Skip to content

T dist

T.DIST Function

The T.DIST function in Excel is used to return the probability density (p-value) of the left-tailed Student's t-distribution, which is widely utilized in hypothesis testing and statistical significance analysis.

This function calculates probabilities based on a t-value, degrees of freedom, and specifies whether the distribution is cumulative (area under the curve) or the probability density function (PDF) value.

Key Features of T.DIST:

  • Left-Tailed Probability: Computes the cumulative probability from negative infinity to the given t-value for the Student's t-distribution.
  • Degrees of Freedom: Requires the degrees of freedom to adjust the shape of the t-distribution.
  • Cumulative Option: Allows you to calculate either the cumulative probability or the probability density function value.

Syntax:

T.DIST(x, degrees_freedom, cumulative)
  • x: Required. The numeric value at which to evaluate the distribution (t-value).
  • degrees_freedom: Required. The number of degrees of freedom for the t-distribution.
  • cumulative: Required. A logical value:
    • TRUE if you want the cumulative distribution function (CDF).
    • FALSE if you want the probability density function (PDF).

How It Works:

  • A cumulative distribution function (CDF) computes the probability that the t-distribution is less than or equal to the given t-value.
  • The probability density function (PDF) calculates the height of the probability distribution curve at the given t-value.

Examples:

  1. Cumulative Probability: Find the cumulative probability for a t-value of 2.5 with 10 degrees of freedom:
=T.DIST(2.5, 10, TRUE)

This will return the cumulative probability from negative infinity to 2.5.

  1. Probability Density: Get the probability density value for the same t-value (2.5) and degrees of freedom (10):
=T.DIST(2.5, 10, FALSE)

This calculates the density of the t-distribution at x = 2.5.

  1. Practical Use for Left Tail: Suppose you calculate a t-statistic of -1.96 with 20 degrees of freedom, and you are interested in the left-tailed cumulative probability:
=T.DIST(-1.96, 20, TRUE)

This result can be used in hypothesis testing to determine the p-value.

Notes:

  • Input Validations:
    • The degrees_freedom must be greater than or equal to 1.
    • The cumulative parameter can only accept TRUE or FALSE.
  • Output Details:
    • A TRUE cumulative result gives the area under the curve to the left of x.
    • A FALSE result returns the height of the curve (density) at the specific value of x.
  • Errors:
    • #NUM! if degrees_freedom < 1.
    • #VALUE! if inputs are non-numeric.

Applications:

  • Statistical Testing: Compute p-values for left-tailed t-tests to test hypotheses.
  • Probability Analysis: Evaluate areas under the t-distribution to determine probabilities.
  • Data Science: Use the t-distribution in modeling and analyzing small-sample data or when population variance is unknown.

Tip: Use the T.DIST.RT function to calculate right-tail probabilities, or T.DIST.2T for two-tailed probabilities when required.