Understanding Z-Scores and the Normal Distribution in Statistics
This chapter delves into the concept of Z-scores, defining them as the number of standard deviations a score is above or below the mean. It outlines formulas for converting raw scores to Z-scores and vice versa, along with examples. The chapter explains the characteristics of the normal distribution, including the 50-34-14 rule, which describes the percentages of scores relative to the mean. A section is dedicated to using Z-scores to compare different distributions, including comparisons between SAT and ACT scores, and practical steps for interpreting normal distribution tables.
Understanding Z-Scores and the Normal Distribution in Statistics
E N D
Presentation Transcript
Chapter 3 Z Scores & the Normal Distribution Part 1 Thurs. Aug. 29, 2013
Z Scores • Number of standard deviations a score is above or below the mean • Formula to change a raw score to a Z score: Sign of z indicates whether score is above or below mean. Magnitude of z – how many SDs above/below Example in class:
Z Scores • Formula to change a Z score to a raw score: • Example?
Z scores • Distribution of Z scores • Mean = 0 • Standard deviation = 1 • It’s a standardized scale • z scores can be used to compare 2 scores from same or different distributions • Z= +3 represents score that is very deviant from the mean, no matter the distribution
Z as common unit of comparison • Comparing SAT and ACT scores • Which is a higher score? Getting a 620 on the SAT-verbal or a 27 on the ACT-verbal? • Information about ACT and SAT M and SD: • Did this person do better on the SAT or ACT?
The Normal Distribution • In a normal curve (normal distribution), there will always be a standard % of scores between the mean and 1 and 2 standard deviations from the mean: • 50% all scores fall above the mean, 50% below • 34% betw M and +1 SD, 34% betw M and –1SD • 14% betw +1 and +2 SD, 14% betw –1 and –2SD • 2% above +2 SD and below –2 SD
The Normal Distribution • Can remember this as the ‘50-34-14 rule’ • Appendix A is the normal curve table with Z scores • Gives the precise % of scores between the mean (which has a Z score of 0) and any other Z score • What % of scores are betw M and z = .75? • Table lists only positive Z scores, but since ND is symmetrical, same % for neg z scores (just look up its corresponding pos z)
Using the Z table: From z to % • Steps for figuring the % of scores above or below a particular raw or Z score: 1. Table uses z scores, so convert raw score to Z score (if necessary) 2. Draw normal curve, indicate where the Z score falls on it, shade in the area for which you are finding the % 3. Make rough estimate of shaded area’s percentage (using 50%-34%-14% rule): • what % does it look like the shaded area covers? (use your judgment – we’ll check on this later…)
(cont.) 4. Find exact % using normal curve table • that is, look up the z score you calculated in step 1 and find the % associated with it. 5. If needed, add or subtract 50% from this • since table only gives % between M and Z, if you’re interested in % above Z, subtract from 50% (see example) 6. Check to determine if your answer makes sense given the graph you drew in Step 3
