z-scores

1. z-scores Chapter 5

2. Social comparison • Upward social comparison: you’ve done worse than others in class •  look up at them •  feel badly about yourself • Downward social comparison: you’ve done better than others in class •  look down at them •  feel better about yourself

3. Relevance to statistics? • Upward social comparison = your score is below the mean • Downward social comparison = your score is above the mean

4. But wait – there’s more • Also depends on how spread out scores are • You engage in upward social comparison, and the standard deviation is small •  you feel quite badly • You engage in upward social comparison, and the standard deviation is big •  you feel less bad

5. Comparing a score to other scores • Need to know about that score, about the mean of the group, and the standard deviation •  all this is captured by z-scores (number of standard deviations a score is above or below the mean of the group it’s from) • Remember: every score in a group gets its very own z-score • Positive z-score = above mean • Negative z-score = below mean • Value of z-score = how many standard deviations above or below the mean

6. Calculating a z-score • +/- part, from score minus mean •  positive value if above •  negative value if below •  zero if at the mean • Number of standard deviations away from the mean, from divide difference between score and mean by standard deviation •  more variable distribution, score needs to be farther from the mean to stand out •  less variable distribution, score doesn’t need to be that far from the mean to stand out •  size of z-score = how much score stands out – how weird person with that score is

7. Bringing it all together • Z-score compares score to the group of scores it’s from  use m rather than M and s rather than SD • z = (X – m)/s • Can also calculate raw score from z-score: • X = zs + m

8. Creating a standardized distribution • If turn each raw score of a distribution into a z-score  standardized distribution • Mean = zero • Standard deviation = one • Shape (e.g., skewness) = exactly the same as it was before

9. Fun with z-scores • Compare scores across groups • Figure out percentile ranks (percentage of people who are below a particular score), if the distribution is normal • Very large + z-score will have lots of people below it • Very large – z-score will have few people below it • People with large z-scores are rare, unlikely to find

10. Keep in mind • Z-scores are dependent on one person’s score, and the scores of the group that person is from • The same score in a different distribution would have a different z-score • A different score in the same distribution would have a different z-score • One z-score per score • Z-score tells about one score, compared to the group that that score is from • Z-scores are in units of standard deviation