As a statistician with extensive experience in the field of probability and statistics, I have a deep understanding of various distributions and their properties, including the T distribution. The T distribution, also known as the Student's t-distribution, is a type of probability distribution that arises when estimating the population mean for a normally distributed population when the population standard deviation is unknown. It was first described by William Sealy Gosset under the pseudonym "Student" in 1908.

The T distribution is characterized by its bell-shaped curve, which is similar to the normal distribution but has heavier tails. This means that there is a higher probability of observing values that are far from the mean, compared to the normal distribution. The shape of the T distribution is determined by the degrees of freedom, which is the number of independent observations minus one.

When the sample size is small, the T distribution is quite different from the normal distribution. It is asymmetric and has more pronounced tails. This asymmetry and the heavier tails are due to the fact that the sample mean is being estimated with less information (fewer observations), leading to greater uncertainty.

However, as the sample size increases, the degrees of freedom also increase. With a larger sample size, we have more information about the population mean, which leads to less uncertainty in our estimates. As a result, the T distribution begins to approach the normal distribution. This is because with more data, the sample mean becomes a more accurate estimate of the population mean, and the variability in the estimates decreases.

The reference content you provided is accurate in stating that as the degrees of freedom become large, the T distribution becomes much more normal-looking and much more "tight" around its mean. This is due to the fact that the variance of the T distribution decreases as the degrees of freedom increase. The variance is directly related to the spread of the distribution, so a decrease in variance means that the distribution becomes more concentrated around the mean.

As the degrees of freedom continue to increase, the T distribution's tails become less heavy, and the distribution becomes more symmetric. Eventually, when the degrees of freedom are very large (usually considered to be around 30 or more), the T distribution is virtually indistinguishable from the normal distribution. At this point, the T distribution is often approximated by the normal distribution for practical purposes, especially in hypothesis testing and confidence interval estimation.

In summary, the shape of the T distribution changes as the sample size increases. It starts as a distribution with heavier tails and more asymmetry for small sample sizes, and as the sample size grows, it becomes more symmetric and approaches the normal distribution, with lighter tails and a more concentrated shape around the mean.

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However, as the degrees of freedom become large, the distribution becomes much more normal-looking and much more "tight" around its mean. As such, the effect of dividing by the denominator on the shape of the distribution of the numerator reduces as the degrees of freedom increase.Aug 2, 2014read more >>