The **t-distribution** is a type of probability distribution that arises while sampling a normally distributed population when the sample size is small and the standard deviation of the population is unknown. It is also called the Student’s t-distribution. It is approximately a bell curve, that is, it is approximately normally distributed but with a lower peak and more observations near the tail. This implies that it gives a higher probability to the tails than the standard normal distribution or z-distribution (mean is 0 and the standard deviation is 1).

**Degrees of Freedom** is related to the sample size and shows the maximum number of logically independent values that can freely vary in the data sample. It is calculated as n – 1, where n is the total number of observations. For example, if you have 3 observations in a sample, 2 of which are 10,15 and the mean is revealed to be 15 then the third observation has to be 20. So the Degrees of Freedom, in this case, is 2 (only two observations can freely vary). Degrees of Freedom is important to a t-distribution as it characterizes the shape of the curve. That is, the variance in a t-distribution is estimated based on the degrees of freedom of the data set. As the degrees of freedom increase, the t-distribution will come closer to matching the standard normal distribution until they converge (almost identical). Therefore, the standard normal distribution can be used in place of the t-distribution with large sample sizes.

A **t-test** is a statistical hypothesis test used to determine if there is a significant difference (differences are measured in means) between two groups and estimate the likelihood that this difference exists purely by chance (p-value). In a t-distribution, a test statistic called **t-score **or t-value** **is used to describe how far away an observation is from the mean. The t-score is used in t-tests**,** regression tests and to calculate confidence intervals.

## Student’s t-distribution in R

**Functions used:**

- To find the value of probability density function (pdf) of the Student’s t-distribution given a random variable x, use the
**dt()**function in R.

Syntax: dt(x, df)

Parameters:

- x is the quantiles vector
- df is the degrees of freedom

**pt()**function is used to get the cumulative distribution function (CDF) of a t-distribution

Syntax:pt(q, df, lower.tail = TRUE)

Parameter:

- q is the quantiles vector
- df is the degrees of freedom
- lower.tail – if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].

- The
**qt()**function is used to get the quantile function or inverse cumulative density function of a t-distribution.

Syntax:qt(p, df, lower.tail = TRUE)

Parameter:

- p is the vector of probabilities
- df is the degrees of freedom
- lower.tail – if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].

### Approach

- Set degrees of freedom
- To plot the density function for student’s t-distribution follow the given steps:
- First create a vector of quantiles in R.
- Next, use the dt function to find the values of a t-distribution given a random variable x and certain degrees of freedom.
- Using these values plot the density function for student’s t-distribution.

- Now, instead of the dt function, use the pt function to get the cumulative distribution function (CDF) of a t-distribution and the qt function to get the quantile function or inverse cumulative density function of a t-distribution. Put it simply, pt returns the area to the left of a given random variable q in the t-distribution and qt finds the t-score is of the p
^{th}quantile of the t-distribution.

**Example:** To find a value of t-distribution at x=1, having certain degrees of freedom, say D_{f} = 25,

## R

`# value of t-distribution pdf at ` `# x = 0 with 25 degrees of freedom` `dt` `(x = 1, df = 25)` |

**Output:**

0.237211

**Example:**

Code below shows a comparison of probability density functions having different degrees of freedom. It is observed as mentioned before, larger the sample size (degrees of freedom increasing), the closer the plot is to a normal distribution (dotted line in figure).

## R

`# Generate a vector of 100 values between -6 and 6` `x <- ` `seq` `(-6, 6, length = 100)` ` ` `# Degrees of freedom` `df = ` `c` `(1,4,10,30)` `colour = ` `c` `(` `"red"` `, ` `"orange"` `, ` `"green"` `, ` `"yellow"` `,` `"black"` `)` ` ` `# Plot a normal distribution` `plot` `(x, ` `dnorm` `(x), type = ` `"l"` `, lty = 2, xlab = ` `"t-value"` `, ylab = ` `"Density"` `, ` ` ` `main = ` `"Comparison of t-distributions"` `, col = ` `"black"` `)` ` ` `# Add the t-distributions to the plot` `for ` `(i ` `in` `1:4){` ` ` `lines` `(x, ` `dt` `(x, df[i]), col = colour[i])` `}` ` ` `# Add a legend` `legend` `(` `"topright"` `, ` `c` `(` `"df = 1"` `, ` `"df = 4"` `, ` `"df = 10"` `, ` `"df = 30"` `, ` `"normal"` `), ` ` ` `col = colour, title = ` `"t-distributions"` `, lty = ` `c` `(1,1,1,1,2))` |

**Output:**

**Example:** Finding p-value and confidence interval with t-distribution

## R

`# area to the right of a t-statistic with ` `# value of 2.1 and 14 degrees of freedom` `pt` `(q = 2.1, df = 14, lower.tail = ` `FALSE` `)` |

**Output:**

0.02716657

Essentially we found the one-sided p-value, P(t>2.1) as 2.7%. Now suppose we want to construct a two-sided 95% confidence interval. To do so, find the t-score or t-value for 95% confidence using the qt function or the quantile distribution.

**Example:**

## R

`# value in each tail is 2.5% as confidence is 95%` `# find 2.5th percentile of t-distribution with ` `# 14 degrees of freedom` `qt` `(p = 0.025, df = 14, lower.tail = ` `TRUE` `)` |

**Output:**

-2.144787

So, a t-value of 2.14 will be used as the critical value for a confidence interval of 95%.

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