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School

University of North Dakota
Department

Psychology

Course Code

PSYC 241

Professor

Kristen Leighton

Lecture 5: z-scores

● Distributions

○ How do we describe distribution?

■ Shape (symmetrical, +/- skewed, bimodal, etc.)

■ Central tendency: one number that describes typical score (mean, median,

mode)

■ Variability: quantitative measure of the degree to which scores in a distribution

are spread out or clustered together

● Normal distributions

○ Normal distributions are precisely defined and always the same shape

○ If you randomly choose a score from a normal distribution, it is most likely to be the

mean

○ If you rephrase the question in terms of 'how close would the score be to the mean,' it is

most likely to be within one standard deviation of the mean

● Z-scores

○ Our goal is to identify exactly where any score lies in this distribution

○ Raw score (X value) provides little information about comparison to other scores in

distribution

○ Score may be relatively low, average, or high

○ If raw score is transformed into z-score, the z-score tells us exactly where the score is

located relative to all the other scores

○ Changing and X value into a z-score creates a signed number

■ The sign (+/-) = whether X value is above or below the mean

■ The numerical value of the z-score = the number of standard deviations

between X and the mean of the distribution

○ z-scores are not statistics

○ z-scores are transformed scores

■ Same mathematical operation on every score

■ Transform each individual score to a z-score to see where each score lies in the

distribution

○ z-scores tell us the same things a SD does:

■ Distance from mean

■ Direction from mean

● Z-scores and location

○ A score that is two standard deviations above the mean. . .

○ A score that is one standard deviation below the mean. . .

● Transforming X to Z

○ Two-step process:

■ Compare each score to the mean (deviation scores = distance and direction

score from mean)

■ Divide by one standard deviation

■ Z = (X-μ)/σ

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