e-Statistics
Introduction
  • Advanced User Guide
    Assignment & Quiz
  • User Account
  • Guide to Worksheet 1-2
  • Guide to Worksheet 3-4
  • Guide to Worksheet 5-6
  • Guide to Worksheet 7-8
    Data in Statistical Studies
  • Overview
  • Data from SMDA
  • Data and Story Library
    Exploratory Graphics
  • Overview
  • Histogram
  • Stem and Leaf Plot
  • Boxplot
  • Scatter Plot
  • QQ plot
  • Bar Graph
    Descriptive Statistics
  • Overview
  • Summary Statistics
  • Grouped Data
  • Frequency Table
    Discrete Distributions
  • Frequency Function
  • Binomial Distribution
  • Normal Approximation
    Normal Distribution
  • Normal Density Function
  • Critical Point
  • Z-score
  • Central Limit Theorem
    Inference on Mean
  • t-Distribution
  • Confidence Interval
  • Hypothesis Test
  • Power of Test
  • One-sided CI
    Comparison of Two Groups
  • Overview
  • t-Test
  • Nonparametric Test
  • Test for Paired Data
    Inference on Variances
  • Chi-square Distribution
  • Testing a Standard Deviation
  • F-Distribution
  • F-Test
    Explore Statistics in Drug Safety

    • 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. Since the chi-square distribution is not symmetric, its critical points, denoted by $ \chi_{\alpha,df}$ , is provided separately for the lower tail and the upper tail.

    Level (p-value) $ \alpha =$
    Lower-tailed region $ T < \chi_{1-\alpha,df} =$
    Upper-tailed region $ T > 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.