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finitehelp

Member

Registered: 2009-06-21

Posts: 80

please help on the empirical rule

The 68-95-99.7 rule simply states that 68.26% of values will lie within one standard deviation of the mean. 95.44% of values will lie within 2 standard deviations and 99.73% within 3 standard deviations of the mean. Describe an example using an assumed mean and standard deviation to illustrate the ranges of their normal distribution.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: please help on the empirical rule

Hi;

For a test, a thousand coins were flipped.. 1000 such tests were done and the number of heads in each counted. This was the result:

{503, 514, 521, 491, 492, 506, 501, 512, 519, 530, 508, 514, 466, \
500, 490, 473, 525, 503, 535, 503, 481, 484, 518, 499, 513, 481, 487, \
494, 511, 500, 501, 454, 487, 486, 489, 498, 505, 511, 525, 498, 527, \
539, 536, 485, 514, 499, 501, 499, 482, 499, 512, 504, 495, 508, 498, \
512, 487, 470, 480, 510, 513, 498, 494, 507, 501, 507, 516, 508, 524, \
517, 508, 528, 508, 478, 506, 460, 496, 502, 506, 517, 475, 481, 496, \
500, 521, 476, 477, 499, 494, 488, 493, 503, 501, 487, 472, 497, 510, \
519, 509, 497, 490, 522, 527, 499, 514, 498, 482, 504, 513, 526, 495, \
496, 499, 483, 493, 473, 519, 513, 496, 494, 491, 492, 502, 512, 521, \
462, 497, 539, 502, 499, 505, 475, 516, 497, 494, 497, 495, 494, 480, \
483, 503, 507, 517, 507, 514, 484, 531, 506, 503, 497, 495, 508, 499, \
501, 487, 477, 495, 506, 519, 528, 485, 512, 524, 499, 478, 509, 508, \
496, 485, 475, 500, 507, 488, 498, 497, 503, 508, 467, 503, 501, 503, \
479, 500, 522, 485, 511, 485, 511, 492, 472, 489, 488, 478, 491, 488, \
475, 509, 502, 500, 496, 508, 502, 505, 500, 508, 497, 505, 494, 509, \
503, 513, 506, 520, 495, 508, 506, 494, 507, 537, 489, 515, 500, 549, \
514, 520, 496, 499, 493, 489, 515, 509, 495, 468, 482, 480, 487, 516, \
509, 496, 493, 525, 507, 484, 495, 526, 498, 485, 496, 497, 505, 516, \
511, 495, 497, 496, 535, 519, 489, 485, 523, 460, 497, 458, 486, 504, \
477, 510, 480, 469, 508, 494, 539, 484, 521, 485, 514, 475, 496, 519, \
460, 528, 503, 506, 480, 503, 505, 528, 499, 497, 508, 491, 524, 507, \
500, 491, 494, 506, 530, 506, 479, 497, 504, 493, 500, 506, 519, 492, \
508, 450, 478, 501, 483, 529, 510, 494, 525, 484, 537, 502, 506, 507, \
487, 495, 495, 492, 520, 519, 487, 508, 517, 490, 498, 532, 495, 520, \
512, 500, 495, 497, 497, 477, 528, 504, 507, 505, 483, 501, 496, 520, \
504, 487, 485, 485, 508, 487, 506, 497, 491, 532, 487, 486, 462, 468, \
477, 492, 481, 504, 489, 511, 478, 508, 492, 480, 489, 525, 508, 508, \
516, 485, 508, 527, 477, 504, 504, 506, 516, 504, 493, 493, 478, 493, \
497, 507, 526, 472, 477, 527, 523, 481, 447, 501, 478, 494, 484, 507, \
475, 489, 466, 498, 518, 500, 499, 469, 476, 491, 480, 508, 516, 481, \
525, 493, 488, 523, 493, 516, 492, 508, 512, 503, 486, 488, 480, 496, \
524, 500, 499, 486, 510, 486, 529, 521, 470, 494, 499, 493, 498, 488, \
510, 474, 510, 485, 512, 517, 489, 490, 490, 497, 484, 494, 512, 531, \
496, 528, 516, 505, 497, 512, 475, 497, 508, 499, 507, 507, 507, 495, \
507, 474, 524, 479, 504, 492, 500, 490, 492, 479, 505, 483, 494, 494, \
490, 498, 496, 520, 537, 516, 497, 501, 487, 495, 510, 473, 488, 467, \
507, 510, 490, 523, 509, 495, 507, 487, 493, 525, 500, 513, 501, 502, \
518, 510, 465, 481, 495, 499, 495, 486, 499, 525, 485, 509, 491, 489, \
478, 510, 485, 494, 480, 499, 498, 480, 518, 531, 484, 481, 522, 496, \
505, 507, 511, 489, 489, 509, 503, 497, 489, 485, 522, 515, 517, 497, \
536, 501, 508, 512, 467, 499, 522, 510, 507, 486, 507, 488, 493, 509, \
507, 540, 495, 499, 514, 522, 500, 508, 506, 507, 507, 505, 503, 508, \
500, 501, 498, 512, 513, 472, 515, 468, 473, 507, 505, 498, 497, 478, \
534, 489, 514, 483, 518, 541, 503, 496, 508, 502, 511, 495, 494, 522, \
500, 487, 508, 504, 490, 499, 523, 487, 518, 519, 515, 513, 509, 476, \
516, 528, 521, 485, 484, 488, 504, 508, 524, 501, 522, 478, 502, 491, \
512, 502, 485, 486, 488, 508, 540, 472, 490, 521, 509, 513, 513, 483, \
493, 500, 492, 514, 502, 506, 497, 505, 509, 518, 497, 492, 498, 476, \
481, 483, 498, 520, 496, 508, 498, 507, 520, 498, 524, 480, 514, 504, \
502, 502, 479, 472, 486, 522, 500, 501, 508, 501, 540, 490, 501, 491, \
511, 495, 496, 491, 496, 512, 501, 525, 501, 501, 493, 510, 523, 510, \
478, 505, 512, 514, 499, 508, 500, 511, 489, 495, 484, 514, 478, 506, \
501, 493, 519, 520, 479, 499, 497, 497, 494, 506, 506, 497, 475, 491, \
489, 488, 494, 503, 504, 492, 504, 515, 494, 495, 509, 490, 510, 494, \
518, 522, 487, 524, 485, 500, 515, 475, 517, 508, 486, 509, 497, 511, \
490, 498, 477, 497, 490, 513, 503, 491, 499, 490, 473, 481, 526, 508, \
527, 497, 505, 500, 517, 510, 512, 517, 493, 497, 481, 501, 509, 509, \
504, 508, 478, 487, 505, 491, 473, 483, 479, 513, 519, 493, 518, 503, \
506, 500, 483, 496, 505, 488, 484, 486, 484, 517, 512, 495, 500, 498, \
523, 474, 487, 500, 508, 495, 486, 505, 497, 486, 513, 509, 493, 506, \
459, 505, 485, 473, 493, 497, 508, 500, 504, 510, 494, 495, 497, 497, \
504, 465, 497, 496, 536, 524, 502, 526, 499, 506, 479, 523, 493, 512, \
489, 497, 482, 494, 496, 482, 485, 511, 477, 489, 519, 507, 506, 531, \
496, 506, 513, 490, 511, 528, 479, 482, 507, 519, 509, 492, 494, 500, \
505, 526, 497, 493, 503, 519, 508, 512, 497, 512, 478, 470, 505, 498, \
492, 517, 511, 523, 516, 491, 503, 495, 497, 521, 503, 513, 497, 499, \
539, 508, 519, 487, 488, 517, 524, 489, 486, 504, 489, 509, 467, 508, \
481, 512, 484, 514, 486, 506, 515, 493, 480, 513, 517, 507, 484, 487, \
518, 498, 496, 480, 504, 504, 538, 494, 485, 524, 482, 475, 533, 503, \
504, 520, 493, 517, 508, 488, 505, 515, 511, 507, 487, 480, 512, 532, \
492, 501, 494, 524, 493, 501, 497, 485, 495, 502, 496, 487, 509, 509, \
491, 511, 517, 495, 466, 517, 526}

Now according to theory we expect that 68% of the above list will lie between 1 standard deviation of the mean or expected value. That means between 515.811 and 484.188. Counting the above list we see that approximately 69% of them do, which is quite close.

We would expect that 95% of the above list will lie between 2 standard deviations of the mean or expected value. That means between 531.622 and 468.377. Counting the above list we see that approximately 95% of them do, which again is quite close.

We would expect that 99% of the above list will lie between 3 standard deviations of the mean or expected value. That means between 547.434 and 452.565. Counting the above list we see that approximately 99% of them do, which again is quite close.

End of empirical techniques, we are done.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

finitehelp

Member

Registered: 2009-06-21

Posts: 80

Re: please help on the empirical rule

thanks my friend

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: please help on the empirical rule

Hi;

You are welcome and welcome to the forum.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.