Statistical ThinkingSpace Shuttle O-ring Failure

The night before the disaster, it was predicted that it would be a cold night (30°F). Some NASA engineers advised against a launch because they thought that O-ring failures were related to temperature. They studied data on O-ring failures the night before and no discernable relationship was found between O-ring failure and temperature.

No discernable pattern between number of O-ring failures and temperature is present. Based on this information it was decided to launch.

Hogg-Ledoleter (1992), Applied Statistics for Engineers and Physical Scientist.

Space Shuttle O-ring Failure (continued)

What went wrong? Quite simply, the engineers did not look at all the available data. If they had examined O-ring successes and failures the night before, they would have seen this:

Hogg-Ledoleter (1992), Applied Statistics for Engineers and Physical Scientist.

The extra data added is of flights with no incidents or O-ring failures. While O-ring failures did occur at different temperatures, there is a distinct increased probably of failure at lower temperature, and note that 30°F is not even on this scale.

Old Faithful Geyser

0

10

20

30

40

50

60

70

80

90

100

Wai

t Tim

es (m

inut

es)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

Wait Time in Minutes (Over Time)

Chart

50

60

70

80

90

Wai

t Tim

es

(min

ute

s)

1 2 3

Day

1

2

3

Level

49

50

49

Minimum

49.9

63.5

50.8

10%

51.75

72

64

25%

70.5

77

76

Median

88

79.75

87.25

75%

92.1

86.8

92.1

90%

93

94

93

Maximum

Quantiles

1

2

3

Level

18

18

18

Number

70.0556

75.9444

73.7778

Mean

17.0586

9.0259

14.0945

Std Dev

4.0208

2.1274

3.3221

Std Err

Mean

61.573

71.456

66.769

Lower 95%

78.539

80.433

80.787

Upper 95%

Means and Std Deviations

Centers and Averages

Prison DataWe have a sample of 30 countries reporting the number of people in prison per 100,000 of the population. (JMP Datafile) The data is a SAMPLE from 2007 collected from King’s College, London. http://www.prisonstudies.org.

0 100 200 300 400 500 600 700 800

100.0%

99.5%

97.5%

90.0%

75.0%

50.0%

25.0%

10.0%

2.5%

0.5%

0.0%

maximum

quartile

median

quartile

minimum

714

714

714

514.2

282.25

142.5

88

65.1

54

54

54

Quantiles

Mean

Std Dev

Std Err Mean

Upper 95% Mean

Lower 95% Mean

N

199.73333

162.24352

29.621479

260.31606

139.15061

30

Moments

Prison Population Rate (per 100,000) of national population

Distributions

Questions to Students1. Which is the better measure of central tendency, mean or median?

Why?2. What are the characteristic(s) of countries with high or low prison

population rates?3. Would you rather have the country you live in to have a high or low

prison population rate? Justify your answer.

Natural VariabilityWhen does data indicate ‘Natural Variability’ or a distinct trend?

Exercise - 'Dodgy Dice'We know events such as rolling dice and flipping coins have outcomes within the rules of natural variability. Dice and coins have been used for gambling for thousands of years. Once in a while, a 'loaded' or 'dodgy' coin or dice is used. This item is biased such that it does not have an expected value that a 'fair' item would. For example, a biased coin might have the proportion of expected heads to be greater than 0.5 or a die might have the expected roll of getting a '6' to be greater than 1 in 6. This gives someone an unfair advantage. The dataset below the results from 50 rolls or flips of coins or dice.

What you need to determine is if the results indicate 'natural variation' or something 'dodgy'.

Dataset: https://castatistics.wikispaces.com/file/view/Dodgy+Dice.jmpColumn 1: 50 flips of a coin. A head is indicated with a '1', a tail with a '0'.Column 2: 50 rolls of a die. The score is recorded.Column 3: 50 rolls of two die. The total score is recorded.Column 4: 50 rolls of a die. A '6' is indicated with a '1', a everything else with a '0'.

How do you make a decision whether the die or coins are biased?

NOTE: This stage students have covered Normal, Binomial, Uniform and Skewed distributions

Natural Variability – Exercise D

0 1

0

1

Total

Level

23

27

50

Count

0.46000

0.54000

1.00000

Prob

N Missing 0

2 Levels

Frequencies

Coin Flip

1 2 3 4 5 6

1

2

3

4

5

6

Total

Level

2

10

10

12

12

4

50

Count

0.04000

0.20000

0.20000

0.24000

0.24000

0.08000

1.00000

Prob

N Missing 0

6 Levels

Frequencies

1 Die

2 3 4 5 6 7 8 9 10 11

2

3

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5

6

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11

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Level

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6

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Count

0.06000

0.12000

0.10000

0.08000

0.12000

0.06000

0.12000

0.10000

0.12000

0.12000

1.00000

Prob

N Missing 0

10 Levels

Frequencies

2 Die

0 1

0

1

Total

Level

35

14

49

Count

0.71429

0.28571

1.00000

Prob

N Missing 1

2 Levels

Frequencies

1 Die, 6 indicated

How do you make a decision whether the die or coins are biased?

Students are often divided on which ones are dodgy and which are fair. The simple conclusion is that it can be difficult to determine. However, using statistical techniques we can measure accurately which results are very unlikely once we take into account Natural Variability (i.e. Randomness)

Implications for Teaching(From Thinking and Reasoning Data and Chance, NCTM, 2006)

• Include ‘variability’ as a central issue.– Explore ‘Natural Variability’

• Explore understanding of ‘center’– What do we mean by ‘average’?

• Make connections between sample and populations.– Understand the variability of a sample and how this effects Sampling Distributions