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

    • Normal Density Function

    From the probabilistic point of view a data set of a certain population is merely a collection of values randomly drawn from a common distribution which is not yet known. Such a distribution is characterized by a probability density function (PDF) which is often described by means of parameters. The parameters are used to represent a family of functions having the same general shape, and the introduction of such parameters gave rise to the discipline of statistical modeling.

    A normal density function is determined by two parameters $ \mu =$ and $ \sigma =$ . The common shape of the normal density function is often described as ``bell-shaped'' curve.

    The first parameter $ \mu$ is called mean and it determines the center of the density. The second parameter $ \sigma$ is called standard deviation (SD). The normal density function $ f(x)$ is unimodal and symmetric around $ \mu$ . A small value of $ \sigma$ leads to a high peak with sharp drop, and a larger value of $ \sigma$ leads to a flatter shape of function. The actual function is formulated as

    $\displaystyle f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{(x - \mu)^2}{2 \sigma^2}\right] $

    where $ \pi = 3.14159\ldots$ and $ \exp(u)$ is the exponential function $ e^u$ with the base $ e = 2.71828\ldots$ of the natural logarithm.

    Assuming the normal density function $ f(x)$ for a certain population, we can predict the chance that the observation $ X$ from the population is between $ a$ and $ b$ . Having formed the density function $ f(x)$ , we can obtain such a probability, denoted by $ P(a < X < b)$ , as the area under the curve over the range between $ a =$ and $ b =$ . The calculation of this probability is what is known as ``integration'' in light of calculus, and can be obtained numerically by

    $ \displaystyle P(a < X < b) = \int_a^b f(x) dx =$

    In order to compute the probability $ P(-\infty < X < b)$ or $ P(a < X < \infty)$ in the form above, you must use ``$ a =$ -Inf'' or ``$ b =$ +Inf'' to indicate $ a=-\infty$ or $ b=+\infty$ in appropriate boxes.