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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. 5. 6. 7. 8. 4.  x = 3.5. Deviations from the Mean.

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standard deviation

Standard Deviation

Lecture 18

Sec. 5.3.4

Tue, Oct 4, 2005

deviations from the mean
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

5

6

7

8

4

x = 3.5

deviations from the mean1
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

5

6

7

8

4

x = 3.5

deviations from the mean2
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

5

6

7

8

4

x = 3.5

deviations from the mean3
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

5

6

7

8

4

x = 3.5

deviations from the mean4
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

5

6

7

8

4

x = 3.5

deviations from the mean5
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
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 deviations1
Sum of Squared Deviations
  • SSX = sum of squared deviations
  • 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
The Population Variance
  • Variance of the population – The average squared deviation for the population.
  • The population variance is denoted by 2.
the population standard deviation
The Population Standard Deviation
  • The population standard deviation is the square root of the population variance.
  • We will interpret this as being representative of deviations in the population (hence the name “standard”).
the sample variance
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.
  • This formula for s2 makes a better estimator of 2 than if we had divided by n.
example
Example
  • In the example, SSX = 29.
  • Therefore,

s2 = 29/3 = 9.667.

the sample standard deviation
The Sample Standard Deviation
  • The sample standard deviation is the square root of the sample variance.
  • We will interpret this as being representative of deviations in the sample.
example1
Example
  • In our example, we found that s2 = 9.667.
  • Therefore, s = 9.667 = 3.109.
example2
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
Alternate Formula for the Standard Deviation
  • An alternate way to compute SSXis to compute
  • Note that only the second term is divided by n.
  • Then, as before
example3
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
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
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 deviation1
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 deviation2
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 deviation5
Interpreting the Standard Deviation

s

s

x – s

x

x + s

interpreting the standard deviation6
Interpreting the Standard Deviation

A little closer than normal tox

but not unusual

x – s

x

x + s

interpreting the standard deviation7
Interpreting the Standard Deviation

Unusually close tox

x – s

x

x + s

interpreting the standard deviation8
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 deviation9
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!
  • Let’s Do It! 5.13, p. 329 – Increasing Spread.
  • Example 5.10, p. 329 – There Are Many Measures of Variability.
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