Defining T-distribution
The T distribution is a kind of probability distribution that is almost exactly the same as the normal distribution. The only difference is its bell shape that has heavier tails. They have heavier tails because their chances of getting more extreme values are higher compared to normal distributions.
Other people refer to T distribution as “student’s t distribution.” Why? In 1908, Guinness Breweries’ William Sealy Gosset, who uses a pen name “Student,” discovered it.
Tell me more about T distribution.
We mentioned that T distribution has heavier and fatter tails. The heaviness depends on a T distribution parameter called “degrees of freedom.” Smaller values give heavier tails. Higher values make the T distribution look like a normal distribution with a 0 mean and a 1 standard deviation.
“N” observation samples that are taken from normal distribution population with an M mean and D standard deviation, m sample mean, and d sample standard deviation will not be the same as M and D because the samples are random.
The calculation of the z-score with the standard deviation is: Z= (x – M)/D
This has a normal distribution with 0 mean and 1 standard deviation.
However the t-score has a different calculation when an estimated standard deviation is used: T = (m – M)/{d/sqrt(n)},
The difference between D and d makes a T distribution with (n-1) degrees of freedom’s distribution more like a normal one with a 0 mean and a 1 standard deviation.
How important is the “degrees of freedom” parameter?
“Degrees of freedom” is something that shows us the value sample of size n, and we know that its definition is based on sample size. Hence, the sample size is crucial in the definition of the critical and p values, and they are the ones that dictate the hypothesis tests and confidence interval calculations.
It is more of a concept that people use for different purposes. It tells us how many decisions or variables are dependent. To make it simpler, out of all n observations, one has the freedom to select (n-1) observation in any order. However, if the last order is the only one left, it would be the one chosen.
So, how different is a T distribution from a normal distribution?
As its name suggests, we use a normal distribution if the normal distribution assumption is normal. On the other hand, T distributions are also like normal distributions. The only differences are the heavier and fatter tails. So in layman’s terms, they both have normal distribution population assumption. It’s just that T- distributions have more kurtosis compared to the normal distributions. When we kurtosis, we mean the peak sharpness of a frequency distribution curve. The chances of getting values that are much distant from the mean are high in a T-distribution compared to a normal distribution.
The downsides of T distributions
T distributions can change the accuracy relative to the normal distribution, although this downside of T distribution only happens when perfect normality is needed. All in all, using a normal distribution is not much different from using the T distribution.
