“Worst Bound” of the 90% CI; this is the undesirable end of the range

“Worst Bound” of the 90% CI; this is the undesirable end of the range

“Best Bound” of the 90% CI; this is the desirable end of the range

“Best Bound” of the 90% CI; this is the desirable end of the range

Threshold: Below this point is losing money

Threshold: Below this point is losing money

A

BRelative Threshold

(RT)=B/A

Relative Threshold (RT)=B/A

Computing a “Relative Threshold”

• Use this result as input to the following slide for computing the value of measuring an uncertain range.

• Example: You might invest in a new system if you get a productivity improvement of over 10%. But your current range for this value is 5% to 20%. Compute the RT.

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

10

2

4

68

11

0.2

0.40.6

0.8

.010.1

0.040.06

0.08

100

20

40

60

80

10

100

20

40

60

80

10

24

68

1

10

0.20.40.60.81

0.1 0.05

1. Compute the Relative Threshold (RT)*

1. Compute the Relative Threshold (RT)*

2. Find the RT on the vertical axis

2. Find the RT on the vertical axis

Expected Opportunity Loss Factor (EOLF)

EOLF Curves for Normal Distributions

EOLF Curves for Uniform Distributions

3. Look directly to the right of the RT value, until you get to the appropriate curve (normal or uniform, depending on the probability distribution you are using). This is the EOLF

3. Look directly to the right of the RT value, until you get to the appropriate curve (normal or uniform, depending on the probability distribution you are using). This is the EOLF

4. Compute the Expected Value of Perfect Information (EVPI) = EOLF/1000*units in range*loss per unit

4. Compute the Expected Value of Perfect Information (EVPI) = EOLF/1000*units in range*loss per unit

The EOLF Chart• Use the RT from the

previous slide to compute the value of information.

• Example: You invested in the system in the example on the previous slide. Let’s say if the system does not get a productivity improvement greater than 10%, then you lost $100,000 for each percentage point you are under the threshold. Use this information and the RT to compute the value of reducing uncertainty about the range of potential productivity improvements.

Samples Below Threshold

20%

30%

40%

50%

0.1%

1%

10%

4 5 6 7 8 9 10

2 4 6 8 10 12 16 20Number Sampled

Ch

an

ce t

he

Me

dia

n is

Be

low

th

e T

hre

sho

ld

1 2 3

1814

2%

5%

0.2%

0.5%

0

1. Find the curve beneath the number of samples taken1. Find the curve beneath the number of samples taken

3. Follow the curve identified in step 1 until it intersects the vertical dashed line identified in step 2.

3. Follow the curve identified in step 1 until it intersects the vertical dashed line identified in step 2.

2. Identify the dashed line marked by the number of samples that fell below the threshold

2. Identify the dashed line marked by the number of samples that fell below the threshold

4. Find the value on the vertical axis directly left of the point identified in step 3; this value is the chance the median of the population is below the threshold

4. Find the value on the vertical axis directly left of the point identified in step 3; this value is the chance the median of the population is below the threshold

Measuring to the threshold• Use this chart when using

small samples to determine the probability that the median of a population is below a defined threshold

• Example: You want to determine how much time your staff spends on one activity. You sample 12 of them and only two spend less than 1 hour a week at this activity. What is the chance that the median time all staff spend is more than 1 hour per week? Look up 12 on the top row, following the curve until it intersects the “2” line on the bottom row, and look up the number to the left. The answer is just over 1%.

10%

20%

30%

0%

2%

4%

6%

8%

2 4 6 8

10 15 20

8%

12%

16%

20%

30%

40%

50%

60%

70%

10 15 20

5

2 4

1. Find the total sample size; then find the diagonal line that starts on the small circle beneath it

1. Find the total sample size; then find the diagonal line that starts on the small circle beneath it

2. Follow the diagonal line until it intersects the vertical line that corresponds to the number of samples in the subgroup; at that point lookup the number on the vertical scale to the left labeled “90% CI Lower Bound”

2. Follow the diagonal line until it intersects the vertical line that corresponds to the number of samples in the subgroup; at that point lookup the number on the vertical scale to the left labeled “90% CI Lower Bound”

90

% C

I U

pp

er

Bo

un

d9

0%

CI

Lo

we

r B

ou

nd

# of Samples in subgroup

3. Repeat the process for the upper bound, using the diagonal line above the sample size; follow the diagonal line until it intersect the same vertical line as before; follow it to the number on the vertical axis to the left labeled “90% CI Upper Bound”

3. Repeat the process for the upper bound, using the diagonal line above the sample size; follow the diagonal line until it intersect the same vertical line as before; follow it to the number on the vertical axis to the left labeled “90% CI Upper Bound”

# of Samples in subgroup

10 20 30 408642 8642 8642 8642

80%

40%

Sample Size

Population Proportion Estimate• Use this chart to

estimate the percentage of a population that falls within a subgroup, given a small sample

• Example: you want to measure how many of your customers have shopped at a competitor in the last week. You sample 20 and 10 of them said they did shop at a competitor. The chart shows how to compute the 90% confidence interval for the share of all customers who shopped there.

Population Proportions• This is another version of the first edition Population Proportion chart from the

previous slide.• This table shows the 90% CI for small samples (Lower Bound and Upper Bound

shown as %)• Simply look up the sample size in the column and the row with the number of hits.

Copyright HDR 2010 dwhubbard@hubbardresearch.com

5

1 2 3 4 6 8 10 15 20 300 2.5-78 1.7-63 1.3-53 01.0-45 0.7-35 0.6-28.3 0.5-23.9 0.3-17.1 0.2-13.3 0.2-9.21 22.4-97.5 13.5-87 9.8-75.2 07.6-65.8 05.3-52.1 4.1-42.9 3.3-36.5 2.3-26.4 1.7-20.7 1.2-14.42   36.8-98.3 25-90.3 18.9-81 12.9-65.9 9.8-55 07.9-47.0 5.3-34.4 4.0-27.1 2.7-18.93     47-98.7 34.3-92.4 22.5-78 16.9-66 13.5-57 9.0-42 6.8-33 4.5-234       55-99.0 34.1-87 25.1-75 20-65 13-48 9.9-38 6.6-275         48-94.7 34.5-83 27-73 17.8-55 13.2-44 8.8-316         65-99.3 45-90 35-80 22.7-61 16.8-49 11.1-357           57-95.9 44-87 28-67 21-54 14-388           72-99.5 53-92 33-72 25-58 16-429             64-96.7 39-77 29-63 19-4510             76-99.6 45-82 33-67 21-49

Sample Size

Num

ber

of

“hits

” in

Sam

ple

0.001

0.01

0.1

1

10

100

0 10 20 30 40 50

0.002

0.005

0.02

0.05

0.2

0.5

2

5

20

50

2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8

Upper boundUpper bound

Lower boundLower bound

MeanMean

A

1. Subtract the smallest serial number in the sample from the largest

1. Subtract the smallest serial number in the sample from the largest

2. Find the sample size on the horizontal axis and follow it up to the point where the vertical line intersects the curve marked “Upper Bound”

2. Find the sample size on the horizontal axis and follow it up to the point where the vertical line intersects the curve marked “Upper Bound”

3. Find the value for “A” on the vertical axis closest to the point on the curve and add 1; multiply the result by the answer in step 1. This is the 90% CI UB for total serial numbered items

3. Find the value for “A” on the vertical axis closest to the point on the curve and add 1; multiply the result by the answer in step 1. This is the 90% CI UB for total serial numbered items

4. Repeat steps 2 and 3 for the Mean and Lower Bound4. Repeat steps 2 and 3 for the Mean and Lower Bound

Sample Size

The “Enemy Tank” case

• This chart shows how the WWII statisticians estimated German tank production based on serial numbers of captured tanks