# Normal Distribution’s PDF (Probability Density Function) in Excel

This is one of the following eight articles on the normal distribution in Excel

Overview of the Normal Distribution

Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013

Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013

Solving Normal Distribution Problems in Excel 2010 and Excel 2013

Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013

An Important Difference Between the t and Normal Distribution Graphs

The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean

Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

The normal distribution is a family of distributions with each unique normal distribution being fully described by its two parameters µ (“mu,” population mean) and σ (“sigma,” population standard deviation). The population mean, µ, is a location parameter and the population standard deviation, σ, is a scale parameter. When two different normal distribution curves are plotted on the same set of horizontal and vertical axes, the means determine how shifted one curve is to the left of right of the other curve. The standard deviations detail how much wider or more narrow the first normal curve is than to the second.

The normal distribution is often denoted as N(µ,σ2). σ2 equals the population variance. When a random variable X is normal-distributed with a population mean µ and population variance σ2, it is written in the following form:

X ~ N(µ, σ2)

It is important to note that the two parameters of the normal distribution are population parameters, not measurements taken from a sample. Sample statistics would provide only estimates of population parameters. The t-Distribution is used to analyze normally-distributed data when only sample statistics and/or population parameters are not known. In the real world it is much more common to analyze normally-distributed data with t-Distribution based tests than normal-distribution-based tests because only data from small samples (n<30) are available.

As with all distributions, the normal distribution has a PDF (Probability Density Function) and a CDF (Cumulative Distribution Function).

The normal distribution’s PDF (Probability Density Function) equals the probability that sampled point from a normal-distributed population has a value EXACTLY EQUAL TO X given the population’s mean, µ, and standard deviation, σ.

The normal distribution’s PDF is expressed as f(X,µ,σ).

f(X,µ,σ) = the probability that a randomly-sampled point taken from normally-distributed population with a mean µ and standard deviation σ has the value of X. It is given by the following formula: (Click On Image to See a Larger Version)

exp refers to the value of the mathematical constant e which is the base of the natural logarithm. e is equal to 2.71828 and is the limit of (1 + 1/n)n and n approaches infinity. ea would be expressed in Excel as =exp(a).

The mathematical constant π (“pi”) is equal to 3.14159 and is the ratio of a circle’s circumference to its diameter.

In Excel 2010 and beyond, the normal distribution’s PDF can be calculated directly by the following Excel formula:

f(X,µ,σ) = NORM.DIST(X,µ,σ,FALSE)

The Excel formula parameter “FALSE” indicates that the formula is calculating the normal distribution’s PDF (Probability Density Function) and not its CDF (Cumulative Distribution Function)

Prior to Excel 2010, the normal distribution’s PDF was calculated in Excel by this formula:

f(X,µ,σ) = NORMDIST(X,µ,σ,FALSE)

Statistical formulas that worked in Excel versions prior to 2010 will also work in Excel 2010 and 2013.

The following Excel-generated graph shows the PDF of a normal distribution that has a population mean of 10 and population standard deviation equal to 5. (Click On Image to See a Larger Version)

## Normal Distribution PDF Example in Excel

Determine the probability that a randomly-selected variable X taken from a normally-distributed population has the value of 5 if the population mean equals 10 and the population standard deviation equals 5. The preceding Excel-generated image shows a normal distribution PDF curve with the population mean equaling 10 and the population standard deviation equaling 5.

X = 5

µ = 10

σ = 5

f(X,µ,σ) = NORM.DIST(X,µ,σ,FALSE)

f(X=5,µ=10,σ=5) = NORM.DIST(5,10,5,FALSE) = 0.04834

There is a 4.834 percent chance that randomly-selected X = 5 if X is taken from a normally-distributed population with a population mean µ = 10 and population standard deviation σ = 5. The PDF diagram of this normal distribution curve also shows the probability of X at X = 5 to that value.

Performing the same calculation in Excel using the full normal distribution PDF formula as shown as follows: (Click On Image to See a Larger Version)

f(X=5,µ=10,σ=5) =(1/(SQRT(2*3.14159*5^2)))*EXP((-1)*((5-10)^2)/(2*5^2))

= 0.04834

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