In mid May, Anup47 asked a question in the Chandoo.org forums about the use of a VBA macro to run a number of iterations of a variable against two sets of X values, you can see the post here. It turns out that the number of iterations was 500 columns of data with each column having 27 values.
On examination of the problem, it was going to be a straight forward matter of setting up a statistical function Linest and then using the Data Table command to run each set of data through the function.
The Linest will take the input data and return the statistics that Anup wanted.
The Data Table function will feed in the source data and tabulate the Input and Output data.
This Post follows through a worked example which you can follow along, download the Sample file to suit Sample File 97/2003 or Sample File 2007/10 version. The Sample File contains a worked example of the completed model as well as a Practice Page of the original data. Download the Excel 95/2003 or 2007/10 version above.
Please note that the sample file only contains 14 sets of data as opposed to the 500 Anup47 wanted to process.
Setup
There are a few things that needed setting up before the work starts.
- Headers
- Linest Area
- Link Area
- Data Table Area
Once these areas are setup we simply use the Excel Data Table function.
Once the Data Table function has run, the results can be processed or analysed as required.
Headers
The original data was just that, a tabulation of raw data. The two X sets of Data were in Columns 1 & 2. Each Column from D onwards has a set of Y data that was to be processed.
The first thing that was required was some Headers for the Input Data.
This isn’t strictly required but it is good practice and makes it easier to tabulate and analyse results later.
Insert a Row above the first line
Put X1, X2 in A1, B1 and Y1 in D1 and then drag the lower right Black Handle across top to the right and Excel will autofill the remaining cells.
Linest Area
To get the statistics which Anup wanted we will use the Excel Linest function.
Linest is a Statistical Function that takes a set of data and compares it, in this case to two sets of X Values and produces a set of statistical measure relevant to the correlation between the data sets.
This post isn’t going to explain the intricacies of Linest and I refer you to the Links section at the end where you can read more about the Linest function at your leisure.
For our purposes we need to know that Linest is an Array Formula and requires a 5 Row x 5 Column area to be entered into. For now we will just Array Enter the function =Linest($D$2:$D$28,A2:B28,True, True) into B32:F36.
To do that select the range B32:F36, Press F2 and type/paste the equation in, then Array Enter with Ctrl Shift Enter.
Link Area
To Link the Linest equation to a Data Table we need a link cell, which we will put just above the Linest area.
For now just enter a 1 in it.
We can now go back to the Linest area and link the Linest equation to our link area using the equation, =LINEST(OFFSET($C$2:$C$28,,$B$30),A2:B28,TRUE, TRUE)
To do that select the range B32:F36, Press F2 and type/paste the equation in, then Array Enter with Ctrl Shift Enter.
What this does is allow the Linest formula to access different columns Y1 to Y500 depending on the value of the Link cell B30 which is now 1.
Data Table Area
To setup a Data Table area we need a column of Inputs which will be the Run Numbers and the Row Inputs will be links to the Input and Output Cells.
In a range J33:J46 put the values 1 to 14. These will be the Run Numbers. ie Run No 1, Run No 2 etc (Green in the example below).
Across the top of the Data Table area we can put a number of links and associated labels (Yellow and Blue)
In this case there are 4 Output links =B31, =C31, =B34 and =B33 and their associated labels above them, as well as 2 Input equations and there Labels. The Input equations are simple Offset function that retrieves a value from Rows 1 or 2 based on the value of the Link Cell B30.
These are technically not required but make data analysis and identification of individual results later on a lot simpler.
Run Data Table
We can now run the data Table by selecting the Data Table area: J32:P46
Noting that we will be using a Column Input cell and that it will link to $B$30, the Link cell for the Linest command.
What this does is takes the first value from the Column J32:J46 and puts it into B30, then the Linest command will be calculated and the results put into the Data Table area along with the Inputs.
This is repeated for each cell in J32:J46 automatically.
The final Data Table is now populated as below:
You can see by extending the Data Table input column from 14 to 500 that the full 500 columns of Input Data could easily be processed.
Results
You now have a set-off data that can be analyzed using normal statistics, Min, Max, Std Deviation etc, or can be fed into a Pivot Table/Chart for analysis etc.
References
Linest References
http://chandoo.org/wp/2011/01/26/trendlines-and-forecasting-in-excel-part-2/
http://newtonexcelbach.wordpress.com/2011/01/19/using-linest-for-non-linear-curve-fitting/
Data Table References
http://chandoo.org/wp/2010/05/06/data-tables-monte-carlo-simulations-in-excel-a-comprehensive-guide/
How can the Data Table command help you become a data processing super hero?
How can the Data Table command help you become a data processing super hero?
Let us know in the comments below:
20 Responses to “Simulating Dice throws – the correct way to do it in excel”
You have an interesting point, but the bell curve theory is nonsense. Certainly it is not what you would want, even if it were true.
Alpha Bravo - Although not a distribution curve in the strict sense, is does reflect the actual results of throwing two physical dice.
And reflects the following . .
There is 1 way of throwing a total of 2
There are 2 ways of throwing a total of 3
There are 3 ways of throwing a total of 4
There are 4 ways of throwing a total of 5
There are 5 ways of throwing a total of 6
There are 6 ways of throwing a total of 7
There are 5 ways of throwing a total of 8
There are 4 ways of throwing a total of 9
There are 3 ways of throwing a total of 10
There are 2 ways of throwing a total of 11
There is 1 way of throwing a total of 12
@alpha bravo ... welcome... 🙂
either your comment or your dice is loaded 😉
I am afraid the distribution shown in the right graph is what you get when you throw a pair of dice in real world. As Karl already explained, it is not random behavior you see when you try to combine 2 random events (individual dice throws), but more of order due to how things work.
@Karl, thanks 🙂
When simulating a coin toss, the ROUND function you used is appropriate. However, your die simulation formula should use INT instead of ROUND:
=INT(RAND()*6)+1
Otherwise, the rounding causes half of each number's predictions to be applied to the next higher number. Also, you'd get a count for 7, which isn't possible in a die.
To illustrate, I set up 1200 trials of each formula in a worksheet and counted the results. The image here shows the table and a histogram of results:
http://peltiertech.com/WordPress/wp-content/img200808/RandonDieTrials.png
@Jon: thanks for pointing this out. You are absolutely right. INT() is what I should I have used instead of ROUND() as it reduces the possibility of having either 1 or 6 by almost half that of having other numbers.
this is such a good thing to learn, helps me a lot in my future simulations.
Btw, the actual graphs I have shown were plotted based on randbetween() and not from rand()*6, so they still hold good.
Updating the post to include your comments as it helps everyone to know this.
By the way, the distribution is not a Gaussian distribution, as Karl points out. However, when you add the simulations of many dice together (i.e., ten throws), the overall results will approximate a Gaussian distribution. If my feeble memory serves me, this is the Central Limit Theorem.
@Jon, that is right, you have to nearly throw infinite number of dice and add their face counts to get a perfect bell curve or Gaussian distribution, but as the central limit theorem suggests, our curve should roughly look like a bell curve... 🙂
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I'm afraid to say that this is a badly stated and ambiguous post, which is likely to cause errors and misunderstanding.
Aside from the initial use of round() instead of int(),.. (you've since corrected), you made several crucial mistakes by not accurately and unambiguously stating the details.
Firstly, you said:
"this little function generates a random fraction between 0 and 1"
Correctly stated this should be:
"this little function generates a random fraction F where 0 <= F < 1".
Secondly, I guess because you were a little fuzzy about the exact range of values returned by rand(), you have then been just as ambiguous in stating:
"I usually write int(rand()*12)+1 if I need a random number between 0 to 12".
(that implies 13 integers, not 12)
Your formula, does not return 13 integers between 0 to 12.
It returns 12 integers between 1 and 12 (inclusive).
-- As rand() returns a random fraction F where 0 <= F < 1, you can obviously can only get integers between 1 and 12 (inclusive) from your formula as stated above, but clearly not zero.
If you had said either:
"I usually write int(rand()*12) if I need a random number between 0 to 11 (inclusive)",
or:
"I usually write int(rand()*12)+1 if I need a random number between 1 to 12 (inclusive)"
then you would have been correct.
Unfortunately, you FAIL! -- repeat 5th grade please!
Your Fifth Grade Maths Teacher
Idk if I'm on the right forum for this or how soon one can reply, but I'm working on a test using Excel and I have a table set up to get all my answers from BUT I need to generate 10,000 answers from this one table. Every time, I try to do this I get 10,000 duplicate answers. I know there has to be some simple command I have left out or not used at all, any help would be extremely helpful! (And I already have the dice figured out lol)
Roll 4Dice with 20Sides (4D20) if the total < 20 add the sum of a rerolled 2D20. What is the average total over 10,000 turns? (Short and sweet)
Like I said when I try to simulate 10,000turns I just get "67" 10,000times -_- help please! 😀
@Justin
This is a good example to use for basic simulation
have a look at the file I have posted at:
https://rapidshare.com/files/1257689536/4_Dice.xlsx
It uses a variable size dice which you set
Has 4 Dice
Throws them 10,000 times
If Total per roll < 20 uses the sum of 2 extra dice Adds up the scores Averages the results You can read more about how it was constructed by reading this post: http://chandoo.org/wp/2010/05/06/data-tables-monte-carlo-simulations-in-excel-a-comprehensive-guide/
Oh derp, i fell for this trap too, thinking i was makeing a good dice roll simulation.. instead of just got an average of everything 😛
Noteably This dice trow simulate page is kinda important, as most roleplay dice games were hard.. i mean, a crit failure or crit hit (rolling double 1's or double 6's) in a a game for example dungeons and dragons, if you dont do the roll each induvidual dice, then theres a higher chance of scoreing a crit hit or a crit failure on attacking..
I've been working on this for awhile. So here's a few issues I've come across and solved.
#1. round() does work, but you add 0.5 as the constant, not 1.
trunc() and int() give you the same distributions as round() when you use the constant 1, so among the three functions they are all equally fair as long as you remember what you're doing when you use one rather than the other. I've proven it with a rough mathematical proof -- I say rough only because I'm not a proper mathematician.
In short, depending on the function (s is the number of sides, and R stands in for RAND() ):
round(f), where f = sR + 0.5
trunc(f), where f = sR + 1
int(f), where f = sR + 1
will all give you the same distribution, meaning that between the three functions they are fair and none favors something more than the others. However...
#2. None of the above gets you around the uneven distribution of possible outcomes of primes not found in the factorization of the base being used (base-10, since we're using decimal; and the prime factorization of 10 is 2 and 5).
With a 10-sided die, where your equation would be
=ROUND(6*RAND()+0.5)
Your distribution of possible values is even across all ten possibilities.
However, if you use the most basic die, a 6-sided die, the distributions favor some rolls over others. Let's assume your random number can only generate down to the thousandths (0.000 ? R ? 0.999). The distribution of possible outcomes of your function are:
1: 167
2: 167
3: 166
4: 167
5: 167
6: 166
So 4 and 6 are always under-represented in the distribution by 1 less than their compatriots. This is true no matter how many decimals you allow, though the distribution gets closer and closer to equal the further towards infinite decimal places you go.
This carries over to all die whose numbers of sides do not factor down to a prime factorization of some exponential values of 2 and 5.
So, then, how can we fix this one, tiny issue in a practical manner that doesn't make our heads hurt or put unnecessary strain on the computer?
Real quick addendum to the above:
Obviously when I put the equation after the example of the 10-sided die, I meant to put a 10*RAND() instead of a 6*RAND(). Oops!
Also, where I have 0.000 ? R ? 0.999, the ?'s are supposed to be less-than-or-equal-to signs but the comments didn't like that. Oh well.
How do you keep adding up the total? I would like to have a cell which keeps adding up the total sum of the two dices, even after a new number is generated in the cells when you refresh or generate new numbers.
So, how do you simulate rolling 12 dice? Do you write int(rand()*6) 12 times?
Is there a simpler way of simulating n dice in Excel?
I've run this code in VBA
Sub generate()
Application.ScreenUpdating = False
Application.Calculation = False
Dim app, i As Long
Set app = Application.WorksheetFunction
For i = 3 To 10002
Cells(i, 3).Value = i - 2
Cells(i, 4).Value = app.RandBetween(2, 12)
Cells(i, 5).Value = app.RandBetween(1, 6) + app.RandBetween(1, 6)
Next
Application.ScreenUpdating = True
Application.Calculation = True
End Sub
But I get the same distribution for both columns 4 and 5
Why ?
@Mohammed
I would expect to get the same distribution as you have effectively used the same function