實用參考連結。
Creating a Histogram in Excel
Sales Forecast Example - Part III In Part II of this Monte Carlo Simulation example, we completed the actual simulation. (If you haven't already, Download the example spreadsheet). We ended up with a column of 5000 possible values (observations) for our single response variable, profit. The last step is to analyze the results. We will start off by creating a histogram in Excel, a graphical method for visualizing the results.
We can glean a lot of information from this histogram:
- It looks like profit will be positive, most of the time.
- The uncertainty is quite large, varying between -1000 to 3400.
- The distribution does not look like a perfect Normal distribution.
- There doesn't appear to be outliers, truncation, multiple modes, etc.
The histogram tells a good story, but in many cases, we want to estimate the probability of being below or above some value, or between a set of specification limits.
Creating a Histogram in Excel
Method 1: Using the Histogram Tool in the Analysis Tool-Pak.
This is probably the easiest method, but you have to re-run the tool each to you do a new simulation. AND, you still need to create an array of bins (which will be discussed below).
Method 2: Using the FREQUENCY function in Excel.
This is the method used in the spreadsheet for the sales forecast example. One of the reasons I like this method is that you can make the histogram dynamic, meaning that every time you re-run the MC simulation, the chart will automatically update. This is how you do it:
Step 1: Create an array of bins The figure below shows how to easily create a dynamic array of bins. This is a basic technique for creating an array of N evenly spaced numbers.
To create the dynamic array, enter the following formulas:
B6 = $B$2
B7 = B6+($B$3-$B$2)/5
Then, copy cell B7 down to B11
Figure 2: A dynamic array of 5 bins.
After you create the array of bins, you can go ahead and use the Histogram tool, or you can proceed with the next step.
Step 2: Use Excel's FREQUENCY formula The next figure is a screen shot from the example Monte Carlo simulation. I'm not going to explain the FREQUENCY function in detail since you can look it up in the Excel's help file. But, one thing to remember is that it is an array function, and after you enter the formula, you will need to press Ctrl+Shift+Enter. Note that the simulation results (Profit) are in column G and there are 5000 data points ( Points: J5=COUNT(G:G) ).
The Formula for the Count column:
FREQUENCY(data_array,bins_array)
a) Select cells J8:J48
b) Enter the array formula: {=FREQUENCY(G:G,I8:I48)}
c) Press Ctrl+Shift+Enter
Figure 3: Layout in Excel for Creating a Dynamic Scaled Histogram.
Creating a Scaled Histogram
If you want to compare your histogram with a probability distribution, you will need to scale the histogram so that the area under the curve is equal to 1 (one of the properties of probability distributions). Histograms normally include the count of the data points that fall into each bin on the y-axis, but after scaling, the y-axis will be the frequency (a not-so-easy-to-interpret number that in all practicality you can just not worry about). The frequency doesn't represent probability!
To scale the histogram, use the following method:
Scaled = (Count/Points) / (BinSize)
a) K8 = (J8/$J$5)/($I$9-$I$8)
b) Copy cell K8 down to K48
c) Press F9 to force a recalculation (may take a while)
Step 3: Create the Histogram Chart Bar Chart, Line Chart, or Area Chart:
To create the histogram, just create a bar chart using the Bins column for the Labels and the Count or Scaled column as the Values. Tip: To reduce the spacing between the bars, right-click on the bars and select "Format Data Series...". Then go to the Options tab and reduce the Gap. Figure 1 above was created this way.
A More Flexible Histogram Chart
One of the problems with using bar charts and area charts is that the numbers on the x-axis is actually just labels. This can make it very difficult to overlay data that uses a different number of points or to show the proper scale when bins are not all the same size. However, you CAN use a scatter plot to create a histogram. After creating a line using the Bins column for the X Values and Count or Scaled column for the Y Values, add Y Error Bars to the line that extend down to the x-axis (by setting the Percentage to 100%). You can right-click on these error bars to change the line widths, color, etc.
