Whenever we deal with large amounts of data, one of the goals for analysis is,
How is this data distributed?
This is where a Box plot can help. According to Wikipedia, a box plot is a convenient way of graphically depicting groups of numerical data through their five-number summaries: the smallest observation (sample minimum), lower quartile (Q1), median (Q2), upper quartile (Q3), and largest observation (sample maximum) [more]
Quartile?!? What is that like?
When we say $ 39,000 is the lower quartile of salaries paid in Acme inc. it means, 25% of people make less than or equal to $39,000
Like that Median (Q2) means half the samples are lower than median & the other are more than median.
Example Box Plot
Here is an example box plot depicting salaries of all analysts in USA as per our recent Excel Salary Survey.
The box shows distribution of middle half of data (salaries) while the lines (called as whiskers) show minimum and maximum salaries.
As you can see, 50% of the analysts make between $46,000 to $75,000 while the min is $10k and max is $160k.
Why use Box plots?
Box & whisker plots are an excellent way to show distribution of your data without plotting all the values. They are easy to understand. We can use them whenever we have lots of data or dealing with samples drawn from larger population.
Creating Box plots in Excel – 9 step tutorial
Despite their utility, Excel has no built-in option to make a box plot. Of course you can make a 3D pie chart or stacked horizontal pyramid chart. Lets save them for your last day at work and understand how to create box plots in Excel.
Step 1: Calculate the number summaries
Assuming your data is in list use formulas MIN, MAX & PERCENTILE to calculate summaries like below:
To calculate 25th percentile (Q1) use = PERCENTILE(list, 25%)
Step 2: Make a bar chart from Q1, Median & Q3
Just select the 25th percentile, median & 75th percentile values and create a bar chart.Make sure that your chart shows 3 different colored bars not 3 bars in one color.
Step 3: Set series overlap to 100%
Select any bar, press CTRL+1 (right click > format series) and adjust series overlap to 100%
Step 4: Adjust series order so that you can see all the bars
If you cannot see all the bars, right click on chart, click on “Select data”.
Now, adjust the series order using arrow keys so that you can see all the bars. See this demo:
Step 5: Make 25th percentile (Q1) bar invisible
Select the bar corresponding to Q1 and fill it with white color. If you make it transparent, it will not work. So make it all white.
Step 6: Add error bars to Q1 & Q3 series
Just select Q1 (25th percentile) bar and add error bar (any type) from layout ribbon.
Repeat for Q3 series as well.
Step 7: Set up error values in your data
Add an extra column in your data area and use simple formulas to calculate error values, like below:
Step 8: Set up custom error values for Q1 & Q3
Select the error bar for Q1 (25th percentile) and,
- Press CTRL+1 to format them
- Enable only minus (negative) error bar with no cap.
- Select Custom as error amount and point to the calculated value.
Repeat for Q3, but choose positive error bar instead.
Step 9: Format the box plot to your taste
Remove any legend, axis, labels that you do not need. Change colors to suit your taste and mood. Make the whiskers subtle and knock off the grid lines. You are good to go.
Making Box plots interactive
Since box plots are very useful to understand distribution of values, we use them in dashboards etc. Naturally, you are interested to know how values are distributed for various things.
In this example, we may want to know how analyst salaries compare with manager salaries.
To make things complicated, we have 10 different job types, thus enabling 45 possible comparisons (10c2)
This is where interactive box plots can help. See this demo to understand:
Interactive Box plot in Excel – a Demo
How to make interactive box plot in Excel
Construction of box plot is same as mentioned above. The difference is in adding interactivity.
Step 1: Use combo box form controls to capture comparison criteria
Excuse the tongue twister. Using Developer ribbon > Insert > Form controls, add 2 combo box controls and point them to the list of job types.
Lets assume that these combo boxes are linked to cells D1, D2.
[Related: Introduction to Excel Form Controls]
Step 2: Calculate 5 number summaries using MINIF, MAXIF and PERCENTILEIF formulas
Don’t rush to type the formulas yet. There is no such formula as MINIF (or MAXIF or PERCENTILEIF). Assuming your list of jobs are in joblist, write
=MIN(IF(joblist=”Analyst”, list_of_values,””))
and press CTRL+Shift+Enter
Using MAX(IF(…)) and PERCENTILE(IF(…)) you can calculate remaining 4 summaries.
Step 3: Based on combo box selection, fetch any two sets of values
Using INDEX formula, we can fetch values corresponding to each combo box selection to a set of cells, like this:
Step 4: Connect these values to your box plots
That simple!
Step 5: Format and interact
Format the charts. Play with combo boxes to interactively compare one set of distribution with another. Show it to your boss or client and see them fall off a chair.
Download Box plot tutorial workbook
Click here to download the workbook containing these examples. Play with it. Check out various formulas and chart settings. Learn.
Do you use Box plots?
I love box plots. I have used them several times. Few examples are here: Excel age survey results, Gantt box chart and more.
In our Excel salary survey contest too, many people have used box plots to clearly compare compensation composition. Checkout the entries by Aditya, Allred, Anchalee, Anup, Bryan, Jeanmarc, Joerg, Kostas, Luke, Michael, Nathan, Sergey and Vishwanath. Especially Jeanmarc used interactive version of box plots to allow comparison on demand.
What about you? Do you use Box plots often? How do you prepare them? What is your experience like? Please share using comments.
Create Box plots often? Use Jon’s Add-in
If you need to create box plots often and find the above process tedious, then please consider getting a copy of Jon Peltier’s Box Plot add-in for Excel. It works like a charm and produces what you need. All in a few clicks. Click here to know more.
PS: Link to Jon’s add-in is an affiliate link. It means, when you buy it from Jon thru this link, I will get a few bucks too. I recommend it because I know it is awesome and perfect for box plots.
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