# T-Value Calculator

The t-Value calculator calculates the t-value for a given set of data based on the sample size, hypothesis testing method (one-tail or two-tail), and the significance level.

Many different distributions exist in statistics and one of the most commonly used distributions is the t-distribution.

The normal distribution is the well-known bell-shaped distribution that has an area of 1 under it. The t-distribution is very similar to the normal distribution and, like it, contains an area of 1 underneath the curve; it has a similar shape to the normal distribution but is shorter and flatter than a normal distribution. Like the standard normal (Z) distribution, it is centered at zero, but its standard deviation is proportionally larger compared to the Z-distribution.

As with normal distributions, there is an entire family of different t-distributions. Each t-distribution is distinguished by degrees of freedom, which are related to the sample size of the data. The degrees of freedom equals the sample size minus 1 (df= n-1).

Smaller sample sizes have flatter t-distributions than larger sample sizes. The larger the sample size, the more a t-distribution curve approaches the shape and values of a normal distribution curve. When the sample size approaches the value of 30, the values of the t-distribution are about equal to the value of the normal distribution curve.

So, again, each t-distribution has its own shape and its own set of probabilities. T-distributions have degrees of freedom ranging from 1 to 30.

If you look at the t-table which gives T values, the horizontal represents right tail probabilities. The numbers below any given column represent the values on each t-distribution having those right tail probabilities.

Going down each of the rows is the degrees of freedom (1 to 30). Below the df=30 row is the df= ∞ (infinity) row. This means that if df > 30, the t-values for each of the respective right probability values will be equal to this row.

Let's go over an example.

Let's say that for a given data set, the sample size is 12, the hypothesis testing method is one-tail and the significance level is 0.05.

The degrees of freedom, df= n-1= 12-11. Therefore, the degrees of freedom equals 11. The significance level is 0.05 (or 5%). Therefore, we look for the row where df=11 and the column where the significance level is 0.05.

Check this on the t-table, we get the value of 1.795885.

Now let's do the same example now with only the hypothesis testing method changed to two-tail testing. So we have the same sample size of 12 and the same significance of 0.05 (or 5%).

So, again, the degrees of freedom, df= n-1= 12-1= 11.

The thing that changes for the two-tail testing method is that because it's divided into 2 parts, a right and left side, you divide the significance level into 2. So, here we want a significance level of 0.05. Divided into 2, this produces a significance level of 0.025 on each side, left and right side.

So instead of looking up the significance level for 0.05, we look up the significance level for 0.025 with a degree of freedom equal to 11; looking this up on the t-table, this gives us the value of 2.20099.

One more example.

Let's say we have a data set where the sample size is 34, the significance level is 0.01, and the hypothesis testing method is two-tail testing.

So remember that t-distribution have degree of freedom rows that go from 1 to 30. After df=30 is the df= ∞ row. Any sample sizes above 31 will refer to this row. This is the row that have values most close to the normal standard distribution.

So since our sample size is 34, our degrees of freedom is, df= n-1=34-1=33. Therefore, we refer to the df= ∞ row.

Since we dealing with two-tail hypothesis testing, we take the significance level and divide it by 2. This gives us a significance level of 0.01/2= 0.005. This, again, is because with two-tail hypothesis testing, the total significance level is 0.01 and this is divided into 2 sides, a left side and a right side. Therefore, the total significance level is 0.01 but the significance level on each side is 0.005. We look up the 0.005 significance level to find the t-value.

So we look at the df= ∞ row with a significance level of 0.005. This gives us a t-value of 2.57583.

So this is how t-distribution works.

With the hypothesis testing method, sample size, and significance level, we can find the t-value for a given data set.

To use this calculator, a user simply enters in the hypothesis testing method (either
one-tail or two-tail), the sample size, and the significance level and clicks
the 'Calculate' button. The resultant t-value will automatically be computed
and displayed.

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