Chapter 6: The Normal Probability Distribution: Properties, Applications, and Normality Assessment

Continuous Random Variables and the Uniform Distribution

• A continuous random variable has an infinite number of possible outcomes, meaning the probability of selecting any single exact point is zero ($P(X = a) = 0$).

• A probability density function (pdf) models continuous probabilities and satisfies two requirements:

  • The total area under the graph equals $1$.

  • The function value is non-negative for all values: $f(x) \ge 0$.

• For a uniform probability distribution over an interval from $a$ to $b$, the probability density function is:   $f(x) = \frac{1}{b - a}, \quad a \le x \le b$

• Probabilities for uniform distributions correspond to the rectangular area over a specific sub-interval:   $\text{Probability} = \frac{\text{Interval Width}}{\text{Total Range}} = \frac{x_2 - x_1}{b - a}$

Properties of the Normal Distribution

• The normal distribution is a continuous, symmetric, bell-shaped probability distribution defined by the mean ($\mu$) and standard deviation ($\sigma$).

• Adjusting $\mu$ shifts the curve's horizontal position, whereas adjusting $\sigma$ alters the spread and height of the curve.

• Key characteristics of the normal curve include:

  • Symmetry about the center, where $\text{Mean} = \text{Median} = \text{Mode}$.

  • Inflection points occurring at $\mu - \sigma$ and $\mu + \sigma$.

  • Total area under the curve equals $1$, with an area of $0.5$ on each side of $\mu$.

  • Asymptotic tails that approach $0$ as $x$ moves toward positive or negative infinity.

• The Empirical Rule defines distribution proportions:

  • Approximately $68\%$ of observations lie within $1$ standard deviation of the mean ($\mu \pm 1\sigma$).

  • Approximately $95\%$ of observations lie within $2$ standard deviations of the mean ($\mu \pm 2\sigma$).

  • Approximately $99.7\%$ of observations lie within $3$ standard deviations of the mean ($\mu \pm 3\sigma$).

• Area under any interval of the curve equates to both the proportion of the population within that range and the probability of a randomly chosen individual having a value in that range.

Applications and Standard Normal Calculations

• Any normal random variable $X$ can be converted into the standard normal random variable $Z$ with mean $\mu = 0$ and standard deviation $\sigma = 1$ via the standardization formula:   $z = \frac{x - \mu}{\sigma}$

• Then take that z value and use standard normal distribution table to find value.

• Common area calculation procedures using standard normal tables:

  • Area to the left of $x$:     $P(X < x) = P(Z < z)$

  • Area to the right of $x$:     $P(X > x) = 1 - P(Z < z)$

  • Area between two points $x_1$ and $x_2$:     $P(x_1 < X < x_2) = P(Z < z_2) - P(Z < z_1)$

Finding Values of a Normal Random Variable

• To find an unknown variable value $x$ from a given percentile or area:

  • Determine the cumulative area situated to the left of the desired point.

  • Locate the corresponding $z$-score from the standard normal distribution table.

  • If probability is less than 50% look in positive table

  • If probability is greater than 50% look in the negative table

  • Solve for the raw score using the inverse transformation formula:     $x = \mu + z\sigma$

Assessing Normality

Given data, how to assess normality…

• Frequency Histogram Method:

  • Data should appear unimodal, bell-shaped, roughly symmetric, and contain minimal outliers.

• Normal Probability Plot Method (QQ plot)

  • Plots observed data values against their expected normal z-scores.

  • A linear, straight-line pattern confirms that the data is well-approximated by a normal distribution.