# Normal Distribution vs. T-Student Distribution

## What's the Difference?

Normal distribution and T-student distribution are both probability distributions commonly used in statistics. Normal distribution is a symmetric bell-shaped curve that is characterized by its mean and standard deviation, while T-student distribution is similar to the normal distribution but has heavier tails, making it more suitable for smaller sample sizes. Normal distribution is often used when the sample size is large and the population standard deviation is known, while T-student distribution is used when the sample size is small or the population standard deviation is unknown. Overall, both distributions are important tools in statistical analysis and have their own unique characteristics and applications.

## Comparison

Attribute | Normal Distribution | T-Student Distribution |
---|---|---|

Shape | Bell-shaped curve | Bell-shaped curve |

Mean | μ | μ |

Variance | σ^2 | σ^2 * (df / (df - 2)) |

Parameter | μ, σ | μ, σ, df |

Use | For continuous data | For small sample sizes |

## Further Detail

### Introduction

Normal distribution and T-Student distribution are two of the most commonly used probability distributions in statistics. While both distributions are used to describe the behavior of random variables, they have distinct characteristics that make them suitable for different types of data analysis. In this article, we will compare the attributes of Normal Distribution and T-Student Distribution to understand their differences and similarities.

### Definition

Normal distribution, also known as Gaussian distribution, is a continuous probability distribution that is symmetric around its mean. It is characterized by a bell-shaped curve where the mean, median, and mode are all equal. The distribution is defined by two parameters - the mean (μ) and the standard deviation (σ). On the other hand, T-Student distribution, also known as Student's t-distribution, is a family of distributions that arise from the estimation of the mean of a normally distributed population when the sample size is small. It is similar to the normal distribution but has heavier tails, which makes it more suitable for small sample sizes.

### Shape

The shape of the normal distribution is symmetrical and bell-shaped, with the majority of the data clustered around the mean. The tails of the distribution extend infinitely in both directions, which means that extreme values are possible but rare. In contrast, the T-Student distribution has heavier tails compared to the normal distribution. This means that there is a higher probability of observing extreme values in a T-Student distribution, especially when the sample size is small. As the sample size increases, the T-Student distribution approaches the normal distribution in shape.

### Parameters

The normal distribution is defined by two parameters - the mean (μ) and the standard deviation (σ). These parameters determine the center and spread of the distribution, respectively. In contrast, the T-Student distribution is defined by a single parameter - the degrees of freedom (df). The degrees of freedom in a T-Student distribution represent the number of independent observations in the sample. As the degrees of freedom increase, the T-Student distribution approaches the normal distribution.

### Use Cases

The normal distribution is commonly used in statistical analysis to model a wide range of natural phenomena, such as heights, weights, and test scores. It is also used in hypothesis testing, confidence intervals, and regression analysis. On the other hand, the T-Student distribution is used when the sample size is small and the population standard deviation is unknown. It is commonly used in t-tests, analysis of variance (ANOVA), and confidence intervals for small samples.

### Robustness

The normal distribution is robust to outliers and deviations from normality when the sample size is large. This means that even if the data is not perfectly normally distributed, the results of statistical tests based on the normal distribution are still valid. In contrast, the T-Student distribution is less robust to outliers and deviations from normality, especially when the sample size is small. In such cases, non-parametric tests may be more appropriate to use.

### Conclusion

In conclusion, Normal Distribution and T-Student Distribution are two important probability distributions in statistics with distinct characteristics. While the normal distribution is symmetrical and bell-shaped, the T-Student distribution has heavier tails and is more suitable for small sample sizes. The normal distribution is defined by the mean and standard deviation, while the T-Student distribution is defined by the degrees of freedom. Both distributions have their own use cases and are valuable tools in statistical analysis. Understanding the differences between these distributions is essential for choosing the appropriate distribution for a given data set.

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