Z-Scores and Normal Distributions Cheat Sheet

Z-scores standardize data by measuring how many standard deviations a data point is from the mean, allowing for comparisons across different distributions and enabling probability calculations using the normal distribution.

Core Principles

Normal Distribution: A bell-shaped curve where data is symmetrically distributed around the mean, with most values clustering near the center and fewer values at the extremes.
Standard Deviation: A measure of the dispersion or spread of data points around the mean.
Z-Score: A standardized score representing the number of standard deviations a raw score is from the mean (z = (x - μ) / σ).

Action Steps

1. Calculate the mean (μ) and standard deviation (σ) of your dataset.
2. For each data point (x), calculate its z-score using the formula z = (x - μ) / σ.
3. Use z-scores to find the probability of a data point falling within a certain range by referencing a standard normal distribution table or using statistical software.
4. Compare data points from different distributions by converting them to z-scores.

Formulas

Z-Score: $z = \frac{x - \mu}{\sigma}$
Raw Score from Z-Score: $x = \mu + z\sigma$

Key Terms

Mean (μ): The average of a dataset.
Standard Deviation (σ): A measure of the spread of data around the mean.
Z-Score: A standardized score indicating the number of standard deviations a data point is from the mean.
Normal Distribution: A probability distribution characterized by its bell shape and symmetry around the mean.
Empirical Rule (68-95-99.7 Rule): A guideline stating that for a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Real World Examples

Comparing student scores on two different exams with different means and standard deviations.: Convert each student's score to a z-score to determine relative performance on each exam.
Determining the likelihood of a specific measurement (e.g., height, weight) occurring in a population.: Calculate the z-score for the measurement and use a z-table to find the corresponding probability.