The **chi-square distribution** is a standard application in hypothesis testing. It is a constant probability dispersal whose shape is based on the parameter k.

This blog post discusses more about the chi-square distribution.

## Definition: Chi-square distribution

The chi-square distribution is a constant hypothetical dispersal of values for a population. It is commonly applied in **statistical hypothesis tests**.^{1}

The parameter k, which denotes the **degrees of freedom**, governs the outline of a chi-square distribution. The chi-square distribution applies to theoretical distributions. In contrast, normal and Poisson distributions apply in real-world distributions.^{2}

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## Chi-square distribution vs. standard normal distribution

The **chi-square distribution** is closely related to the standard regular dispersal; hence its application in hypothesis testing.

The **standard normal distribution** is a normal distribution with a **mean** and **variance** of zero and one, respectively.

### Evaluating the k degrees of freedom

Generally, if you are ample from k autonomous average nominal distributions and square and quantify their values, you will yield a chi-square distribution in the existence of k degrees of freedom.

**Therefore:**

X^{2}_{k}= (Z_{2})^{2}+XXX+(Z_{k})^{2}

## Chi-square distribution formula

Chi-square assessments are types of **hypothesis tests** whose **test statistics** tail a chi-square distribution in the null proposition. The most common chi-square test is **Pearson’s chi-square **assessment, which was also the first to be discovered.^{1}

The table explains **Pearson’s chi-square test figure**:

Formula |
Description |

X2= (Ʃ (0-E)2)/E | X2 = chi-square test statistic Ʃ = “take the summation of” O = observed frequency E = anticipated frequency |

The test measurement of a sampled population usually follows a chi-square distribution if the **null hypothesis** is correct. This applies where you sample a population severally and find the Pearson’s **test statistic** for every trial.

## Chi-square distribution shape

Graphs of the chi-square probability compactness function illustrate how the chi-square distribution alters as k increases. A concentration function defines a continuous prospect dispersal.

### K = 1 or K = 2

The outline of a chi-square distribution alters depending on the value of k. When k is ½, the shape curves back into a “J.”

This implies a high probability of the **X ^{2}** being close to zero.

### K greater than 2

When the parameter is better than two, the chi-square distribution appears **hump-shaped**. This implies that it starts out low followed by an increase, then a decrease. This denotes a low probability of **X ^{2}** being close to zero.

^{1}

In contrast, when the parameter (k) is slightly more than two, the chi distribution will be longer on the **right peak side** than on the left.^{2}

The chi distribution tends to resemble the normal distribution as k increases. In some instances (when k is 90 or more) the normal distribution applies as an estimate of a chi-square distribution.

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## Properties of a chi-square distribution

Chi-square distribution usually has some standard properties. Here are its **properties**:

Property |
Symbol |

Type (Discrete/ continuous) |
Continuous |

Variance |
2k |

Mean |
K |

Standard deviation |
Square root of 2k |

Mode |
K – 2 (if k less than 2) |

Range |
Zero to infinity |

Symmetry |
Rightly-skewed, increasing as k increases |

## Chi-square distribution example

The chi-square distribution is applied in many statistical and theoretical tests. Here are its **most frequent applications**:

### Pearson’s chi-square test

This is a statistical test for definite statistics used to determine the significance of the difference between the data and your expectations.

There are **two** categories of Pearson’s chi-square tests:^{2}

- Chi-square test of good fit
- Chi-square independence test

### Population variance inferences

The chi-square distribution comes in handy when making extrapolations about standard deviation or variance.

It helps with hypothesis testing to determine if the population variance equates to a specific value or to calculate its confidence intervals.

### F distribution

The chi-square distribution helps with defining the incidence distribution, especially in **ANOVAs**.

## Non-central chi-square distribution

This is an overall kind of chi-square distribution used in some forms of power analyses. It features an extra lambda and non-central parameter, which changes its shape. Its **peak shifts to the right** and increases as the variance grows. The lambda parameter defines the mean figure of the normal dispersal.^{3}

## FAQs

The curve shifts from **downward** to **hump-shaped**. The more the k increases, the more right-skewed it gets.

The chi-square distribution is a **constant probability dispersal** commonly applied in hypothesis testing.

There are **two types**. They are:

- the chi-square test of good fit
- the chi-square freedom test

The chi-square distribution is used to define the **quantity** of a **squared random variable**.^{3}

## Sources

^{1} ScienceDirect. “Chi-Square Distribution.” Accessed March 22, 2023. https://www.sciencedirect.com/topics/mathematics/chi-square-distribution.

^{2} Statistics Knowledge Portal. “The Chi-Square Distribution.” JMP. Accessed March 22, 2023. https://www.jmp.com/en_us/statistics-knowledge-portal/chi-square-test/chi-square-distribution.html.

^{3} Primas, Mia “Chi-Square Distribution: Definition & Examples.” Study.com. December 07, 2018. https://study.com/academy/lesson/chi-square-distribution-definition-examples.html.