God Teaches the Lesson of the Sample Mean

Squared Error

€

(xi − guess)2

Which Squared Error?

€

(x1 − guess)2, (x2 − guess)2, or (x3 − guess)2 ?

Measuring Best

minimize

€

(x1 − guess)2 +(x2 − guess)2 +(x3 − guess)2

Calculati Form a Plan

€

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

€

−2(x1 − x) − 2(x2 − x) − 2(x3 − x)

Calculati Set the Derivative to Zero

€

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

€

x1 =x1 + x2

2

Means for Scientists Two and Three

€

x3 =x1 + x3

2

€

x2 =x2 + x3

2

Algebraists to the Fore

• Compare Three Sample Means

€

x1, x2 , and x3

With the Population Mean

€

x1 + x2 + x3

3

€

x1 + x2 + x3

3=

x1 + x2

2+x2 + x3

2+x1 + x3

23

€

=

2x1 + 2x2 + 2x3

23

€

=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

€

=x1 + x2 + x3

3

€

SSE = (x1 − μ )2 +(x2 − μ )2 +(x3 − μ )2

Scientist One’s Plan

€

x =x1 + x2

2

€

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}.

€

x1 = 12

€

x2 = 6

€

x3 = 6

First Practice Test

• From this generation of answers was begat the three Sample SSE values

€

SSE1 = (12−12)2 +(12−12)2 = 0

€

SSE2 = (12− 6)2 +(0− 6)2 = 72

€

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

€

=12+12+ 0

3= 8

€

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