Do Now, Calculate these values

• Customer waiting times, in seconds, for two banks

• Mean =• Median =• Mode =• Midrange =

Jefferson Providence

6.5 4.2

6.6 5.4

6.7 5.8

6.8 6.2

7.1 6.7

7.3 7.7

7.4 7.7

7.7 8.5

7.7 9.3

7.7 10

Measures of Variation

• We need to measure the variation in such a way that– Data in which values that are close together should have a

small measurement

– Values that are spread apart should have a high measurement

• Range• Standard deviation• Variance

Range

• maximum - minimum• Jefferson:• Providence:

Standard Deviation

• “Measurement of variation of values about their mean”

• Standard deviation based on a sample:– Abbreviated s

2

1

1

n

ii

x xs

n

• Same units as the original values• Sample variance = s2

Population Standard Deviation

• Standard deviation of the entire population:– Abbreviated σ (lower-case sigma)

2

1

n

ii

x

N

• Same units as the original values• Population variance = σ2

Standard Deviation

• Interpretation– “Average distance from the mean”

– About 2/3 of the values should fall within 1 standard deviation of the mean

• Can be estimated with the Range rule of thumb:

4

ranges

Calculating a Standard Deviation

1. Find the mean

2. For each valuea. Subtract the mean from the value

b. Square the difference

3. Add the squares from step 2b.

4. Divide that sum buy one less than number of samples

5. Find the square root of the quotient

Live Example: Jefferson Valley

6.5

6.6

6.7

6.8

7.1

7.3

7.4

7.7

7.7

7.7

Your turn: Providence

4.2

5.4

5.8

6.2

6.7

7.7

7.7

8.5

9.3

10

Rounding

• Round to one more decimal place that the original data.– E.g., the mean of 14 15 9 16 and 12 would be 13.2

– E.g., the mean of 1.04, 2.17, and 3.79 would be 2.333

– Except for ‘well-know’ decimals

• 0.25, 0.75, 0.5

• Round the standard deviation to one more place that the mean

Homework

• Find the standard deviation of Super Bowl Points• Use the range rule of thumb to estimate the standard

deviation of motor vehicle deaths and Sunspot number

Uses Standard Deviation

• Interpretation– “Average distance from the mean”

– About 2/3 of the values should fall within 1 standard deviation of the mean

• Uses– Identify outliers and determining what is “unusual”

– Frequency Tables

– Curving

Estimating Frequencies

meanmean + s

mean + 2smean + 3smean – 3s

mean – 2smean – s

34% 34%17% 17%2.4% 2.4%0.1% 0.1%

68%

95%

99.7%

• In a large sample of normally distributed data a specific percent of values should fall into frequency classes whose boundaries are based on the mean and standard deviation

Distribution Spread of Bank Data

7.15

7.15

2 0.48

8.11

7.15

(3)0.48

8.59

0 (mean)

+1s +2s + 3s-3s -2s -1s

7.15

0.48

7.63

7.15

0.48

6.67

7.15

2 0.48

6.19

7.15

(3)0.48

5.71

Outliers and Unusual Values

• 5% is considered the boundary for being unusual• Values are considered outliers or unusual if they are

beyond 2 standard deviations from the mean.

mean mean + 2smean – 2s

2.5% 2.5%

95%

Outliers and Unusual Values

• Outliers and/or unusual values are usually investigated– Different category

– Bad data

– “Bad day”

• Outliers and/or unusual values might be trimmed from the list

Frequency Table

• We can use the mean and standard deviation as the boundary of our frequency classes

• If the data is normally distributed– 68% of our samples should be within one standard

deviation of the mean

– 95% should be within two standard deviations

– 99.7 should be within three standard deviations

Building a Frequency Table

1. Create a table of six classes2. Find the mean and standard deviation3. Use the mean as the lower class fourth class4. Add the standard deviation to the mean to get the

LCL for the fifth classes, add it again for the LCL for the sixth the mean.

5. Subtract the standard deviation from the mean to get the LCL for the third class, subtract it again to get the LCL for the second class. The first class has no LCL

6. Fill in the UCLs; the sixth class has no UCL7. Fill in the frequencies

IQ Scores

• The mean and standard deviation for IQ scores is 100 and 15, respectively.

• Create a frequency table for the following IQ scores: 100 111 122 95 83 95 109 93 102 110 86 98 120 108 117 94 101 105 104 72

IQ Frequency Table

LCB UCB Frequency

Using Mean and Std Dev to curve a tests

C+ B B+ AF D D+ C

• 25 55 59 59 63 71 71 74 80 80 80 83 84 84 87 88 95 95 100 100

• X-bar is 77, s is 18

• Use s divided by 3 for grade boundaries

A+

Your Turn

• Define distribution spread and a frequency table using the following values:50 140 104 75 55 22 75 39 90 45 110 62 90 35 200 90

• Are there any “unusual values?• What is the “theoretical” percent of values that fall

within 1 sigma?• What is the actual percent of values that fall within 1

sigma?

IQ Frequency Table

LCB UCB Frequency

Homework

• The mean adult male height is 69 inches; the standard deviation is 3.5

• The mean adult female height is 64 inched; the standard deviation is 3

• On the next slide use these values to create a frequency table of your opinion poll data.

• Identify outliers/unusual values

Frequency Table: Male

LCB UCB Frequency

Frequency Table: Female

LCB UCB Frequency