when a sample of n observations is taken from a normally distributed population having mean m and standard deviation d the sample mean m and the sample standard deviation d will differ from m and d because of the randomness of the sample 
a z score can be calculated with the population standard deviation as z x m d and this value has the normal distribution with mean 0 and standard deviation 1 but when using the estimated standard deviation a t score is calculated as t m m d sqrt n and the difference between d and d makes the distribution a t distribution with n 1 degrees of freedom rather than the normal distribution with mean 0 and standard deviation 1 2example of how to use a t distributiontake the following example for how t distributions are put to use in statistical analysis first remember that a confidence interval for the mean is a range of values calculated from the data meant to capture a population mean this interval is m t d sqrt n where t is a critical value from the t distribution for instance a 95 confidence interval for the mean return of the dow jones industrial average djia in the 27 trading days prior to sept 11 2001 is 0 33 2 055 1 07 sqrt 27 giving a persistent mean return as some number between 0 75 and 0 09 the number 2 055 the amount of standard errors to adjust by is found from the t distribution 3because the t distribution has fatter tails than a normal distribution it can be used as a model for financial returns that exhibit excess kurtosis which will allow for a more realistic calculation of value at risk var in such cases 4t distribution vs normal distributionnormal distributions are used when the population distribution is assumed to be normal the t distribution is similar to the normal distribution just with fatter tails both assume a normally distributed population t distributions thus have higher kurtosis than normal distributions the probability of getting values very far from the mean is larger with a t distribution than a normal distribution 5limitations of using a t distributionthe t distribution can skew exactness relative to the normal distribution its shortcoming only arises when there s a need for perfect normality the t distribution should only be used when the population standard deviation is not known if the population standard deviation is known and the sample size is large enough the normal distribution should be used for better results 6