Normal Distributions

The 68% Rule: Approximately $68\%$ of all data points in a normal distribution fall within one standard deviation ($\pm 1\sigma$) of the mean.

The 95% Rule: Approximately $95\%$ of the data points fall within two standard deviations ($\pm 2\sigma$) of the mean, covering the vast majority of the population.

The 99.7% Rule: Nearly all data ($99.7\%$) falls within three standard deviations ($\pm 3\sigma$) of the mean, meaning values outside this range are considered highly unusual or extreme outliers.

Predictive Power: This rule allows researchers to calculate the probability of a specific score occurring based solely on the mean and standard deviation of the dataset.

Standardization: To compare different datasets, researchers convert raw scores into Z-scores, which represent how many standard deviations a value is from the mean.

The Z-Score Formula: The transformation is calculated as $z = \frac{x - \mu}{\sigma}$, where $x$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Standard Parameters: A 'Standard Normal Distribution' always has a mean of $0$ and a standard deviation of $1$, allowing for the use of universal probability tables.

Interpretation: A positive Z-score indicates a value above the mean, while a negative Z-score indicates a value below the mean.

Identify the Peak: Always look for the highest point on the frequency graph; in a normal distribution, this must be the center of the $x$-axis range.

Check the Tails: Ensure the curve approaches but never touches the $x$-axis; if it touches or crosses, it is technically not a theoretical normal distribution.

Standard Deviation Logic: If a question asks for the percentage of people scoring between two values, check if those values are exactly $1$, $2$, or $3$ standard deviations away to use the Empirical Rule.

Common Trap: Do not assume every symmetrical graph is 'normal'; a normal distribution must follow the specific mathematical proportions of the bell curve (e.g., the $68-95-99.7$ ratios).