What does it mean to say that the t-distribution provides an "adjustment" or "estimate" for a normal distribution? I have heard several times in classes that the t-distribution allows us to more accurately model data when the sample size is small or when we do not know the variance. I am wondering what is meant by this and why the t-distribution is considered to be a "readjustment" in a way. 
With respect to what is a t-distribution readjusting? I understand the t-distribution was originally developed to handle small sample sizes (6 I believe in Gossett's paper), and that there is a notion of correcting for the tails. Could someone shine light on the correction? Thanks.
 A: If the data are normally distributed, the test statistic with the sample standard deviation in the denominator will have a $t$ distribution with $n-1$ degrees of freedom.  For large $n$, the $t$ distribution is approximately normal.  But in small samples, it will be symmetric with heavier tails than those of the normal distribution.  The fact that the tails are heavier than for the standard normal could be viewed as adjusting the normal in the tails.
A: It is conventional to have the sample variance calculated to be an unbiased estimator of the population variance by dividing by $n-1$ rather than $n$.  
But then the sample standard deviation is not an unbiased estimator of the population standard deviation, and the reciprocal of the sample standard deviation is even more biased as an estimator of the reciprocal of the population standard deviation.  In particular, under-estimates of the variance will make the estimate of reciprocal of the standard deviation much much too big, and this is not fully offset by over-estimates of the variance. 
Dividing the value of interest by the population standard deviation would lead to a normally distributed statistic.  
But the $t$ statistic divides the value of interest by the estimate of the standard deviation from the sample (i.e. multiplies by its reciprocal).  The possibility of this result being much more than it would have been using the population standard deviation helps drive the heavier tails of the $t$ distribution, and large distortions of this type are more likely with small samples. In a sense the $t$ distribution can be seen as an adjustment of the normal distribution to take this into account.
