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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

- 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 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 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 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 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
- 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.

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

- 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

- 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.
- 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 SSXis to compute
- Note that only the second term is divided by n.
- Then, as before

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 fromx by much more than s are unusually far from the mean.
- Observations that deviate fromx by much less than s are unusually close to the mean.

Interpreting the Standard Deviation

A little closer than normal tox

but not unusual

x – s

x

x + s

Interpreting the Standard Deviation

A little farther than normal fromx

but not unusual

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.

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