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z-Scores: Location of Scores and Standardized Distributions, image, image,…
z-Scores: Location of Scores and Standardized Distributions
5-1 Introduction
In this chapter, we should be learning a statistical technique that uses the mean and the standard deviation to transform each score (X Value) into a z-score or a standard score.
The purpose of
z-scores
, is to identify and describe the exact location of each score in a distribution.
5-2 z-Scores and Locations in a Distribution
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.
The formula for transforming scores into z-scores is:
z = X - μ / σ
. The numerator of the equation is a deviation score. The deviation measures the distance in points between X and μ and the sign of the deviation indicates whether X is located above or below the mean. The formula provides a structured equation to organize the calculations when the numbers are more difficult.
The definition and the purpose of a z-score is the same for a sample as for a population, provided that you use the sample mean and the sample standard deviation to specify each z-score location.
each X value in a sample can be transformed into a z-score through this equation:
z = X - M / s
each z-score can be transformed back into an X value through this equation:
X = M + zs
5-4 Using z-Scores to Standardize a Distribution
z-score transformation
consists of transforming X values into z-scores resulting in a distribution with the same shape as the original, but with a mean of zero and a standard deviation of one. This transformation does not change the relative positions of scores, only their labels, so the distribution remains unchanged in terms of its shape. The z-score provides a useful reference where the numerical value directly represents the number of standard deviations a score is away from the mean.
When all scores in a sample are transformed into z-scores, the resulting distribution of z-scores will maintain the same shape as the original sample. This z-score distribution will have a mean of zero and a standard deviation of one. Despite the transformation, the z-scores are still considered a sample, so sample formulas must be used for computing variance and standard deviation.
A
standardized distribution
is composed of scores that have been transformed to create predetermined values for μ and σ. Standardized distributions are used to make dissimilar distributions comparable.
When any distribution (with any mean or standard deviation) is transformed into z-scores, the resulting distribution will always have a mean of μ = 0 and a standard deviation of σ =1. All z-score distributions have the same mean and the same standard deviation, the z-score distribution is called a
standardized distribution
.
5-3 Other Relationships between z,
X
, the Mean, and the Standard Deviation
Z-scores represent how far a score is from the mean in terms of standard deviations. They help in understanding relationships between a score, the mean, and the standard deviation, which can be used to answer various questions about scores and distributions. In examples, you can determine unknown values (like standard deviation or mean) by analyzing the distance between scores and the mean using z-scores.
5-5 Other Standardized Distributions Based on z-Scores
Z-score distributions can be cumbersome due to negative values and decimals, so scores are often standardized to create a distribution with a predetermined mean and standard deviation that are positive whole numbers. This process maintains individuals' relative positions within the distribution while simplifying the values, commonly used in tests like the SAT or IQ tests, where scores are standardized to specific means and standard deviations. The standardization process involves first transforming scores into z-scores and then converting those z-scores into new values with the desired mean and standard deviation.
5-6 Looking Ahead to Inferential Statistics
As we have learned before
inferential statistics
use sample data to draw conclusions about populations, helping to interpret research results. In research studies, a sample is selected from a population, and a treatment is administered to the sample. The researcher compares the treated sample with the original population to determine if the treatment had an effect, based on whether the sample shows significant differences.
