CHAPTER 5 Z-Scores

Learning Outcomes:

Know the importance of Z-scores and location in a distribution;
Understand and apply how to use z-scores to standardize a distribution;
Understand the importance of Z-scores and their limitations;
Understand other standardized distributions based on Z-scores

CHAPTER 5:

Z-SCORES

5.1 Introduction

z-scores/standard score: the purpose is to identify and describe the exact location of each score in a distribution and standardize an entire distribution

= A score by itself does not necessarily provide much information about its position within a distribution

An example of a Z-score would be if the average score for a group of values is 5, and one value is 10, then the Z-score for that particular value is 5 (10−5)/1

The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.

raw score: original, unchanged scores that are the direct result of measurement

= often transformed into new values that contain more information so they are more meaningful

5.2 Z-scores and Locations in a Distribution

z-score transforms each X value into a signed number (+ or -) so that:

1. the sign tells whether the score is located above (+) or below (-) the mean

2. the number tells the distance between the score and the mean in terms of the number of standard deviations

z = (X-mean)/standard deviation
X = mean + (z-score x standard deviation)

5.3 Other Relationships between z, X, the Mean, and the Standard Deviation

be able to ﬁnd the standard deviation using the X, the mean, and the z-score

5.4 Using z-Scores to Standardize a Distribution

Population and Sample Distributions

if every X value is transformed into a z-score, the distribution will have the following properties:

Shape: same shape as the original distribution of scores
Mean: always have a mean of zero, which makes it a convenient reference point
Standard Deviation: always have a standard deviation of 1 so the numerical value of a z-score is the same as the number of standard deviations from the mean.

Using z-Scores for Making Comparisons

standardized distribution: composed of scores that have been transformed to create predetermined values for mean and standard deviation.
used to make dissimilar distributions comparable.

5.5 Other Standardized Distributions Based on z-Scores

Transforming z-scores to a Distribution with a Predetermined Mean and SD

1. Original scores are transformed into z-scores

2. z-scores are transformed into new X values so that the speciﬁc mean and SD are attained

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