Z-scores

What do you need to calculate z-score?

• Individual score (X)

• Population mean

• Population standard deviation (not estimated)

Z-score intervals + area coverage

• ±1 SD = 68%

• ±2 SD = 95%

• ±3 SD = 99.7%

Assumptions (3)

• Normality

• Independence of observations

• Known (population) standard deviation

3 question types

• Probability of a z-score

• Raw score from probability

• Representativeness of sample mean compared to population

How do you calculate the probability of a z-score? (3)

• Calculate z (using formula)

• Look up z in table (using z-score + corresponding alpha level)

  • If z-score is positive, use positive z-table

  • If z-score is negative, use negative z-table

• That’s the probability of getting that z-score

How do you answer this kind of question? (2)
“What is the IQ score above which 5% of the population fall?”

• “Above which 5% of the population fall” → 95th percentile score

  • Look this up in z-table → find corresponding z-score

• Which IQ score corresponds to obtained Z-score? → Use X = mean + z x SD to calculate raw IQ score

“How likely is it that Sample A is representative of the population?”: required info (4)

• Population: mean, SD

• Sample: sample size, mean

“How likely is it that sample A is representative of the population?”: steps (4)

1. Calculate standard error of the distribution of sample means

2. Calculate the z-score for sample mean (using sample mean - population mean / standard error (which is your standard deviation))

3. Find probability (i.e. p), using obtained z-score and alpha level = 0

4. Is the obtained probability <.05? If so, sample is unlikely to be representative of population

Important note (1)

• Careful with “score below” vs “score higher than”: with latter, you have to subtract probability from 1