Standard deviation
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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

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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 deviation3

Interpreting the Standard Deviation

x


Interpreting the standard deviation4

Interpreting the Standard Deviation

s

s

x


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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