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God Teaches the Lesson of the Sample Mean
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Transcript of God Teaches the Lesson of the Sample Mean
God Teaches the Lesson of the Sample Mean
Squared Error
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(xi − guess)2
Which Squared Error?
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(x1 − guess)2, (x2 − guess)2, or (x3 − guess)2 ?
Measuring Best
minimize
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(x1 − guess)2 +(x2 − guess)2 +(x3 − guess)2
Calculati Form a Plan
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SSE = (x1 − x)2 +(x2 − x)2 +(x3 − x)2
Algebrati Form a Plan
• Vertex of y = ax2 + bx + c
• is at x = -b/(2a)
Calculati Compute the Derivative
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−2(x1 − x) − 2(x2 − x) − 2(x3 − x)
Calculati Set the Derivative to Zero
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0 = −2(x1 − x) − 2(x2 − x) − 2(x3 − x)
Calculati Find Best Representative
• (x1+x2+x3)/3
The Lesson of Uncertainty
• Three Scientists Receive Three Samples
Mean for Scientist One
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x1 =x1 + x2
2
Means for Scientists Two and Three
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x3 =x1 + x3
2
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x2 =x2 + x3
2
Algebraists to the Fore
• Compare Three Sample Means
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x1, x2 , and x3
With the Population Mean
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x1 + x2 + x3
3
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x1 + x2 + x3
3=
x1 + x2
2+x2 + x3
2+x1 + x3
23
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=
2x1 + 2x2 + 2x3
23
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=x1 + x2 + x3
3= μ
The Mean of the Sample Means is the Population Mean
• End of Act One
God Teaches the Mystery of n-1
Least Squared Error
• SSE = (x1-)2 + (x2-)2 + (x3-)2 . . . until the end of the Population
God Challenges Man Again
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=x1 + x2 + x3
3
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SSE = (x1 − μ )2 +(x2 − μ )2 +(x3 − μ )2
Scientist One’s Plan
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x =x1 + x2
2
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SSE = (x1 − x)2 +(x2 − x)2
Scientist Two’s Plan
• Guess = 3/2 Times SSE of Sample
The First Practice Test
• And the first practice test was {12, 12, 0}.• the three Samples would be {12,12}, {12,0}, and
{12,0}.
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x1 = 12
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x2 = 6
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x3 = 6
First Practice Test
• From this generation of answers was begat the three Sample SSE values
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SSE1 = (12−12)2 +(12−12)2 = 0
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SSE2 = (12− 6)2 +(0− 6)2 = 72
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SSE3 = (12− 6)2 +(0− 6)2 = 72
Strategy of Scientist One
• SSE guesses as 0, 72 and 72.
Multiply-by-3/2 strategy Recommended by Scientist Two
• the three guesses as 0, 108 and 108
Computations for God and the SSE of the Population
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=12+12+ 0
3= 8
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SSE = (12− 8)2 +(12− 8)2 +(0− 8)2 = 96
Then God would compute the mean of their guesses as(0 +72+72)/3 = 48 for the Scientist One strategyand(0+108+108)/3 = 72 for the Scientist Two strategy.
More Practice Tests
Population Samples mean SSE SSE’s Mean SSE
Scaling Required
0,6,12 6 72 18, 18, 72 36 2 0,9,12 12 378 40.5, 162, 364.5 189 2 1,2,3 2 2 0.5,0.5,2 1 2
1,7,10 6 42 18,4.5,40.5 21 2
Sacred Relics
• The URL for the proof is• http://www.mr-ideahamster.com/stat-think/n-minus-one.pdf
• Excel Spreadsheet• http://www.mr-ideahamster.com/stat-think/nm1study.xls
Scaling N=6
• N n scale factor
• 3 2 2/1
• 6 5 5/4
• 6 4 5/3
• 6 3 5/2
• 6 2 5/1
Normalize by n-1
• General Scale Factor is (N-1)/(n-1)
• Use SSE/(n-1) Eliminates all scale factors
• SSE/(n-1) is called the Variance
Go Forth and Nevermore ask Why Divide by n-1
The End