Overview
This lecture explains how to determine which day's cash register receipts were more "unusual" using z-scores and discusses how to compare the unusualness of two different observations.
Calculating Z-Scores for Each Day
- To assess unusualness, calculate the z-score for each day's receipts.
- Monday: $10,700 taken in; Thursday: $7,640 taken in.
- Population mean (average) is $9,200, population standard deviation is $400.
- Z-score formula: ( z = \frac{x - \mu}{\sigma} ), where ( x ) is the observed value, ( \mu ) is the mean, ( \sigma ) is the standard deviation.
- Monday: ( z = \frac{10,700 - 9,200}{400} = 3.75 )
- Thursday: ( z = \frac{7,640 - 9,200}{400} = -3.9 )
Interpreting Z-Scores and Unusualness
- Z-scores beyond ±3 are considered definitely unusual.
- Both Monday and Thursday have z-scores beyond 3 standard deviations from the mean.
- Both days’ receipts are considered definitely unusual.
Comparing Unusualness
- To determine which day is more unusual, compare the absolute values of the z-scores.
- Ignore the positive or negative sign and focus on the magnitude.
- Monday’s absolute z-score: 3.75; Thursday's: 3.9.
- The larger absolute value (Thursday) indicates a more unusual event.
- Thursday's receipt is more unusual because its z-score is further from zero.
Key Terms & Definitions
- Z-score — Number of standard deviations an observation is from the mean.
- Standard deviation (( \sigma )) — Measure of the spread of data values around the mean.
- Mean (( \mu )) — The average value in a data set.
- Unusual (Statistical) — Observation that is more than 3 standard deviations away from the mean.
Action Items / Next Steps
- Practice calculating and interpreting z-scores for various data points.
- Review definitions of mean, standard deviation, and z-score for next class.