The t-distribution, also known as the Student's t-distribution, is a probability distribution that is used in statistical inference for estimating population parameters when the sample size is small and the population standard deviation is unknown. The t-distribution is similar to the normal distribution but has heavier tails, making it more robust to outliers.

Key characteristics of the t-distribution include:

1. Shape: The t-distribution has a bell-shaped curve similar to the normal distribution, but with heavier tails. As the degrees of freedom increase, the t-distribution approaches the shape of the standard normal distribution.
2. Parameters: The shape of the t-distribution is determined by a parameter called degrees of freedom (df). The degrees of freedom represent the number of observations in the sample that are free to vary. The larger the degrees of freedom, the closer the t-distribution is to the normal distribution.
3. Application: The t-distribution is commonly used in situations where the population standard deviation is unknown, and the sample size is small. It is especially relevant for making inferences about the population mean based on a sample mean.
4. Comparison to the Normal Distribution: When the sample size is large, the t-distribution approaches the normal distribution. In practice, a common rule of thumb is to use the t-distribution when the sample size is less than 30. However, the choice depends on the specific context and assumptions of the statistical analysis.

The t-distribution is useful in hypothesis testing, confidence interval estimation, and other statistical analyses. When dealing with small sample sizes, using the t-distribution allows for a more accurate estimation of the population parameters and provides wider confidence intervals due to the increased uncertainty associated with smaller samples. It is a key tool in scenarios where assumptions about normality and known population standard deviation may not hold.