Basic Concepts of t Distribution

The one sample hypothesis test described in Hypothesis Testing using the Central Limit Theorem using the normal distribution is fine when one knows the standard deviation of the population distribution and the population is either normally distributed or the sample is sufficiently large that the Central Limit Theorem applies.

The problem is that the standard deviation of the population is generally not known. One approach for addressing this is to use the standard deviation s of the sample as an approximation for the standard deviation σ for the population. In fact, as is described below, such an approach is possible using the t distribution.

Definition 1: The (Student’s) t distribution with k degrees of freedom, abbreviated T(k) has probability distribution function given by

Observations: Key statistical properties of the t distribution are:

  • Mean = 0 for k > 0
  • Median = 0
  • Mode = 0
  • Range = (-∞, ∞)
  • Variance = k ⁄ (k – 2) for k > 2
  • Skewness = 0 for k > 3
  • Kurtosis = 6 ⁄ (k – 4) for k > 4

The overall shape of the probability density function of the t distribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the t distribution approaches the standard normal distribution, and in fact the approximation is quite close for k ≥ 30.

t distribution chart

Figure 1 – Chart of t distribution by degrees of freedom

Theorem 1: If x has normal distribution N(μ, σ), then for samples of size n, the random variable


has distribution T(n – 1).

Click here for a proof of Theorem 1.

Corollary 1: For samples of sufficiently large size n, the random variable


has distribution T(– 1).

Proof: This follows from the theorem by the Central Limit Theorem.

Observation: The test statistic in the theorem and corollary are the same as


from Central Limit Theorem with the population standard deviation σ replaced by the sample standard deviation s. What makes this useful is that usually the standard deviation of the population is unknown while the standard deviation of the sample is known.

Excel Functions: Excel provides the following functions regarding the t distribution:

TDIST(x, df, tails) = the right tail at x of the Student’s t cumulative probability distribution function with df degrees of freedom when tails = 1 (for a one-tailed test). When tails = 2 (for a two-tailed test), TDIST(x, df, tails) is the sum of the right and left tails.

Since the t distribution is symmetric about x = 0, TDIST(x, df, 2) is simply 2 * TDIST(x, df, 1). Also note that x must be non-negative, but since the t distribution is symmetric about x = 0, the left tail when x < 0 is TDIST(-x, df, tails). Thus we can use the formula TDIST(ABS(x), df, tails) for any x. The cumulative probability distribution function is given by 1 – TDIST(x, df, 1) when x ≥ 0 and by TDIST(-x, df, 1) when x < 0.

TINV(p, df) = x such that TDIST(x, df, 2) = p; i.e. TINV is the inverse of TDIST in the two-tailed case. For the one-tailed case simply double p; i.e. TINV(2*p, df) = x such that TDIST(x, df, 1) = p.

With Excel 2010/2013 there are a number of new functions (T.DIST, T.INV, T.DIST.RT, T.INV.RT and T.INV.2T) that provide equivalent functionality to TDIST and TINV, but whose syntax is more consistent with other distribution functions. These functions are described in Built-in Statistical Functions.

In Excel 2010/2013 T.DIST(x, df, TRUE) is the cumulative distribution function for the t distribution with df degrees of freedom and T.DIST(x, df, FALSE) is the pdf for the t distribution. For prior versions of Excel, the supplemental function T_DIST(x, df, cum) contained in the Real Statistics Resource Pack provides the equivalent functionality.

8 Responses to Basic Concepts of t Distribution

  1. Gilbert says:

    Hi Sir Charles, my professor told me that in running a correlation in excel there must be three tables, i think one of them is a t-test table. He told me that in the final table it shows which variable has the most significant correlation. I’m asking if you happen to know how to make these three tables?

  2. Heta says:

    Hello Charles,

    I’m having problems calculating my degrees of freedom as I have to work out VaR using the t-distribution on Excel. I tried to rearrange the formula to d=2v/v-1, but I am getting -0.000314724, which doesn’t seem write. I got an error when using the T.INV function, showing #NUM! .

    If you could figure this problem out, please help!

    Thank you


    • Charles says:

      Hello Heta,
      Sorry, but I don’t understand the situation you are describing. T.INV will produce an error if the df is less than 1.
      If you send me an Excel file with your data and calculation I will try to figure out what is going on. You can get my email address at Contact Us.

  3. peter says:

    What makes this useful is that usually the standard deviation of the population is unknown while the standard devastation (sic) of the sample is known.

    • Charles says:

      Thanks for identifying this “devastating” typo. I have now corrected the webpage, although I am devastated that I didn’t catch it sooner. All joking aside, thanks for your help.

  4. Gopi says:

    hey, charles Good job. i have a doubt in my project, i need to find out t- test for each of my sample. In order to get the final analysis of my project i need it, but i cant make it out can u help me please.

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