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# Standard Deviation PowerPoint PPT Presentation

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.

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

• 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

### 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 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.

x

s

s

x

s

s

x – 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.