Question

If the standard deviation of a \(x^{2}\) distribution in 10 , then its degree of freedem is ___ .

Short answer

The degrees of freedom is 50.

Step-by-step solution

Understand the Distribution

Recognize that a \( \chi^2 \) (chi-square) distribution is defined by its degrees of freedom (df). This distribution is used in statistical tests and inference for categorical data.

Recall the Formula for Standard Deviation

For a chi-square distribution with degrees of freedom \( k \), the standard deviation formula is given by \( \sqrt{2k} \).

Set Up the Equation

We are given the standard deviation \( \sqrt{2k} = 10 \). To find \( k \), start by squaring both sides to remove the square root: \( 2k = 100 \).

Solve for Degrees of Freedom

Divide both sides of the equation \( 2k = 100 \) by 2 to isolate \( k \). This yields \( k = 50 \).

Conclusion

The degrees of freedom of the chi-square distribution is 50.

Key concepts

Standard Deviation

Standard deviation is a fundamental concept in statistics that tells us how much variation or "spread" exists from the average or mean of a set of data. When discussing distributions, the standard deviation is a way to understand how data is dispersed, making it crucial for interpreting statistical results. For the chi-square distribution, which is primarily used in statistical tests involving categorical data, the standard deviation is notably different compared to other distributions.

• The formula for the standard deviation of a chi-square distribution with degrees of freedom, denoted by \( k \), is \( \sqrt{2k} \).
• Standard deviation can help assess whether the spread of the data is compact or diverse.
• It is always a non-negative number, since it is derived from the square of differences.

The standard deviation that we discussed is specifically used here to solve for the degrees of freedom when certain values are known.

Degrees of Freedom

Degrees of freedom is a concept used in various areas of statistics to describe the number of values in a calculation that are free to vary. It is a critical component of chi-square tests and many other statistical methods. In simple terms:

• Degrees of freedom refer to the amount of freedom a given dataset has to vary.
• For a chi-square distribution, the degrees of freedom \( k \) determine the roughness of the distribution. More degrees of freedom lead to a smoother curve.
• The degrees of freedom determine the skewness and peak of the chi-square curve.

In our exercise case, we calculated the degrees of freedom \( k \) by using the formula related to standard deviation and isolating \( k \) in the equation \( \sqrt{2k} = 10 \), which gave us a solution of \( k = 50 \).

Statistical Inference

Statistical inference encompasses the methods that allow us to make generalizations or predictions about a larger population based on sample data. It is a fundamental part of using statistics in real-world scenarios and involves drawing conclusions beyond the immediate data observations.

• One primary tool of statistical inference is the chi-square test, which utilizes the chi-square distribution to test hypotheses.
• This involves estimating population parameters and testing hypotheses using sample statistics.
• Inference allows us to consider the uncertainty and variability present in sample data.

In our context, understanding the degrees of freedom, and subsequently finding it using chi-square statistics, is part of making informed inferences about the dataset we are analyzing.

Chi-Square Statistics

Chi-square statistics are used in statistical hypothesis testing, particularly for categorical data. It's a powerful tool for determining whether there is a significant difference between expected and observed data.

• The chi-square distribution itself is used in questions related to variances, tests of independence, and goodness-of-fit tests.
• It depends heavily on the degrees of freedom \( k \), which shape the distribution's curve.
• Chi-square statistics require large sample sizes for the test results to be valid, as small samples might not adequately reflect population characteristics.

This statistical test is an integral part of analyzing data and inferring relationships and characteristics within a dataset. By understanding chi-square statistics, we’re equipped to test hypotheses and uncover insights that might not be immediately apparent.