Chapter 3

3.3

Measures of Variation

3.3 Measures of Variation Bank waiting times In the first bank, the manager carefully controls waiting

times by changing the number of tellers as needed. In the second bank, customers all wait in a single line that feeds to the available tellers. In the third bank, customers wait in separate lines for each of the different tellers.

Bank 1: Variable waiting times

6 6 6

Bank 2: Single Waiting line 4 7 7

Bank 3: Multiple Waiting Lines

1 3 14

Bank Waiting Times Observe the Variation from the mean

0

5

10

15

1 2 3

0

5

10

15

1 2 30

5

10

15

1 2 3

Bank 1: NO variation from the mean

Bank 2: Small variation from the mean

Bank 3: Large variation from the mean

3.3 Measures of Variation

Range = (highest value) – (lowest value) Standard deviation of a sample

See the formulas on pg. 101 Denoted as “s” S = 0 means that all of the data are the same number

The standard deviation of a sample gives us the

measure of how much the data values vary from the mean

We will NOT use the formulas to find the standard deviation. Use your calculator!

3.3 Calculating Standard Deviation in TI 30XII calculator

1. Press 2nd DATA and arrow over to CLRDATA, press ENTER

2. Press 2nd DATA and arrow over to 1VARSTATS and press ENTER

3. Press DATA, and enter the 1st value, then press the down arrow key

4. Enter the frequency of that number, press down arrow key

5. Enter 2nd value in the data set, etc.6. When finished entering data, press

STATVAR key and arrow over to the Sx

symbol.7. This is the standard deviation of a sample.

3.3 Standard deviation of a population

We won’t be calculating this very often, because we don’t often have an entire population’s data. Most often we have sample data

When we know the standard deviation of the population, we call it σ (small letter sigma)

On the TI=30XIIS calculator, the symbol for the population standard deviation is σx. This will not be our answer in our class.

Formula: N

x 2

The mean of the population

The number of the total population

3.3 Measures of Variation

Variation describes the amount that values vary among themselves

The terms dispersion and spread are sometimes used instead of variance

Variance Variance = (standard deviation)2

Sample variance is denoted as s2

Population variance is denoted as σ2

Because variation also squares the units, it is a very abstract concept. What is a square minute? For that reason, our test will focus on standard deviation.

Find the standard deviation

Use your calculator to find the standard deviation of the three different bank waiting times: 1 minute, 3 minutes, and 14 minutes.

You should get Sx = 7 minutes

1. Press 2nd DATA and arrow over to CLRDATA, press ENTER

2. Press 2nd DATA and arrow over to 1VARSTATS and press ENTER

3. Press DATA, and enter the 1st value, then press the down arrow key

4. Enter the frequency of that number, press down arrow key

5. Enter 2nd value in the data set, etc.

6. When finished entering data, press STATVAR key and arrow over to the Sx symbol.

7. This is the standard deviation of a sample.

3.3 Variance Example

In a preceding example, we used the customer waiting times of 1 min, 3 min, and 14 min to find that the standard deviation is given by s = 7.0 min.

Find the variance of that same sample using your calculator.

Sample variance = s2 = 49.0 min2

Once you have the standard deviation on your calculator, hit the x-squared button to square the number. This is the variance.

3.3 Range Rule of Thumb

Understanding Standard Deviation using the Range Rule of Thumb

To estimate a standard deviation if you know the range,

In many data sets, 95% of the sample values lie within two standard deviations of the mean

To estimate the high value in the data set if you know the standard deviation: high value ≈ mean + 2s

To estimate the low value in the data set if you know the standard deviation: low value ≈ mean – 2s

We consider values “unusual” if they are not within 2 standard deviations of the mean.

4

ranges

3.3 Exampleselect the link to watch a helpful demonstration!http://www.screencast.com/users/mwittmer/folders/Jing/media/a260900f-cfab-4c78-a091-37bd70129268

Past results from the National Health Survey suggest that the pulse rates (beats per minute) have a mean of 76.0 and a standard deviation of 12.5. Use the range rule of thumb to find the minimum and maximum “usual” pulse rates that might be the result of some disorder. Then determine whether a pulse rate of 110 could be considered “unusual.”

Minimum “usual” value = (mean) – 2 x (s) = 76.0 – 2(12.5) = 51 beats per minute

Maximum “usual” value = (mean ) + 2 x (s) = 76.0 + 2(12.5) = 101 beats per minute

Based on these results, we expect that typical women have pulse rates between 51 bpm and 101 bpm. Because 110 beats per minute does not fall within those limits, it would be considered unusual.

3.3 Interpreting and Understanding Standard Deviation

1. The Empirical Rule (Pg. 106)

In a bell-shaped distribution, approximately

68% of all values fall within 1 standard deviation of the mean

95% of all values fall within 2 standard deviations of the mean

99.7% of all values fall within 3 standard deviations of the mean

2.4%2.4%

3.3 Empirical Rule Exampleclick on the link:http://www.screencast.com/users/mwittmer/folders/Jing/media/d59f7fd1-3381-4fff-923c-6da5ac2e4082

IQ scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 15.

a) What percent of adults have a score higher than 130? 2.4% + .1% = 2.5%

b) What scores are in the middle 95%? Between 70 and 130

100 115 130 145857035

2.4%2.4%

3.3 Example

Find the range, variance, and standard deviation for the given sample data.

3 24 30 47 43 7 47 13 44 39

a) Range (47 – 3) = 44

b) Standard deviation = (in calculator)= Sx = 17.02

c) Variance = (Sx)2 = (17.02)2 = 289.57