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

    • Frequency Function

    When an outcome is a numerical value, outcome(s) determined by an experiment is referred as random variable (r.v.). We conventionally denote random variables by uppercase letters $ X, Y, Z, U, V, \ldots$ from the end of the alphabet. Particularly if the random variable $ X$ takes ``discrete'' values such as 1,2,3,$ \ldots$ (or 0,1,2,3,$ \ldots$ ), we call it a discrete random variable. The statement such as ``$ X = 1$ '' is an event, and therefore, is associated to the probability $ P(X = 1)$ . We can assign $ P(X = i)$ for all the possible values $ i = 1,2,3,\ldots$ (or $ i = 0,1,2,3,\ldots$ ), which will completely describe the ``probabilistic nature of the random variable $ X$ '', that is, the probability distribution of $ X$ . $ P(X = i)$ is often called a frequency function, and simply written by $ P(i)$ .

    The frequency function $ P(i)$ is graphically presented. The bar graph may be able to indicate a particular outcome with the highest frequency, the existence of mode of the distribution of interest.

    To find the probability from the frequency function $ P(i)$ , it is important to know the exact range $ \{j \le X \le k\}$ of the random variable $ X$ . Then it is computed as

    $ P\big($ $ \le X \le$ $ \big) = P(j) + P(j+1) + \cdots + P(k) =$