Overview
This lecture introduces z-scores, the normal curve, and how standardized scores are used to interpret data, including applications to sample means and the central limit theorem.
Z-Scores and Their Purpose
- Z-scores standardize raw scores, showing how far and in what direction a score is from the mean.
- The sign of a z-score indicates if it is above (positive) or below (negative) the mean.
- Z-scores are based on the assumption of a normal distribution.
Calculating Z-Scores
- For a sample: z = (raw score - sample mean) / sample standard deviation.
- For a population: z = (raw score - population mean μ) / population standard deviation σ.
- Given a z-score, mean, and standard deviation, you can recover the original raw score.
Interpreting Z-Scores
- Z-scores allow comparison across different distributions and help locate a score’s position relative to others.
- Negative z-scores are below the mean; positive are above; absolute value shows distance from the mean.
- The distribution of z-scores (z-distribution) has a mean of 0 and standard deviation of 1.
Standard Normal Curve and Percentiles
- The standard normal curve is a perfectly symmetric, bell-shaped curve with mean 0 and standard deviation 1.
- Specific percentages of scores fall within certain z-score ranges (e.g., ~34% between z = 0 and z = 1).
- Percentiles can be found by summing the area under the curve to the left of a z-score.
Applications to Sample Means & Central Limit Theorem
- Sampling distribution of means: repeated samples’ means form a normal distribution.
- The mean of the sampling distribution equals the population mean.
- Standard error of the mean (SEM) = σ / √n, where σ is population standard deviation and n is sample size.
- Z-score for a sample mean: z = (sample mean - population mean) / SEM.
Key Terms & Definitions
- Z-score — number of standard deviations a score is from the mean.
- Standard deviation (σ or s) — measure of spread in a distribution.
- Standard normal curve — normal distribution with mean 0, standard deviation 1.
- Percentile — percentage of scores below a given value.
- Sampling distribution of means — distribution of sample means from repeated samples.
- Standard error of the mean (SEM) — estimate of spread in the sampling distribution of means (σ/√n).
- Central Limit Theorem — sampling distribution of the mean approaches normality as sample size increases.
- Absolute value — distance from zero, ignoring sign.
Action Items / Next Steps
- Practice calculating z-scores and raw scores from z-scores.
- Be prepared for guided z-score calculation examples in class.
- Review the concept of standard error and sampling distributions for upcoming topics.