Excel Tip of the Week #162 - Normal distribution functions

Hello all and welcome back to the Excel Tip of the Week!  This week, we have a Creator post in which we are looking at the functions for calculating various statistics related to the normal distribution - namely NORM.DIST, NORM. INV, NORM.S.DIST, NORM.S.INV, and STANDARDIZE.

Note that this post is intended to talk about how to use these functions in Excel - it is not a probability primer!

Also note that the functions we discuss here are those present in Excel 2010 and later - very similar functions (without the full stops) are present in Excel 2007 and earlier, and are available for compatibility reasons only in later versions.

Throughout this Tip, I will be using a well-known normal distribution as an example to illustrate each of the points - namely women's heights in inches.  I am using a mean of 63.6 inches and a standard deviation of 2.5 inches.

How to write these functions

=NORM.DIST(x, mean, standard deviation, cumulative)

=NORM.INV(probability, mean, standard deviation)

=NORM.S.DIST(z, cumulative)


=STANDARDIZE(x, mean, standard deviation)

x - The value being tested
mean - The mean of the normal distribution in question (assumed to be 0 for the NORM.S functions)
standard deviation - The standard deviation of the normal distribution in question (assumed to be 1 for the NORM.S functions)
cumulative - An option setting - TRUE or 1 will return the cumulative distribution function (i.e. the probability that the value will be less than or equal to x), and FALSE or 0 will return the probability density function (the probability that the value will be exactly x).
probability - The probability that is being worked back from for the inverse functions
z - The z-value for the standard normal distribution - essentially the same as the x-value for NORM.DIST

NORM.DIST tells us the probability for a result at a particular value - either the spot probability of hitting that exact value, or the cumulative probability of seeing that value or lower.  For example, we could calculate the probability that a woman is 5'7" (67 inches) as:
     =NORM.DIST(67, 63.6, 2.5, FALSE) = 6.3%
And the probability that a woman is 5'7" or shorter would be:
     =NORM.DIST(67, 63.6, 2.5, TRUE) = 91.3%
The probability that a woman is between 5' and 6' is the difference between these two functions:
     =NORM.DIST(72, 63.6, 2.5, TRUE) - NORM.DIST(60, 63.6, 2.5, TRUE) = 99.96% - 7.49% = 92.47%

NORM.INV does the reverse - that is, if we want a given likelihood of a certain event happening, what threshold should we set?  For example, if we want to find the lower quartile of women's heights:
     =NORM.INV(25%, 63.6, 2.5) = 61.9 inches
We can invert this by considering symmetry - for example, if we want a threshold that 60% of women are taller than, we can find the one that 40% of women are shorter than.

NORM.S.DIST and NORM.S.INV are essentially the same functions if the mean is 0 and the standard deviation is 1; these functions are used more in raw statistics.  STANDARDIZE counts how many standard deviations from the mean a certain value is.

Generating random numbers that conform to a known distribution

If you want to generate random numbers that follow a given mean and standard deviation, you can do so by using a RAND() function as the input to a NORM.INV function.  This is far fairer for creating dummy data than creating truly random data.  Check out the example in the attached file.

Previous post
TOTW index 
Next post

This blog is brought to you by the Excel Community where you can find additional blogs, extended articles and webinar recordings on a variety of Excel related topics. In addition to live training events, Excel Community members have access to a full suite of online training modules from Filtered. There is also an online forum where you can ask questions and share ideas with other community members.