Monday, June 2, 2014

An Important Difference Between t and Normal Distribution Graphs

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

An Important Difference Between t and Normal Distribution Graphs

t-Distribution

The t-Distribution is a family of distribution curves that are all symmetrical about a mean of zero. Each unique curve of the t-Distribution is completely described and fully differentiated from all other t-Distribution curves by its degrees of freedom, which is designated by either df or v (Greek letter “nu”). Degrees of freedom is related to sample size by df = n – 1.

The t-Distribution’s PDF formula calculates the PDF at the t Value of point X. The t Value of point X is the number of standard errors that point X is from the sample mean. When calculating the t-Distribution’s PDF (or CDF) at point X, the t Value of point X must be computed for that point X. The t Value of point X is the required input of the t-Distribution’s PDF (and CDF) formula.

The t Value of point X must be determined as follows:

(Click On Image To See a Larger Version)

before the t-Distribution’s PDF can be calculated by the following formula:

(Click On Image To See a Larger Version)

The Excel formula to calculate the t-Distribution’s PDF is shown here as well.

f(t,v) = T.DIST(t, df, FALSE)

The t Value of the point X must also be calculated before calculating the t-Distribution’s CDF, which can be found with the following Excel formula:

F(t,v) = T.DIST(t, df, TRUE)

Normal Distribution

The normal distribution is also a family of distribution curves. Each distribution curve is fully described and completely differentiated from all other normal curves by only two parameters: its mean, μ, and standard deviation, σ.

The normal distribution’s PDF at point X is calculated directly by using point X as one of the inputs of the normal distribution’s PDF formula as follows:

(Click On Image To See a Larger Version)

The Excel formula to calculate the normal distribution’s PDF is shown here as well.

f(X,μ,σ) = NORM.DIST(X, μ, σ, FALSE)

The t-Distribution requires that point X be converted to a t Value before calculating the PDF or CDF at point X. The normal distribution does not require such a conversion because point X is one of the inputs to and part of its PFD and CDF formulas.

The normal distribution’s CDF formula in Excel is as follows:

F(X,μ,σ) = NORM.DIST(X, μ, σ, TRUE)

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