The 't' distribution is a theoretical probability distribution. The 't' distribution is symmetrical, bell-shaped, and to some extent similar to the standard normal curve. It has an additional parameter called degree of freedom and is centered at zero. The shape of 't' distribution changes due to the degree of freedom. Degrees of freedom (df) can be any real number greater than zero. Consider the equation       X + Y = 4. In this equation once we fix the value of X the value of Y is set automatically so the degree of freedom for this equation is said to be one. 

t distribution with n-1 degree of freedom is defined as

t

Where,

               =    The sample mean

         m       =    Population mean

         S       =    Sample standard deviation

         n       =    The sample size

As shown in the figure below, it is symmetrical like the normal distribution, but its peak is lower than the normal curve and its tail is a little higher above the abscissa than the normal curve.

Figure 

The 't' distributions with a smaller degree of freedom have more area in the tails of the distribution than one with a larger degree of freedom. As the degrees of freedom for a 't' distribution get larger and larger, the 't' distribution gets closer and closer to the standard normal distribution. As the df increase, the 't' distribution approaches the standard normal distribution. The standard normal curve is a special case of the 't' distribution when df =  ∞ . For practical purposes, the 't' distribution approaches the standard normal distribution relatively quickly, such that when degree of freedom = 30 the two are almost identical. So the best use of 't' distribution is when the degree of freedom is less than 30. It is used instead of the normal distribution whenever the standard deviation is estimated. The 't' distribution has relatively more scores in its tails than does the normal distribution. One more purpose for using 't' distribution is when the population standard deviation is unknown.

Example 

Consider the t-distribution with df = 13. What is the area to the right of 1.771?

From the tables, it can be seen that the area is 0.05.