The Student t Distribution

  If one adds a given set of normal random numbers with the same mean μ and standard deviation σ together Student showed in 1908 that the sum deviated from a normal distribution. The square of the standard deviation of a sum is the sum of the squares of the standard deviation of each number and since each is the same σ_Σ =σ√n̅ and for an average σ_avg =σ/√n̅. Student used the z-value z=(x̄-μ)/σ_avg to define the probability density distribution for the average of the sum which has become known as the Student t distribution. The variable t for which the distribution is defined is a z-value. A formula for the probability p(t) can be found in the Wikipedia article Student's t-distribution. We can evaluate this function for a number of degrees of freedom (DF) in Excel.

As the number of degrees of freedom increases the t-distribution approaches a normal distribution with mean μ=0 and standard deviation σ=1.

Using Excel's t.dist function we get the same results.