Statistics, Probability and Decision Making 1

Statistics, Probability, and Decision Making

Statistics, Probability and Decision Making

Statistics, Probability and Decision Making 2

Which trial represents the length?

Most feel the mean is the best estimate.

Trial Length

1 25.45

2 25.40

3 25.50

4 25.42

5 25.38

Mean

25.44

Statistics, Probability and Decision Making 3

How Precise is the Estimate?

You decide that the length is 25.43.

But look at the measurements. Is 25.50 a misfit?

Statistics, Probability and Decision Making

Statistics, Probability and Decision Making 4

What about an unexpected value?

• Get rid of it…

• No, you need a statistical reason !

• Only if it was a mistake.

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Is it a mistake?

An outlier: A single observation "far away" from the rest.

Q: How far away is “far away”?

A: It depends on whether the value differs from the

rest within a “reasonable” range.

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Decisions, decisions…

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Rejecting Data in a Small Data Set

Run the “Q-test.”

To test 25.50, calculate Q.

Q = (The suspect - the value closest to it)

Range

Q = 0.05 ÷ 0.12 = ≈ 0.42

Trial Length

1 25.45

2 25.40

3 25.50

4 25.42

5 25.38

Mean

25.44

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Compare Qcalculated with Qcritical

• If Qcalc > Qcritical, reject.

• If Qcalc < Qcritical, keep .

Qcritical90% confidence

0.94 0.76 0.64 0.56 0.51 0.47 0.44 0.41

Number oftrials

3 4 5 6 7 8 9 10

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From the previous example…

Qcalc = 0.42

N = 5, Qcritical = 0.64

• If Qcalc > Qcritical

• If Qcalc < Qcritical

Statistics, Probability and Decision Making

Statistics, Probability and Decision Making 10

Rejecting data in a large set

• Find the confidence interval

µ ± 3 σ

• Does measurement falls outside the confidence interval?

Use a Normal Distribution

95% of the data falls withintwo standard deviations of the mean.

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Outliers…Q: Why worry about them?

A: Values may not be properly distributed.

Q: Where do they come from?A: Possible sources:

1. Recording and measurement errors

2. Incorrect distribution3. Unknown data structure

Note: Outliers are in red

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Managing Outliers If the data is a normal distribution:

1. Calculate the mean and the standard deviation.

2. Find the ±3 standard deviation range for imposing limits on the data.

3. Identify outliers (greater ± 3 standard deviations).

4. Get rid of them!!!