How to use the BETA.DIST function

What is the BETA.DIST function?

The BETA.DIST function calculates the beta distribution. This function was introduced in Excel 2010.

1. Introduction

What is the beta distribution?

The beta distribution is a continuous probability distribution defined over the interval [0, 1] and parameterized by two positive shape parameters, alpha (α) and beta (β).

What is a continuous probability distribution?

A continuous probability distribution is a function defined over a range of continuous values that provides the probability of a random variable falling between any two points, having a density described by an equation rather than discrete probabilities.

2. Syntax

BETA.DIST(x,alpha,beta,cumulative,[A],[B])

x	Required.
alpha	Required. A parameter which determines the shape of the distribution.
beta	Required. A parameter which determines the shape of the distribution.
cumulative	Required. A boolean value that determines the form of the function. TRUE - Cumulative beta probability distribution function FALSE - Beta probability density function
[A]	Optional. Lower bound, default value 0 (zero).
[B]	Optional. Upper bound, default value 1.

What is alpha and beta in a beta distribution?
The beta distribution is very flexible due to how alpha and beta shape its variance and skew. Changing their values gives a wide range of distribution shapes. In the beta distribution, alpha and beta are shape parameters that control the form of the distribution:

Alpha (α): Controls the height of the peak. Higher alpha = taller, more concentrated peak. Alpha mainly changes the peak height.
Beta (β): Controls the tails of the distribution. Higher beta = shorter, thinner tails. Beta mainly changes the tail thickness.

As α & β > 1, shape becomes more symmetric and bell-shaped.
As α & β < 1, shape becomes asymmetrical and J or U shaped.
When α = β = 1, the distribution becomes uniform.

They allow modeling different levels of variance and asymmetry.

What is a cumulative beta probability distribution?

The cumulative beta distribution function gives the probability that a beta-distributed random variable with parameters α and β will be less than or equal to a given value x, providing the accumulated area under the probability density curve from 0 to x.

What is a beta probability density distribution?

A beta probability density distribution is a function whose shape over [0,1] depends on parameters α and β that gives the relative likelihood of a beta-distributed random variable occurring at different points, whose total area under the curve integrates to 1.

When to use the beta distribution?

The beta distribution is used to model random variables limited to intervals of 0 to 1, such as binomial success probabilities, percentage or fraction outcomes, and measurements constrained between limits, making it useful in Bayesian statistics, experimental design, weather forecasting, and other applications.

What are continuous values?

Continuous values are numbers that can take on any quantity within a range and can have infinitely many possibilities, unlike discrete values which have distinct separated values; continuous values can use intervals and ranges to describe events rather than fixed outcomes.

What are discrete probabilities?

Discrete probabilities are individual separated probabilities assigned to each of a countable number of possible outcomes that sum to 1, like rolling a die where each number has its own exact probability, as opposed to continuous distributions.

What are binomial success probabilities?

Binomial success probabilities describe the chance of a certain number of “successes” occurring in a fixed number of independent binary trial events modeled by the binomial distribution, like the probability of getting 3 heads in 10 coin flips.

What is Bayesian statistics?

Bayesian statistics is an approach to statistics using Bayes' theorem where prior beliefs about probabilities are updated as new evidence is acquired to determine conditional probabilities and update understanding of likelihood.

3. Example 1

The BETA.DIST function calculates the cumulative beta distribution representing an outcome in the form of probability between 0 and 1.  It can also calculate the probability density function, the third argument lets you select which you want to calculate. The beta distribution is often used to model the uncertainty about the value of a probability or proportion when there is some prior information available.

Alpha is the number of successes plus one and beta is the number of failures plus one. Successes and failures is a generalization, it can be satisfied customers vs not satisfied customers or what ever you want.

A company manufactures a new product. The first 10 products has 7 working and 3 faulty. Assuming that the proportion of working products follows a beta distribution with parameters α = 8 and β = 4, what is the probability that more than 90% are working for the next products in line?

Alpha is the number of working products plus one and beta is the number of faulty products plus one. The ratio between the number of working products and the total number of products is 7/10 equals 0.7 The blue line has it's highest point at x-axis value 0.7.

The image above has argument

• x in cell C18 and that value is 0.9
• alpha in cell C19
• beta in cell C20
• the cumulative argument is TRUE

Formula in cell C24:

The formula in cell C24 returns approx. 0.981 which is the cumulative beta distribution meaning the area below the blue density probability function in the chart above up to x value 0.9

The value we are looking for is the area to the right of 0.9 which we can calculate by subtracting 0.947 with 1. 1 - 0.947 equals approx. 0.0185 or 1.85%

The probability that more than 90% are working is 1.85% The orange line in the chart displayed in the image above shows the cumulative value. As time goes by and more and more products are manufactured you can update the parameters of the beta distribution model to get a better probability value.

4. Example 2

This example continues on example 1 above. The company has now manufactured 100 products, 90 working and 10 defect. What is the probability that more than 90% are working for the next products in line?

The BETA.DIST function returns approx 0.570 which gives us 1 - 0.570 equals 0.430 or 43 %. This number 43% is the probability that the products to be manufactured in the future have 90% working and 10% defect.

This example shows that as the number of observations increases the probability density curve (blue) gets more narrow meaning the uncertainty is also decreasing. In other words, the function gets better at predicting the probability as the number of observations increases.

5. Example 3

How is the upper [B] and lower [A] arguments related to the percentage value?

The optional arguments [A] and [B] are lower and upper limits. The BETADIST function uses these limits to assist you calculating the percentage value for you which is a number between 0 and 1.

You will get the same result if you calculate the percentage yourself, here is an example.

The arguments are:

• x = 30
• alpha = 90
• beta = 110
• [A] = 15
• [B] = 45

Formula in cell C7:

The formula in cell C7 returns 0.922 which is 92.2 %. To calculate the percentage (x) we calculate the ratio between x and the total of the upper and lower limit.

30/(15+45) equals 30/60 = 0.5 We can use this number in the BETADIST function without the upper and lower limit and check if we get the same result.

=BETA.DIST(0.5,90,110,TRUE)

returns 0.921957262116145 which is the exact same value as above (0.921957262116145).

6. The function is not working?

The BETADIST function returns

• #VALUE! error value if any argument is non-numeric.
• #NUM! error value if:
  • alpha <= 0
  • beta <= 0
  • x < A
  • x >B
  • A = B

6.1 Troubleshooting the error value

When you encounter an error value in a cell a warning symbol appears, displayed in the image above. Press with mouse on it to see a pop-up menu that lets you get more information about the error.

1. The first line describes the error if you press with left mouse button on it.
2. The second line opens a pane that explains the error in greater detail.
3. The third line takes you to the "Evaluate Formula" tool, a dialog box appears allowing you to examine the formula in greater detail.
4. This line lets you ignore the error value meaning the warning icon disappears, however, the error is still in the cell.
5. The fifth line lets you edit the formula in the Formula bar.
6. The sixth line opens the Excel settings so you can adjust the Error Checking Options.

Here are a few of the most common Excel errors you may encounter.

#NULL error - This error occurs most often if you by mistake use a space character in a formula where it shouldn't be. Excel interprets a space character as an intersection operator. If the ranges don't intersect an #NULL error is returned. The #NULL! error occurs when a formula attempts to calculate the intersection of two ranges that do not actually intersect. This can happen when the wrong range operator is used in the formula, or when the intersection operator (represented by a space character) is used between two ranges that do not overlap. To fix this error double check that the ranges referenced in the formula that use the intersection operator actually have cells in common.

#SPILL error - The #SPILL! error occurs only in version Excel 365 and is caused by a dynamic array being to large, meaning there are cells below and/or to the right that are not empty. This prevents the dynamic array formula expanding into new empty cells.

#DIV/0 error - This error happens if you try to divide a number by 0 (zero) or a value that equates to zero which is not possible mathematically.

#VALUE error - The #VALUE error occurs when a formula has a value that is of the wrong data type. Such as text where a number is expected or when dates are evaluated as text.

#REF error - The #REF error happens when a cell reference is invalid. This can happen if a cell is deleted that is referenced by a formula.

#NAME error - The #NAME error happens if you misspelled a function or a named range.

#NUM error - The #NUM error shows up when you try to use invalid numeric values in formulas, like square root of a negative number.

#N/A error - The #N/A error happens when a value is not available for a formula or found in a given cell range, for example in the VLOOKUP or MATCH functions.

#GETTING_DATA error - The #GETTING_DATA error shows while external sources are loading, this can indicate a delay in fetching the data or that the external source is unavailable right now.

6.2 The formula returns an unexpected value

To understand why a formula returns an unexpected value we need to examine the calculations steps in detail. Luckily, Excel has a tool that is really handy in these situations. Here is how to troubleshoot a formula:

1. Select the cell containing the formula you want to examine in detail.
2. Go to tab “Formulas” on the ribbon.
3. Press with left mouse button on "Evaluate Formula" button. A dialog box appears.
   The formula appears in a white field inside the dialog box. Underlined expressions are calculations being processed in the next step. The italicized expression is the most recent result. The buttons at the bottom of the dialog box allows you to evaluate the formula in smaller calculations which you control.
4. Press with left mouse button on the "Evaluate" button located at the bottom of the dialog box to process the underlined expression.
5. Repeat pressing the "Evaluate" button until you have seen all calculations step by step. This allows you to examine the formula in greater detail and hopefully find the culprit.
6. Press "Close" button to dismiss the dialog box.

There is also another way to debug formulas using the function key F9. F9 is especially useful if you have a feeling that a specific part of the formula is the issue, this makes it faster than the "Evaluate Formula" tool since you don't need to go through all calculations to find the issue..

1. Enter Edit mode: Double-press with left mouse button on the cell or press F2 to enter Edit mode for the formula.
2. Select part of the formula: Highlight the specific part of the formula you want to evaluate. You can select and evaluate any part of the formula that could work as a standalone formula.
3. Press F9: This will calculate and display the result of just that selected portion.
4. Evaluate step-by-step: You can select and evaluate different parts of the formula to see intermediate results.
5. Check for errors: This allows you to pinpoint which part of a complex formula may be causing an error.

The image above shows cell reference C3 converted to hard-coded value using the F9 key. The BETA.DIST function requires numerical values in the first argument which is not the case in this example. We have found what is wrong with the formula.

Tips!

• View actual values: Selecting a cell reference and pressing F9 will show the actual values in those cells.
• Exit safely: Press Esc to exit Edit mode without changing the formula. Don't press Enter, as that would replace the formula part with the calculated value.
• Full recalculation: Pressing F9 outside of Edit mode will recalculate all formulas in the workbook.

Remember to be careful not to accidentally overwrite parts of your formula when using F9. Always exit with Esc rather than Enter to preserve the original formula. However, if you make a mistake overwriting the formula it is not the end of the world. You can “undo” the action by pressing keyboard shortcut keys CTRL + z or pressing the “Undo” button

6.3 Other errors

Floating-point arithmetic may give inaccurate results in Excel - Article

Floating-point errors are usually very small, often beyond the 15th decimal place, and in most cases don't affect calculations significantly.

7. How is the function calculated mathematically?

The general equation to calculate the beta distribution:

The text representation of the beta distribution math formula is: x^(α-1)(1 - x)^(β-1) / B(α - β)

Where:

• x is the variable
• α and β are the shape parameters
• B(α - β) is the beta function evaluated at (α - β)

Useful resources

Functions in 'Statistical' category

The BETA.DIST function function is one of 73 functions in the 'Statistical' category.

Excel function categories

Excel categories

Leave a Comment