Normal Distributions and Z-Scores

Statistical Terminology and Notation

• Population: All measurements or observations of interest.

• Sample: A subset of the population.

• Parameter: A numerical value describing a population (e.g., $\mu$, $\sigma$, $p$).

• Statistic: A numerical value describing a sample (e.g., $\bar{x}$, $S_x$, $\hat{p}$, $n$).

• Notation Breakdown:

  • Mean: Sample $= \bar{x}$, Population $= \mu$

  • Standard Deviation: Sample $= S_x$, Population $= \sigma$

  • Proportion: Sample $= \hat{p}$, Population $= p$

  • Probability is represented by $P$.

Properties of the Normal Distribution

• Shape: Symmetric and unimodal with no outliers ($\text{mean} = \text{median} = \text{center}$).

• Asymptote: The mathematical model uses the $x$-axis as a horizontal asymptote.

• Total Area: The total area beneath the normal curve is $1.00$, representing $100\%$ of data.

• Area Interpretation: The area under any segment of the curve corresponds to the proportion or percentage of data along that interval.

• Variable Notation: A normally distributed quantitative variable $x$ is denoted as $x \sim N(\mu, \sigma)$.

Empirical Rule (68-95-99.7 Rule)

• Applies strictly to normal distributions:

  • $\approx 68\%$ of data lie within one standard deviation of the mean ($\mu \pm 1\sigma$).

  • $\approx 95\%$ of data lie within two standard deviations of the mean ($\mu \pm 2\sigma$).

  • $\approx 99.7\%$ of data lie within three standard deviations of the mean ($\mu \pm 3\sigma$).

Standard Normal Distribution and Z-Scores

• Standard Normal Distribution ($z$): A standardized normal model with mean $\mu = 0$ and standard deviation $\sigma = 1$.

• $z$-score Definition: The number of standard deviations a given value lies above (positive) or below (negative) the mean; $z = 0$ at the mean.

• Formula:

  • $\text{z-score} = \frac{\text{value} - \text{mean}}{\text{SD}}$

  • Population distribution formula: $z = \frac{x - \mu}{\sigma}$

• Equivalence: The area bordering a raw value on its specific normal distribution equals the area bordering its corresponding $z$-score on the standard normal curve.

TI-84 Calculator Functions

• normalcdf (accessed via 2ND VARS 2):

  • Purpose: Calculates the area, probability $P(\text{LB} < \text{value} < \text{UB})$, or proportion between bounds.

  • Syntax: normalcdf(LB, UB, mean, SD)

  • Defaults: If omitted, mean $= 0$ and SD $= 1$.

  • Infinite Bounds: Extreme lower bound ($-\infty$) is entered as -1E99; extreme upper bound ($\infty$) is entered as 1E99 (enter E via 2ND ,).

• invnorm (accessed via 2ND VARS 3):

  • Purpose: Finds the cutoff value or percentile corresponding to a specified cumulative left-tail area.

  • Syntax: invnorm(Area, mean, SD, LEFT)

  • Input Requirement: Area must be entered in decimal form as left-tail cumulative area (e.g., for the top $10\%$, use $\text{Area} = 1 - 0.10 = 0.90$).