 Add them up
Sum of Squared Deviations

SSX = sum of squared deviations
SSX   x  x 
For example, if the sample is {1, 4, 7, 8, 10},
then
2

SSX = (1 – 6)2 + (4 – 6)2 + (7 – 6)2 + (8 – 6)2 + (10 – 6)2
= (–5)2 + (–2)2 + (1)2 + (2)2 + (4)2
= 25 + 4 + 1 + 4 + 16
= 50.
The Population Variance
 Variance
of 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
 x  x 
2
N
The Sample Variance
 Variance
of a sample
 The sample variance is denoted by
s2.
 x  x 
SSX
2
s 

n 1
n 1
2
The Sample Variance
shows that if we divide by n –
1 instead of n, we get a better
estimator of 2.
 Therefore, we do it.
 Theory
Example
In the example, SSX = 50.
 Therefore,
s2 = 50/4 = 12.5.

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 = 12.5.
 Therefore, s = 12.5 = 3.536.
 How does that compare to the individual
deviations?

Alternate Formula for the
Standard Deviation

An alternate way to compute SSX is to
compute

x 

2
SSX  x

2
Then, as before
SSX
s 
n 1
2
n
Example



Let the sample be {1, 4, 7, 8, 10}.
Then  x = 30 and
 x2 = 1 + 16 + 49 + 64 + 100 = 230.
So
SSX = 230 – (30)2/5
= 230 – 180
= 50,
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
= 3.535533906.
 x = 3.16227766.
Interpreting the Standard
Deviation
The standard deviation is directly
comparable to actual deviations.
 How does 3.536 compare to -5, -2, +1, +2,
and +4?

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