social statistics ch 5 z scores

A distribution of scores has a mean of μ= 100 and a standard deviation of σ=10. What z-score corresponds to a score of X= 130 in this distribution?

3.00

A distribution of scores has a mean of μ=86 and a standard deviation of σ=7. What z-score corresponds to a score of X=95 in this distribution?

9/7 = 1.29

a z score

A z-score specifies the precise location of each X value within a distribution. The sign of the z-score (+ or −) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and μ.

how to find the mean In a sample with a standard deviation of s=6, a score of X=33 corresponds to z= +1.50. What is the mean for the sample?

Again, we begin with the z-score value. In this case, a z-score of +1.50 indicates that the score is located above the mean by a distance corresponding to 1.50 standard deviations. With a standard deviation of s=6, this distance is (1.50)(6) = 9 points. Thus, the score is located 9 points above the mean. The score is X=33, so the mean must be 24.

What information is provided by the magnitude (numerical value) of the z-score?

How many standard deviations the score is away from the mean

purpose of a z score

The purpose of z-scores, or standard scores, is to identify and describe the exact location of each score in a distribution. second purpose is to standardize an entire distribution. the sign tells whether the score is located above (+) or below (−) the mean, and the number tells the distance between the score and the mean in terms of the number of standard deviations.

in a distribution of IQ scores with μ and σ, a score of would be transformed into .

The z-score value indicates that the score is located above the mean (+) by a distance of 2 standard deviations (30 points).

raw scores

These original, unchanged scores that are the direct result of measurement are called raw scores.

finding the standard deviation (σ) In a population with a mean of μ=65, a score of X=59 corresponds to z = - 2.00. What is the standard deviation for the population?

To answer the question, we begin with the z-score value. A z-score of −2.00 indicates that the corresponding score is located below the mean by a distance of 2 standard deviations. You also can determine that the score (X=59) is located below the mean (μ=65) by a distance of 6 points. Thus, 2 standard deviations correspond to a distance of 6 points, which means that 1 standard deviation must be σ=3 points

a z-score always consists of two parts:

a sign (+ or −) and a magnitude (the numerical value). Both parts are necessary to describe completely where a raw score is located within a distribution.

determining a raw score from a z score

converting a z score into an x

X- μ

deviation score

the deviation measures the

distance in points between X and m and the sign of the deviation indicates whether X is located above or below the mean

Is a standardized score necessarily a z-score?

no

formula for transforming scores into z scores

z = X-μ / σ

For a distribution with a mean of μ=60 and σ=8, what X value corresponds to a z-score of z = -1.50?

X= 48

formula for converting Z into X

X= μ + Zσ

The deviation score is then divided by σ because

we want the z-score to measure distance in terms of standard deviation units.