Limited Time Offer: Save 10% on all 2022 Premium Study Packages with promo code: BLOG10 # T-distribution

The student’s T-distribution is a bell-shaped probability distribution symmetrical about its mean. It is considered the best distribution to use for the construction of confidence intervals when:

1. dealing with small samples of less than 30 elements;
2. the population variance is unknown; and
3. the distribution involved is either normal or approximately normal.

In the absence of outright normality of a given distribution, the T-distribution may still be appropriate for use if the sample size is large enough to allow the application of the central limit theorem. In this case, the distribution is considered approximately normal.

The T-statistic, also called the T-score, is given by:

$$t = \cfrac {(x – \mu)}{\left(\cfrac {S}{\sqrt n} \right)}$$

Where:

x is the sample mean,

μ is the population mean,

S is the sample standard deviation,

n is the sample size.

The T-distribution allows us to analyze distributions that are not perfectly normal. It has the following properties:

1. it has a mean of zero;
2. its $$\text {variance}= \frac {v}{ \left(\frac {v}{2} \right) }$$, where v represents the number of degrees of freedom and v ≥ 2;
3. although it’s very close to one when there are many degrees of freedom, the variance is greater than 1 at all times. With a large number of degrees of freedom, the T-distribution resembles the normal distribution; and
4. its tails are fatter than those of the normal distribution, indicating more probability in the tails.

## The Degrees of Freedom

The T-distribution, just like several other distributions, has only one parameter: the degrees of freedom (d.f.). The number of degrees of freedom refers to the number of independent observations (total number of observations less 1):

$$v = n-1$$

Hence, a sample of 10 observations or elements would be analyzed using a T-distribution with 9 degrees of freedom. Similarly, a 6 d.f. distribution would be used for a sample size of 7 observations.

### Notations

It is standard practice for statisticians to use tα to represent the T-score with a cumulative probability of (1 – α). Therefore, if we were interested in a T-score with a 0.9 cumulative probability, α would be equal to 1 – 0.9 = 0.1. We would denote the statistic as t0.1.

However, the value of tα depends on the number of degrees of freedom. For example,

$$t_{0.05,2}= 2.92$$ where the second subscript (2) represents the number of d.f., and

$$t_{0.05,20} = 1.725$$

### Important Relationships

$$t_{\alpha}= -t_{1 – \alpha} \text{ and } t_{1 – \alpha} = -t_{\alpha}$$

The above relationships are true because the T-distribution is symmetrical about the mean.

The T-distribution has thicker tails relative to the normal distribution. The shape of the T-distribution is dependent on the number of degrees of freedom so that as the number of d.f. increases, the distribution becomes more ‘spiked,’ and its tails become thinner.

The table below represents one-tailed confidence intervals and various probabilities for a range of degrees of freedom.

$$\begin{array}{c|c|c|c|c} \textbf{r} & \textbf{90%} & \textbf{95%} & \textbf{97.5%} & \textbf{99.5%} \\ \hline {1} & {3.07768} & {6.31375} & {12.7062} & {63.6567} \\ \hline {2} & {1.88562} & {2.91999} & {4.30265} & {9.92484} \\ \hline {3} & {1.63774} & {2.35336} & {3.18245} & {5.84091} \\ \hline {4} & {1.53321} & {2.13185} & {2.77645} & {4.60409} \\ \hline {5} & {1.47588} & {2.01505} & {2.57058} & {4.03214} \\ \hline {10} & {1.37218} & {1.81246} & {2.22814} & {3.16927} \\ \hline {30} & {1.31042} & {1.69726} & {2.04227} & {2.75000} \\ \hline {100} & {1.29007} & {1.66023} & {1.98397} & {2.62589} \\ \hline {\infty} & {1.29007} & {1.66023} & {1.98397} & {2.62589} \\ \end{array}$$

Shop CFA® Exam Prep

Offered by AnalystPrep Level I
Level II
Level III
All Three Levels
Featured Shop FRM® Exam Prep FRM Part I
FRM Part II
FRM Part I & Part II
Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success
Shop Actuarial Exams Prep Exam P (Probability)
Exam FM (Financial Mathematics)
Exams P & FM
Shop GMAT® Exam Prep Complete Course Sergio Torrico
2021-07-23
Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar. diana
2021-07-17
So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings. Kriti Dhawan
2021-07-16
A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep. nikhil kumar
2021-06-28
Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures. Marwan
2021-06-22
Great support throughout the course by the team, did not feel neglected Benjamin anonymous
2021-05-10
I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way! michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.