## Monday, June 2, 2014

### Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013 (Click On Image To See a Larger Version)

This is one of the following eleven articles on creating user-interactive graphs of statistical distributions in Excel

Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013

Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013

Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013

Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013

Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013

Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

# Overview of t-Distribution

The t-distribution is a family of continuous probability distributions. This is evidenced by the smooth shape of the above graph of a t-distribution’s PDF (Probability Density Function) curve.

The t-distribution (also called the Student’s t-distribution) describes the distribution of a sample taken from a normally-distributed population when the population standard deviation is unknown.

The t-distribution closely resembles the standard normal distribution (the normal distribution when the mean equals zero and the standard deviation equals one) except that the t-distribution’s outer tails have more weight (are thicker) and its mean has a lower peak than the standard normal distribution.

As sample size increases, the t-distribution converges to (more closely resembles) the standard normal distribution. When the sample size becomes large (n > 30), the t-Distribution almost exactly resembles the standard normal distribution.

The t-distribution has only one parameter which is the degrees of freedom. Degrees of freedom is usually designated as df or ν (Greek letter “nu”). This differs from the normal distribution because the normal distribution is described by the following two parameters: mean (µ - Greek letter “mu”) and standard deviation (s - Greek letter “sigma”).

The graph of the t-distribution is symmetrical about a mean of 0 and the units on its horizontal axis describing the distance from the mean of 0 are units of standard errors. This differs from the normal distribution because the normal distribution can be symmetrical about a mean of any real number and the units of its horizontal axis describing the distance from its mean use the same real number scale in which the mean was measured.

The t-distribution’s PDF or CDF at any real number X requires that the X value be converted to the number of standard errors that the X value is from the sample mean. The standard error is equal to the sample standard deviation divided by the square root of the sample size.

The t-distribution more closely describes the distribution of a small sample (n < 30) taken from a normally-distributed population than the normal distribution does. Small samples taken from a normally-distributed population have a slightly higher probability that sample values will occupy the outer tails than do larger samples. The t-distribution has slightly thicker tails and a lower peak than does the normal distribution. The t-distribution is therefore used to describe the distribution of small samples taken from a normally-distributed population.

The extra weight in the outer tails of the t-distribution accounts for the additional uncertainty of having to use the sample standard deviation to estimate the population standard deviation. This estimate becomes more uncertain as sample size decreases. The t-distribution’s shape reflects that as its outer tails become thicker as sample size decreases.

The t-distribution is used to analyze samples taken from a normally-distributed population when either of the following is true:

1) Small size is small (n < 30).

2) The population standard deviation is not known, which is often the case.

The t-distribution is used much more often than the normal distribution to perform several basic parametric statistical tests such as hypothesis tests of a population mean and confidence intervals of a population mean. Requirements for statistical tests are generally less rigorous when a statistical test can be based upon the t-distribution instead of the normal distribution.

## Graphing the t-Distribution’s PDF – Probability Density Function

### Step 1 – Create the t-Values

The X-axis of the standardized t-distribution does not change. It is always centered about a t value of 0 and the units on the X axis are units of standard errors of the t distribution. In this case the t values are calculated from 3 standard errors below t = 0 and 3 standard errors above t = 0 in increments of 0.1 standard errors. This t value count is accomplished by the Excel formula -3 + (ROW() – ROW(B))/10 (Click On Image To See a Larger Version)

### Step 2 – Create the X-Axis Values

The X-axis values are the same as the t-values. (Click On Image To See a Larger Version)

### Step 3 – Create the Y Values

One Y value is created for each increment of the Count. The Y value of each data point is its PDF value. The t-distribution’s PDF value requires only one parameters, the degrees of freedom df and nothing else. The X axis of the standardized t-distribution PDF curve is never shifted of expanded. (Click On Image To See a Larger Version)

The following Excel-generated graph shows the t-distribution’s PDF (Probability Density Function) for as the t value goes from -3 to 3 with degrees of freedom df = 30.

The PDF value of a statistical distribution (the Y value) at a specific X value equals the probability that the value of a random sample will be equal to that X value if the population of data values from which the sample was taken is distributed according the stated distribution. The CDF value of a statistical distribution (the Y value) at a specific X value equals the probability that the value of a random sample will be up to that X value. (Click On Image To See a Larger Version)

The process of creating an Excel area chart and connecting the user inputs to the chart is shown in detail in the section of this manual that provides instructions on how to create an interactive normal distribution PDF curve with outer tails.

### Effect of Changing the Degrees of Freedom

Reducing the degrees of freedom makes the t-distribution’s PDF curve wider and flatter. The t-distribution’s PDF curve converges to the standard normal distribution’s PDF curve when df exceeds 30. (Click On Image To See a Larger Version)

## Graphing the t-Distribution’s CDF – Cumulative Distribution Function

The following Excel-generated graph shows the t-distribution’s CDF (Cumulative Distribution Function) for df = 30.

The CDF value of a statistical distribution (the Y value) at a specific X value equals the probability that the value of a random sample will be up to that X value if the population of data values from which the sample was taken is distributed according the stated distribution. The PDF value of a statistical distribution (the Y value) at a specific X value equals the probability that the value of a random sample will be equal to that X value. (Click On Image To See a Larger Version)

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