e-Statistics
e-Statistics
For Students
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Data in Statistical Studies
Textbook Data Sets
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Data and Story Library
Exploratory Graphics
Stem and Leaf Plot
Histogram
Boxplot
Scatter Plot
QQ plot
Bar Graph
Descriptive Statistics
Summary Statistics
Frequency Table
Grouped Data
Transformed Data
Normal Distribution
Normal Density Function
Z-score
Critical Value
Central Limit Theorem
Discrete Distributions
Binomial Distribution
Frequency Function
Normal Approximation
Inference on Mean
t-Distribution
Confidence Interval
Hypothesis Test
Power of Test
One-sided CI
Comparison of Two Groups
Test for Independent Groups
Nonparametric Test
Test for Paired Data
Simple Linear Regression
Regression Line
R Square
Residual Plot
Inference on Parameters
Predicting Responses
One-Way ANOVA Model
F-Distribution
Analysis of Variance
Strip Chart
Multiple Comparisons
Multiple Linear Regression
Model Estimate and Residuals
Model Selection and F-test
Two-Way Additive Model
Test of Effect
Analysis of Variance
Inference on Variances
Testing a Standard Deviation
Test for Two Groups
Test for Homogeneity
Inference on Proportions
Testing a Proportion
Comparison of Two Proportions
Chi-square Test
Chi-square Distribution
Goodness of Fit
Test for Homogeneity
Logistic Regression
Inference on parameters
Adjusted log OR

Chi-square Distribution

The chi-square distribution has the number of degrees of freedom (df) = . The curve lies on the positive line, and its shape is skewed to the right particularly when df is small. The critical point, denoted by $ \chi_{\alpha,df}$, is provided for the upper tail.

Level (p-value) $ \alpha =$
Upper-tailed region $X > \chi_{\alpha,df} =$

When the sample variance $ S^2$ is obtained from the data of n observations which satisfies the normality assumption, the statistic $ X = \displaystyle\frac{(n-1)S^2}{\sigma^2}$ with true variance $ \sigma^2$ has the chi-square distribution with df = n-1 degrees of freedom.

Conversely when the statistic X = is given, we can find the corresponding $ \alpha$ so that the value X belongs to the upper-tailed critical region, and call it p-value.

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