Intro

Page 1

Intro to MonteCarlo.xlsThis workbook introduces Monte Carlo Simulation with a simple example.

In this book, we use Microsoft Excel to simulate chance processes.

Typically, we use Excel to draw a sample, then compute a sample statistic, e.g., the sample average. We can repeat this process, again and again, keeping track of the result each time. We then look at the empirical histogram and summary statistics of the results in order to discover or illustrate the statistical properties of the chance process. This method of studying a chance process is called Monte Carlo simulation.

La hoja OneFreeThrow muestra cómo Excel se puede utilizar dibujar un número al azar y simular un tiro libre.

La Sample sheet tiene el resultado de 100 intentos de tiros libres.

There are other hidden sheet in this workbook.Hay otra hoja oculta en este libro.

The OneFreeThrow sheet shows how Excel can be used draw a random number and simulate a free throw.

The Sample sheet has the outcome of 100 free throw attempts.

Intro

Page 2

Typically, we use Excel to draw a sample, then compute a sample statistic, e.g., the sample average. We can repeat this process, again and again, keeping track of the result each time. We then look at the empirical histogram and summary statistics of the results in order to discover or illustrate the statistical properties of the chance process. This method of studying a chance process is called

ALEATORIOLa hoja OneFreeThrow muestra cómo Excel se puede utilizar dibujar un número al azar y simular un tiro libre.

sheet shows how Excel can be used draw a random number and simulate a free throw.

OneFreeThrow

Page 3

To simulate a free throw, we use a random draw in an IF statement. Scroll down to see how.

The random draw uses the formula, "=RANDOM()", to get a uniformly distributed number in the 0,1 interval.El sorteo se utiliza la fórmula "= ALEATORIO ()", para obtener un número uniformemente distribuida en el intervalo de 0,1.Click in cell B5 to see the formula. Hit F9 (or CTRL - =) to draw a new number. For more on random number generation, see sections 9.2 and 9.3.Haga clic en la celda B5 para ver la fórmula. Pulse F9 (o CTRL - =) para dibujar un nuevo número. Para más información sobre la generación de números aleatorios, ver secciones 9.2 y 9.3.

Random draw #MACRO? Sorte Aleatorio

The RANDOM() function is the foundation of the Monte Carlo Simulation because it's our roulette wheel, but we need one more concept to complete the simulation. We need a way to tell Excel that Larry has a 90% chance of making the free throw.Fortunately, this is easily done with an IF statement.

Take a look at the formulas in cells B18, D18, and D18. Cell D18 says, "if the value in cell B18 is less than 0.9, then the value of this cell is 1, else the value is 0." The structure is "expression to be evaluated," "value if true," "value if false." Notice the difference between cells D18 and D19--yet they end up doing the same thing. While you are at it, try to figure out what's going on in cells C18 and C19.

Random draw #MACRO? #MACRO? #MACRO?

#MACRO? #MACRO?

Combining the RAND and IF statements into a single formula, we get a virtual 90% free throw shooter:La combinación de la RAND y sentencias IF en una sola fórmula, obtenemos un 90% virtual de tiros libres:

#MACRO?Notice that this outcome will occasionally differ from the one above. The reason is that it's based on a different draw from the uniform distribution.Tenga en cuenta que este resultado ocasionalmente diferir de la de arriba. La razón es que se basa en un consumo diferente de la distribución uniforme

Hit F9 (or CTRL - =) repeatedly (say 30 times in a row) to see how Larry's doing.As long as the value stays at 1, Larry hit the free throw. When it changes to 0, he missed.

Question:"I noticed that sometimes he misses in cell B23, but makes it in cells C18 through D19. What's going on?"

La función random () es la base de la simulación de Monte Carlo, porque es nuestra rueda de la ruleta, "pero necesitamos un concepto más para completar la simulación. Necesitamos una manera de decirle a Excel que Larry tiene un 90% de probabilidades de hacer el tiro libre. Afortunadamente, esto es fácil de hacer con una instrucción IF.

Echa un vistazo a las fórmulas de las celdas B18, D18, D18 y. D18 de la célula dice, "si el valor de celda B18 es menor que 0,9, entonces el valor de esta celda es 1, de lo contrario el valor es 0." La estructura es "la expresión para ser evaluados", "el valor de ser cierto," "valor si es falso." Note la diferencia entre las células D18 y D19 - sin embargo, terminan haciendo lo mismo. Mientras que usted está en él, tratar de averiguar lo que está pasando en las células C18 y C19

Larry Bird Simulated

Pulse F9 (o CTRL - =) varias veces (digamos 30 veces seguidas) para ver cómo hacer de Larry. Mientras el valor esté a 1, Larry golpeó el tiro libre. Cuando se cambia a 0, se perdió.

OneFreeThrow

Page 4

"Me di cuenta de que a veces echa de menos en B23 celular, pero lo hace en las celdas C18 a D19. ¿Qué está pasando?"

Answer:

RANDOM() in a cell, Excel draws a new random number.

Question:"Sometimes I see the word "Calculate" at the bottom of the screen. What does that mean?"

Answer:Excel can be set up to automatically calculate the values of every formula wheneverthe user enters new information anywhere in the workbook OR it can be in Manual Calculatemode, which means that it will not automatically recalculate. Manual Calculation is needed for Monte Carlo simulation to prevent constant redrawing of random numbers. The "Calculate" message in the Status Bar at the bottom of the screen alerts the reader that Excel has notautomatically recalculated the cells in the workbook. If you hit F9, the "Calculate" messagewill disappear. Section 9.3 explains about Calculation in more detail and shows how to set the calculation mode.

Excel is drawing different random numbers in cells B18 and B23. Every time it sees

Excel está dibujando diferentes números aleatorios en las celdas B18 y B23. Cada vez que ve ALEATORIO () en una celda, Excel dibuja un nuevo número aleatorio.

OneFreeThrow

Page 5

To simulate a free throw, we use a random draw in an IF statement. Scroll down to see how.

The random draw uses the formula, "=RANDOM()", to get a uniformly distributed number in the 0,1 interval.El sorteo se utiliza la fórmula "= ALEATORIO ()", para obtener un número uniformemente distribuida en el intervalo de 0,1.Click in cell B5 to see the formula. Hit F9 (or CTRL - =) to draw a new number. For more on random number generation, see sections 9.2 and 9.3.Haga clic en la celda B5 para ver la fórmula. Pulse F9 (o CTRL - =) para dibujar un nuevo número. Para más información sobre la generación de números aleatorios, ver secciones 9.2 y 9.3.

The RANDOM() function is the foundation of the Monte Carlo Simulation because it's our roulette wheel, but we need one more concept to complete the simulation. We need a way to tell Excel that Larry

Cell D18 says, "if the value in cell B18 is less than 0.9, then the value of this cell is 1, else the value is 0." The structure is "expression to be evaluated," "value if true," "value if false." Notice the difference between cells D18 and D19--yet they end up doing the same thing. While you are at it, try to figure out what's going on in cells C18 and C19.

Sorte Aleatorio

Combining the RAND and IF statements into a single formula, we get a virtual 90% free throw shooter:La combinación de la RAND y sentencias IF en una sola fórmula, obtenemos un 90% virtual de tiros libres:

Notice that this outcome will occasionally differ from the one above. The reason is that it's based on a different draw from the uniform distribution.Tenga en cuenta que este resultado ocasionalmente diferir de la de arriba. La razón es que se basa en un consumo diferente de la distribución uniforme

Hit F9 (or CTRL - =) repeatedly (say 30 times in a row) to see how Larry's doing.As long as the value stays at 1, Larry hit the free throw. When it changes to 0, he missed.

"I noticed that sometimes he misses in cell B23, but makes it in cells C18 through D19. What's going on?"

La función random () es la base de la simulación de Monte Carlo, porque es nuestra rueda de la ruleta, "pero necesitamos un concepto más para completar la simulación. Necesitamos una manera de decirle a Excel que Larry tiene un 90% de probabilidades de hacer el tiro libre. Afortunadamente, esto es fácil de hacer con una instrucción

Echa un vistazo a las fórmulas de las celdas B18, D18, D18 y. D18 de la célula dice, "si el valor de celda B18 es menor que 0,9, entonces el valor de esta celda es 1, de lo contrario el valor es 0." La estructura es "la expresión para ser evaluados", "el valor de ser cierto," "valor si es falso." Note la diferencia entre las células D18 y D19 - sin embargo, terminan haciendo lo mismo. Mientras que usted está en él, tratar de averiguar lo que

Pulse F9 (o CTRL - =) varias veces (digamos 30 veces seguidas) para ver cómo hacer de Larry. Mientras el valor esté a 1, Larry golpeó el tiro libre. Cuando se cambia a 0, se perdió.

OneFreeThrow

Page 6

"Me di cuenta de que a veces echa de menos en B23 celular, pero lo hace en las celdas C18 a D19. ¿Qué está pasando?"

"Sometimes I see the word "Calculate" at the bottom of the screen. What does that mean?"

Excel can be set up to automatically calculate the values of every formula wheneverthe user enters new information anywhere in the workbook OR it can be in Manual Calculatemode, which means that it will not automatically recalculate. Manual Calculation is needed for Monte Carlo simulation to prevent constant redrawing of random numbers. The "Calculate" message in the Status Bar at the bottom of the screen alerts the reader that Excel has notautomatically recalculated the cells in the workbook. If you hit F9, the "Calculate" messagewill disappear. Section 9.3 explains about Calculation in more detail and shows how to set the calculation mode.

random numbers in cells B18 and B23. Every time it sees

Excel está dibujando diferentes números aleatorios en las celdas B18 y B23. Cada vez que ve ALEATORIO () en una

OneFreeThrow

Page 7

Click in cell B5 to see the formula. Hit F9 (or CTRL - =) to draw a new number. For more on random number generation, see sections 9.2 and 9.3.Haga clic en la celda B5 para ver la fórmula. Pulse F9 (o CTRL - =) para dibujar un nuevo número. Para más información sobre la generación de números aleatorios, ver secciones 9.2 y 9.3.

Tenga en cuenta que este resultado ocasionalmente diferir de la de arriba. La razón es que se basa en un consumo diferente de la distribución uniforme

Sample

Page 8

#MACRO?

1 #MACRO?2 #MACRO?3 #MACRO?4 #MACRO?5 #MACRO?6 #MACRO?7 #MACRO?8 #MACRO?9 #MACRO?10 #MACRO?11 #MACRO?12 #MACRO?13 #MACRO?14 #MACRO?15 #MACRO? Buscador de Racha16 #MACRO?17 #MACRO?

18 #MACRO?19 #MACRO?20 #MACRO?21 #MACRO?22 #MACRO?23 #MACRO?24 #MACRO?25 #MACRO?26 #MACRO?27 #MACRO?28 #MACRO?29 #MACRO?30 #MACRO?31 #MACRO?32 #MACRO?33 #MACRO?34 #MACRO?35 #MACRO?

Percentage MadePorcentaje Hecho

This sheet contains a single repetition of 100 draws (free throw attempts). Esta hoja contiene una sola repetición de 100 dibuja (intentos de tiros libres).

AttemptIntento

ResultResultado

Larry Bird hits 90% of his free throws. The formula in Cell B4 informs Excel of this chance process. Larry Bird golpea el 90% de sus tiros libres. La fórmula en la celda B4 informa en Excel de este proceso de oportunidad.The exact same formula is used in the 100 cells from B4 to B103. Click on a few of these cells to see for yourself. La misma fórmula exacta se utiliza en las células 100 de B4 a B103. Haga clic en algunas de estas células para ver por sí mismo.Hit F9 to simulate another 100 free throws. How did Larry do this time? ¿Cómo salió Larry esta vez?

Hit F9 again and again. Watch cell B1. How much spread or bounce is there?

Wouldn't it make sense to keep track of Larry's sample percentage? Click on the button below to do so.

Curious about streaks? Click on the button below. Click this button several times.

Do you think Max Streaks follow a normal distribution? Click on the button below to see a Monte Carlo simulation that answers this question. Haga clic en el botón de abajo para ver una simulación de Monte Carlo que responde a esta pregunta.

Sample

Page 9

36 #MACRO?37 #MACRO?38 #MACRO?39 #MACRO?40 #MACRO?41 #MACRO?42 #MACRO?43 #MACRO?44 #MACRO?45 #MACRO?46 #MACRO?47 #MACRO?48 #MACRO?49 #MACRO?50 #MACRO?51 #MACRO?52 #MACRO?53 #MACRO?54 #MACRO?55 #MACRO?56 #MACRO?57 #MACRO?58 #MACRO?59 #MACRO?60 #MACRO?61 #MACRO?62 #MACRO?63 #MACRO?64 #MACRO?65 #MACRO?66 #MACRO?67 #MACRO?68 #MACRO?69 #MACRO?70 #MACRO?71 #MACRO?72 #MACRO?73 #MACRO?74 #MACRO?75 #MACRO?

Sample

Page 10

76 #MACRO?77 #MACRO?78 #MACRO?79 #MACRO?80 #MACRO?81 #MACRO?82 #MACRO?83 #MACRO?84 #MACRO?85 #MACRO?86 #MACRO?87 #MACRO?88 #MACRO?89 #MACRO?90 #MACRO?91 #MACRO?92 #MACRO?93 #MACRO?94 #MACRO?95 #MACRO?96 #MACRO?97 #MACRO?98 #MACRO?99 #MACRO?100 #MACRO?

Sample

Page 11

Buscador de Racha

This sheet contains a single repetition of 100 draws (free throw attempts). Esta hoja contiene una sola repetición de 100 dibuja (intentos de tiros

Larry Bird hits 90% of his free throws. The formula in Cell B4 informs Excel of this chance process. Larry Bird golpea el 90% de sus tiros libres. La fórmula en la celda B4 informa en Excel de este proceso de oportunidad.The exact same formula is used in the 100 cells from B4 to B103. Click on a few of these cells to see for yourself. La misma fórmula exacta se utiliza en las células 100 de B4 a B103. Haga clic en algunas de estas células para ver por sí mismo.Hit F9 to simulate another 100 free throws. Pulse F9 para simular lanzamientos otros 100 libre.How did Larry do this time? ¿Cómo salió Larry esta vez?

Hit F9 again and again. Watch cell B1. Una y otra vez Pulse F9. Vea la celda B1.How much spread or bounce is there? ¿Cuánto propagación o rebote es allí?

Wouldn't it make sense to keep track of Larry's sample percentage? ¿No tendría más sentido para llevar un registro del porcentaje de la muestra de Larry?Click on the button below to do so. Haga clic en el botón de abajo para hacerlo

Curious about streaks? Click on the button below. Tengo curiosidad de las rachas? Haga clic en el botón de abajo.Click this button several times. Haga clic en este botón varias veces

Do you think Max Streaks follow a normal distribution? ¿Crees que Max Rayas siguen una distribución normal?Click on the button below to see a Monte Carlo simulation that answers this question. Haga clic en el botón de abajo para ver una simulación de Monte Carlo que responde a esta pregunta.

Sample

Page 12

Esta hoja contiene una sola repetición de 100 dibuja (intentos de tiros

Larry Bird hits 90% of his free throws. The formula in Cell B4 informs Excel of this chance process. Larry Bird golpea el 90% de sus tiros

The exact same formula is used in the 100 cells from B4 to B103. Click on a few of these cells to see for yourself. La misma

¿No tendría más sentido para llevar un registro del porcentaje de la muestra de Larry?

Haga clic en el botón de abajo para ver una simulación

Q&A for MonteCarlo.xls1) Run a Monte Carlo with 1,000 repetitions of the percentage made of 100 free throws.

Use the Take a Picture button to take a picture of your results.Copy and paste the picture in a Word document.

2) Run a Monte Carlo with 10,000 repetitions of the percentage made of 100 free throws.Take a picture, copy it, and paste it in the Word document below the 1,000 repetition Monte Carlo.

3) In what sense is the 10,000 repetition Monte Carlo better than the 1,000 repetition Monte Carlo in this example?

4) Run a Max Streak Monte Carlo.Take a picture, copy it, and paste it in the Word document.

5) What are the chances that the virtual Larry Bird gets a max streak of 51 in a row or more?Explain how you arrived at your answer.

1) Run a Monte Carlo with 1,000 repetitions of the percentage made of 100 free throws.Use the Take a Picture button to take a picture of your results.

2) Run a Monte Carlo with 10,000 repetitions of the percentage made of 100 free throws.Take a picture, copy it, and paste it in the Word document below the 1,000 repetition Monte Carlo.

3) In what sense is the 10,000 repetition Monte Carlo better than the 1,000 repetition Monte Carlo in this example?

5) What are the chances that the virtual Larry Bird gets a max streak of 51 in a row or more?

Monte Carlo Simulation of Sample Percentage with 10000 Repetitions

1 87%2 93% Time Elapsed 2 secs3 92%4 94%5 94%6 93%7 93%8 91%9 91%10 88%11 94%12 92%13 91%14 86%15 87%16 88%17 87%18 91%19 87%20 87%21 87%22 94%23 88%24 92%25 93%26 89%27 87%28 92%29 91%30 92%31 92%32 93%33 90%34 92%35 86%

Sample Number

Sample Percentag

eExcel Tip:Excel Tip: Detenga el cursor sobre una de las barras, se muestra el número de muestras (o altura de la barra).

78%79%80%81%82%83%84%85%86%87%88%89%90%91%92%93%94%95%96%97%98%99%100%

0

200

400

600

800

1000

1200

1400

1600 Empirical Histogram for 10000 Repetitions

Min = 76%

Average = 89.98%

Summary Statistics

Max = 99%

0

20

40

60

80

100

120

140

160

78% 80% 82% 84% 86% 88% 90% 92% 94% 96% 98% 100%

Em pirical Histogram for 1000 Repetitions

M in = 81%

A verage = 89.90%SD = 3.034%

Summary Statistics

M ax = 98%

36 91%37 88%38 92%39 88%40 87%41 87%42 83%43 92%44 89%45 86%46 89%47 87%48 87%49 87%50 87%51 89%52 94%53 93%54 92%55 88%56 90%57 92%58 86%59 88%60 94%61 86%62 85%63 88%64 93%65 87%66 88%67 90%68 86%69 89%70 85%71 89%72 91%73 88%74 87%75 85%76 87%

0

20

40

60

80

100

120

140

160

78% 80% 82% 84% 86% 88% 90% 92% 94% 96% 98% 100%

Em pirical Histogram for 1000 Repetitions

M in = 81%

A verage = 89.90%SD = 3.034%

Summary Statistics

M ax = 98%

77 90%78 87%79 90%80 91%81 98%82 90%83 93%84 87%85 85%86 93%87 87%88 91%89 90%90 91%91 89%92 91%93 85%94 91%95 88%96 91%97 93%98 91%99 92%100 90%101 93%102 92%103 93%104 89%105 89%106 86%107 89%108 92%109 92%110 93%111 86%112 90%113 88%114 93%115 90%116 91%117 92%

118 94%119 88%120 89%121 90%122 90%123 85%124 90%125 93%126 94%127 90%128 89%129 85%130 93%131 90%132 96%133 88%134 91%135 89%136 91%137 89%138 91%139 95%140 92%141 90%142 93%143 89%144 85%145 93%146 85%147 93%148 92%149 83%150 91%151 94%152 91%153 93%154 88%155 89%156 86%157 91%158 88%

159 92%160 87%161 91%162 87%163 87%164 91%165 89%166 93%167 90%168 95%169 90%170 86%171 94%172 90%173 93%174 89%175 96%176 93%177 86%178 84%179 87%180 93%181 92%182 86%183 90%184 85%185 95%186 90%187 91%188 95%189 91%190 88%191 89%192 88%193 90%194 92%195 95%196 90%197 92%198 90%199 96%

200 90%201 94%202 82%203 89%204 84%205 86%206 90%207 92%208 84%209 93%210 95%211 90%212 88%213 91%214 93%215 90%216 90%217 95%218 93%219 86%220 87%221 84%222 86%223 90%224 90%225 91%226 92%227 87%228 85%229 87%230 88%231 91%232 91%233 95%234 96%235 88%236 87%237 91%238 93%239 92%240 89%

241 92%242 95%243 90%244 87%245 81%246 96%247 93%248 91%249 90%250 93%251 91%252 87%253 90%254 90%255 93%256 86%257 94%258 92%259 83%260 91%261 89%262 87%263 91%264 90%265 87%266 92%267 93%268 94%269 91%270 90%271 93%272 90%273 87%274 89%275 89%276 91%277 93%278 93%279 84%280 89%281 92%

282 91%283 91%284 94%285 88%286 91%287 87%288 94%289 92%290 88%291 88%292 89%293 91%294 86%295 92%296 94%297 91%298 92%299 89%300 93%301 90%302 89%303 84%304 91%305 89%306 89%307 93%308 94%309 90%310 85%311 83%312 93%313 90%314 94%315 91%316 83%317 91%318 96%319 89%320 90%321 92%322 91%

323 86%324 89%325 92%326 89%327 92%328 95%329 93%330 93%331 96%332 89%333 85%334 94%335 88%336 88%337 89%338 91%339 91%340 92%341 87%342 93%343 89%344 94%345 96%346 88%347 90%348 88%349 86%350 87%351 85%352 93%353 93%354 92%355 89%356 90%357 88%358 86%359 92%360 82%361 87%362 89%363 93%

364 89%365 90%366 94%367 88%368 93%369 91%370 86%371 88%372 91%373 90%374 85%375 86%376 91%377 94%378 90%379 87%380 92%381 90%382 90%383 93%384 89%385 93%386 91%387 88%388 91%389 93%390 84%391 92%392 88%393 89%394 91%395 93%396 94%397 87%398 94%399 90%400 90%401 90%402 90%403 91%404 89%

405 92%406 92%407 87%408 84%409 86%410 90%411 94%412 89%413 90%414 88%415 91%416 92%417 89%418 89%419 90%420 93%421 88%422 93%423 94%424 90%425 89%426 93%427 89%428 88%429 90%430 93%431 87%432 85%433 94%434 91%435 94%436 93%437 89%438 90%439 87%440 88%441 92%442 87%443 86%444 89%445 87%

446 87%447 93%448 89%449 94%450 88%451 90%452 91%453 90%454 92%455 84%456 90%457 88%458 91%459 95%460 87%461 91%462 88%463 94%464 87%465 89%466 94%467 89%468 90%469 85%470 93%471 84%472 92%473 89%474 89%475 94%476 88%477 91%478 92%479 86%480 93%481 86%482 90%483 91%484 91%485 88%486 87%

487 90%488 87%489 91%490 87%491 85%492 87%493 88%494 93%495 93%496 84%497 87%498 91%499 92%500 93%501 89%502 88%503 91%504 87%505 94%506 90%507 91%508 95%509 93%510 92%511 86%512 94%513 87%514 92%515 94%516 90%517 93%518 86%519 90%520 91%521 93%522 87%523 86%524 94%525 92%526 94%527 87%

528 88%529 90%530 87%531 86%532 91%533 93%534 90%535 85%536 88%537 90%538 89%539 84%540 88%541 93%542 91%543 92%544 90%545 94%546 91%547 90%548 87%549 89%550 85%551 91%552 94%553 86%554 90%555 89%556 85%557 93%558 89%559 93%560 91%561 90%562 92%563 90%564 87%565 92%566 88%567 93%568 87%

569 90%570 89%571 88%572 91%573 86%574 89%575 92%576 91%577 93%578 92%579 84%580 90%581 92%582 89%583 89%584 88%585 89%586 90%587 90%588 92%589 93%590 94%591 86%592 93%593 91%594 88%595 91%596 88%597 89%598 91%599 90%600 91%601 91%602 87%603 84%604 83%605 87%606 89%607 87%608 89%609 91%

610 89%611 93%612 91%613 87%614 93%615 92%616 93%617 90%618 84%619 92%620 89%621 94%622 82%623 88%624 88%625 93%626 89%627 87%628 89%629 92%630 89%631 90%632 90%633 89%634 95%635 86%636 94%637 89%638 89%639 91%640 89%641 88%642 92%643 87%644 89%645 91%646 93%647 91%648 92%649 91%650 89%

651 95%652 90%653 88%654 86%655 88%656 91%657 92%658 86%659 88%660 92%661 91%662 91%663 93%664 93%665 89%666 93%667 93%668 88%669 94%670 93%671 90%672 92%673 89%674 98%675 89%676 89%677 85%678 85%679 90%680 87%681 91%682 89%683 89%684 89%685 85%686 93%687 95%688 88%689 97%690 85%691 84%

692 91%693 90%694 93%695 92%696 80%697 92%698 90%699 91%700 86%701 91%702 85%703 95%704 89%705 89%706 84%707 89%708 92%709 90%710 86%711 89%712 88%713 95%714 86%715 92%716 88%717 93%718 88%719 80%720 92%721 86%722 94%723 89%724 83%725 91%726 89%727 95%728 91%729 93%730 93%731 94%732 92%

733 88%734 91%735 93%736 88%737 93%738 89%739 89%740 93%741 90%742 93%743 90%744 94%745 83%746 97%747 87%748 94%749 85%750 89%751 93%752 92%753 90%754 93%755 87%756 86%757 94%758 86%759 91%760 90%761 92%762 87%763 90%764 88%765 91%766 90%767 92%768 82%769 89%770 90%771 91%772 89%773 91%

774 92%775 92%776 82%777 88%778 87%779 88%780 94%781 89%782 84%783 92%784 91%785 94%786 89%787 90%788 89%789 92%790 84%791 97%792 89%793 86%794 88%795 90%796 91%797 91%798 88%799 87%800 92%801 88%802 94%803 85%804 90%805 92%806 88%807 90%808 94%809 86%810 89%811 89%812 94%813 93%814 90%

815 92%816 83%817 85%818 93%819 95%820 88%821 91%822 95%823 90%824 85%825 93%826 89%827 91%828 90%829 92%830 90%831 93%832 92%833 89%834 92%835 88%836 92%837 89%838 90%839 92%840 93%841 85%842 91%843 85%844 93%845 94%846 92%847 87%848 82%849 85%850 91%851 96%852 92%853 91%854 89%855 93%

856 93%857 90%858 85%859 89%860 91%861 86%862 91%863 89%864 87%865 90%866 88%867 88%868 89%869 89%870 91%871 86%872 88%873 87%874 86%875 91%876 91%877 94%878 88%879 90%880 94%881 90%882 87%883 85%884 90%885 89%886 90%887 91%888 91%889 88%890 89%891 91%892 90%893 91%894 88%895 92%896 86%

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95%85%89%92%89%93%90%94%93%89%94%87%90%86%91%88%82%92%89%93%89%93%89%93%89%94%90%93%88%91%92%90%91%88%89%92%89%92%92%86%87%

89%90%86%88%86%88%90%95%89%90%90%86%88%85%93%91%92%89%92%87%88%87%91%87%83%89%92%86%92%92%92%88%90%90%91%89%85%90%88%91%94%

85%90%94%92%92%88%91%87%89%91%95%89%87%91%89%91%92%94%90%86%91%93%93%90%87%91%92%88%89%86%96%90%90%90%90%94%89%91%90%94%89%

88%88%85%87%86%91%91%89%91%91%84%89%94%86%84%88%92%92%95%89%86%85%90%87%87%92%92%87%91%86%88%87%86%94%90%91%93%88%89%90%89%

89%89%88%92%89%90%89%91%91%89%93%92%88%93%94%85%86%82%93%92%92%92%90%89%92%86%90%91%92%89%89%92%88%88%88%94%94%91%93%85%90%

90%93%94%93%88%92%88%93%90%89%88%89%87%94%90%91%89%92%88%96%90%85%91%91%92%89%91%81%93%86%89%93%85%87%89%88%92%91%89%88%96%

91%89%95%92%87%88%92%91%88%92%90%89%87%91%89%92%88%93%89%91%87%91%88%88%90%90%91%87%95%88%92%88%95%94%94%89%90%87%92%89%93%

93%88%87%93%89%89%87%88%91%95%88%94%94%92%89%90%89%91%89%87%94%84%87%92%95%91%89%91%90%91%93%90%89%81%89%90%90%81%92%84%90%

89%87%96%85%94%93%92%88%89%94%87%94%86%87%90%92%91%88%86%86%88%93%91%91%84%92%88%89%92%86%86%88%88%90%86%93%82%93%92%91%91%

90%89%89%91%92%91%89%90%94%94%90%92%90%85%92%90%91%96%92%89%87%85%95%88%92%91%95%94%87%88%91%92%83%90%91%86%97%90%86%89%92%

94%86%91%91%86%88%90%95%93%94%91%92%89%87%91%91%93%89%94%91%87%88%82%93%96%83%91%85%89%88%95%90%89%91%95%87%89%86%82%93%94%

92%90%87%91%87%85%91%92%81%94%88%90%88%91%87%86%92%92%89%86%88%93%93%90%88%90%92%91%89%87%91%88%91%95%91%87%93%90%89%85%89%

86%97%91%93%90%85%93%88%85%93%88%87%87%94%85%91%81%92%92%93%89%88%87%91%87%89%92%91%91%89%87%88%88%91%85%92%88%95%84%88%91%

88%87%89%91%94%91%94%89%91%91%96%89%87%88%92%91%89%95%91%88%90%87%92%92%89%91%90%91%90%91%88%87%91%91%92%89%89%89%94%89%90%

93%90%85%91%92%90%89%92%93%94%94%91%87%91%87%89%95%89%86%90%92%91%93%90%92%88%89%90%92%91%90%86%93%86%89%93%90%86%85%92%87%

95%93%93%87%92%92%89%90%89%92%86%91%92%88%87%90%89%89%90%87%92%92%94%94%87%85%87%87%92%95%92%89%86%91%93%88%96%92%88%86%91%

93%93%86%82%92%86%91%87%92%80%89%95%92%89%93%94%92%88%88%91%89%86%93%90%88%94%92%87%91%88%91%93%87%89%91%93%87%90%90%88%88%

91%95%90%86%93%91%93%90%89%87%93%90%95%93%93%92%89%91%93%93%91%87%89%92%91%92%91%88%90%90%93%87%88%90%87%87%85%89%88%87%89%

91%87%91%89%91%90%89%91%89%88%90%90%86%90%93%91%92%90%88%96%91%87%93%91%92%85%91%86%91%90%92%93%86%85%94%93%91%94%89%90%93%

89%92%87%94%94%91%85%94%89%93%88%86%90%94%87%93%88%83%89%85%91%87%93%93%85%90%93%97%92%90%88%92%90%90%89%95%95%84%88%89%92%

89%85%90%94%89%89%91%89%85%86%87%92%91%92%90%85%92%94%93%89%93%91%84%92%92%91%94%92%94%91%91%92%92%87%96%93%83%89%86%90%91%

85%89%89%88%93%91%90%88%96%89%84%82%87%90%92%92%90%86%92%92%90%84%91%86%82%92%87%89%90%91%89%93%90%87%90%89%92%90%93%91%88%

93%88%85%95%89%91%92%89%85%96%91%96%90%92%95%95%93%90%84%91%91%88%86%86%92%97%91%93%91%90%87%90%91%84%87%89%92%89%91%89%93%

90%91%91%92%92%89%88%90%87%90%94%91%92%85%91%91%89%88%97%88%91%88%92%87%86%92%90%91%91%94%92%91%90%90%87%90%93%91%93%86%89%

90%92%90%94%91%92%93%91%87%87%92%89%86%90%88%88%92%90%87%86%89%88%87%88%94%88%90%92%90%88%91%90%95%85%89%89%87%91%93%92%94%

93%87%83%91%92%94%88%91%87%85%85%88%88%85%92%91%91%90%94%93%92%93%91%95%84%88%87%88%82%90%86%93%91%87%89%95%87%91%94%95%89%

91%91%91%90%88%92%91%93%89%90%90%92%91%93%90%89%94%91%91%90%90%90%90%93%92%89%88%90%92%91%91%90%86%91%95%89%98%90%88%93%96%

92%89%86%91%84%92%92%86%82%89%91%95%94%93%93%87%90%88%88%96%91%93%91%93%90%85%85%87%93%92%86%93%87%90%96%92%88%90%90%89%93%

93%87%86%82%91%92%93%88%95%93%88%90%89%84%84%86%91%92%93%90%85%91%84%90%89%89%95%92%88%91%91%89%90%85%87%85%90%91%93%94%92%

92%85%93%93%93%87%89%91%89%82%91%87%86%94%91%95%88%93%86%92%85%87%93%93%90%89%85%92%90%89%84%86%93%89%97%87%86%96%91%87%92%

90%91%92%90%87%86%95%92%89%89%94%98%89%92%90%91%87%96%88%84%90%92%89%89%90%87%85%86%89%92%90%94%94%86%91%91%91%86%95%89%90%

94%88%89%83%91%92%91%89%90%89%90%89%91%87%87%90%90%93%86%92%90%88%93%90%88%93%96%91%88%92%85%92%92%92%91%92%85%91%88%93%88%

87%92%94%95%95%90%90%93%90%89%91%95%91%95%90%88%91%93%86%92%93%87%96%93%91%89%88%87%85%87%92%94%86%93%92%95%87%87%86%90%91%

94%91%89%89%92%92%91%92%91%89%91%92%88%89%93%91%91%96%86%91%89%89%95%92%89%89%87%85%87%92%89%93%88%89%91%94%91%88%91%93%90%

92%95%86%93%87%91%93%92%89%90%89%90%90%90%93%91%89%92%89%88%87%89%88%94%91%92%87%95%91%92%92%91%93%86%89%91%88%88%87%90%88%

89%91%

Monte Carlo Simulation of Sample Percentage with 10000 Repetitions

Excel Tip: Pause the cursor over one of the bars, the number of samples (or height of the bar) is displayed.Excel Tip: Detenga el cursor sobre una de las barras, se muestra el número de muestras (o altura de la barra).

78%79%80%81%82%83%84%85%86%87%88%89%90%91%92%93%94%95%96%97%98%99%100%

0

200

400

600

800

1000

1200

1400

1600 Empirical Histogram for 10000 Repetitions

Min = 76%

Average = 89.98%

Summary Statistics

Max = 99%

0

20

40

60

80

100

120

140

160

78% 80% 82% 84% 86% 88% 90% 92% 94% 96% 98% 100%

Em pirical Histogram for 1000 Repetitions

M in = 81%

A verage = 89.90%SD = 3.034%

Summary Statistics

M ax = 98%

0

20

40

60

80

100

120

140

160

78% 80% 82% 84% 86% 88% 90% 92% 94% 96% 98% 100%

Em pirical Histogram for 1000 Repetitions

M in = 81%

A verage = 89.90%SD = 3.034%

Summary Statistics

M ax = 98%

Min = 76%

Max = 99%Average = 89.98% Pause the cursor over one of the bars, the number of samples (or height of the bar) is displayed.

Detenga el cursor sobre una de las barras, se muestra el número de muestras (o altura de la barra).

78% 279% 480% 1581% 2382% 5883% 10484% 18685% 33286% 50987% 76388% 96389% 122190% 129091% 136792% 116293% 89594% 55095% 34196% 13797% 5798% 1899% 3

100% 0

Bin Value

Number of

Samples

Monte Carlo Simulation of Sample Percentage with 1000 Repetitions

1 182 32 Time Elapsed 1.00 secs3 184 525 406 367 408 399 3010 3011 2112 2013 1314 2615 3016 2317 3218 3319 27

Sample Number

Máxima de la

Muestra

0 to 5

6 to 10

11 to 15

16 to 20

21 to 25

26 to 30

31 to 35

36 to 40

41 to 45

46 to 50

51 to 55

56 to 60

61 to 65

66 to 70

71 to 75

76 to 80

81 to 85

86 to 90

91 to 95

96 to 100

0

50

100

150

200

250 Empirical Histogram for 1000 Repetitions

Min = 9

Average = 27.0SD = 9.8

Summary Statistics

Max = 82

Monte Carlo Simulation of Sample Percentage with 1000 Repetitions

0 to 5

6 to 10

11 to 15

16 to 20

21 to 25

26 to 30

31 to 35

36 to 40

41 to 45

46 to 50

51 to 55

56 to 60

61 to 65

66 to 70

71 to 75

76 to 80

81 to 85

86 to 90

91 to 95

96 to 100

0

50

100

150

200

250 Empirical Histogram for 1000 Repetitions

Min = 9

Average = 27.0SD = 9.8

Summary Statistics

Max = 82

Resumen EstísticoPromedio = 27.0SD = 9.7Max = 100Min = 8