How to use the PHI function

What is the PHI function?

The PHI function calculates a number of the density function for a standard normal distribution.

1. Introduction

What is a density function?

A density function (PDF) in statistics describes the relative likelihood that a random variable takes on a given value. The area under the entire density function integrates to 1.

A cumulative density function (CDF) represents the probability that the random variable will take a value less than or equal to a specified value.

What is the difference between continuous random variables and discrete random variables?

Continuous random variables can take on any value within a continuous range. Examples: height, weight, temperature.

Discrete random variables can only take on specific discrete values within a finite/countable set. Examples: number of items, dice rolls.

What is a normal distribution?

A normal distribution, also known as a Gaussian distribution, is a continuous probability distribution that is symmetric about the mean and defined by two parameters - the mean (μ) and standard deviation (σ).

The formula for the normal probability density function is:

f(x) = (1 / (σ√(2π))) * e^((-1/2) * ((x - μ) / σ)^2)

Where:
f(x) = probability density function
x = value
μ = population mean
σ = population standard deviation
e = exponential constant (Euler's number)
π = mathematical constant pi

What is a standard normal distribution?

A standard normal distribution is a normal distribution with the mean of 0 (zero) and the standard deviation is 1. You can standardize any normal distribution using the STANDARDIZE function in Excel, it works like this:

z = (x - µ)/σ

z = z-score
µ is the mean.
σ is the standard deviation.

What is the mean?

It is also known as the average. It is calculated by adding up all the values in the data set and dividing by the number of values.

For example, if you have a data set of 5, 7, 9, 11, and 13, the mean is (5 + 7 + 9 + 11 + 13) / 5 = 9.

Use the AVERAGE function to calculate the mean.

What is standard deviation?

Standard deviation is a measure of dispersion that indicates how spread out the values in a dataset are from the mean. It is represented by the Greek letter sigma (σ).

The formula for calculating standard deviation is:

σ = √Σ(x - μ)2 / (N - 1)

Where:
σ = Standard deviation
Σ = Sum of
x = Values in the dataset
μ = Mean of the dataset
N = Number of values in the dataset
(N - 1) = Sample correction factor

Use the STDEV.S function or STDEV.P function to calculate the standard deviation.

What is z-score?

A z-score, also known as a standard score, is a measure of how many standard deviations a data point is away from the mean of a distribution.

The formula to calculate the z-score is:

z = (x - μ) / σ

x = the data point
μ = the mean or average of the distribution
σ = the standard deviation of the distribution

What is the difference between the PHI function and the NORM.DIST function?

PHI has no arguments for mean and standard deviation. It assumes a standard normal distribution (mean 0, standard deviation 1). NORM.DIST allows specifying mean and standard deviation.

PHI(x) is equivalent to NORM.DIST(x,0,1,FALSE) for a standard normal distribution. The NORM.DIST function can calculate both the cumulative and the dnesity functions, the third argument TRUE or FALSE determines the distribution function.

2. Syntax

PHI(x)

3. Arguments

x	Required. x is the number for which you want to calculate the density of the standard normal distribution.

4. Example 1

This example demonstrates the different values of the standard normal distribution between x=-4 and x=4 with an increment of 0.5

Cell range B3:B19 contains the x values starting with -4 with a step value of 0.5. Cell range C3:C19 contains the PHI function, it calculates the density function based on the corresponding x values in B3:B19.

Formula in cell C3:

The image above also shows a chart displaying the standard normal distribution using the PHI function to calculate the actual values.

5. Example 2

A company measures the weight of its product packages. The weights are normally distributed with a mean of 500 grams and a standard deviation of 20 grams. What is the probability that a randomly selected package will weigh exactly 530 grams?

The z-score is calculated like this:

z = (x - μ) / σ

x = the data point
μ = the mean or average of the distribution
σ = the standard deviation of the distribution

(C18-C19)/C20 becomes (530-500)/20 = 1.5 This value is calculated in cell C22.

Formula in cell C24:

The formula returns 0.12952 (12.95%) which means this is the probability of a package weighing 530 based on a mean of 500 and a standard deviation of 20. This is true if the observations follow a normal distribution.

The image above shows a graph displaying the normal distribution. A vertical line shows the x value of 1.5, and the horizontal line shows the y value of approx 0.1295. The intersection between these lines also shows the data point on the blue curve which represents the normal distribution.

6. Example 3

A researcher measured an interesting process in nature, 40 observations were made, shown below. Calculate the mean, standard deviation. What is the probability that an observation is exactly 22? Here are the observations:

23	8	29	33
36	38	36	42
7	20	37	20
26	50	25	31
28	29	38	37
33	27	34	29
32	43	38	31
30	28	23	41
27	26	25	24
28	27	38	44

To calculate the z-score we first need to calculate the mean using the AVERAGE function. Cell C19 calculates the mean:

The calculation in cell C19 returns 30.5 based on the observations in cell range F19:I28. We also need the standard deviation to calculate the z-score. The STDEV.S function allows us to calculate the standard deviation based on a sample of the population. Cell C20 calculates the standard deviation:

To calculate the z-score:

z = (x - μ) / σ

x = the data point
μ = the mean or average of the distribution
σ = the standard deviation of the distribution

Cell C22 calculates the z-score based on the x value in C18 minus the the average in c19, the difference is then divided by the standard deviation in cell C20:

Cell C23 calculates the probability:

The formula returns 0.247 (24.7%) which means this is the probability of an observation is 22 based on a mean of 30.5 and a standard deviation of 8.7. This is true if the observations follow a normal distribution.

The image above shows a graph displaying the normal distribution. A vertical line shows the x value (z-score) of -0.979, and the horizontal line shows the y value of approx 0.25. The intersection between these lines also shows the data point on the blue curve which represents the normal distribution.

7. Function not working

The PHI function returns:

• #NUM# error value if the argument is an invalid numeric value.
• #VALUE! error value if the argument is an invalid data type, like a non numeric value.
• #NAME? error if you misspell the function name.
• propagates errors, meaning that if the input contains an error (e.g., #VALUE!, #REF!), the function will return the same error.

7.1 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.

7.2 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 B3 converted to hard-coded value using the F9 key. The PHI function requires valid numerical values 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

7.3 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.

Functions in 'Statistical' category

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

Excel function categories

Excel categories

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