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Chp. 5 - z-Scores: Location of Scores & Standardized Distributions -…
Chp. 5 - z-Scores: Location of Scores & Standardized Distributions
z-Scores & Locations in a Distribution
z-score transforms each X value into a signed number (+ or -)
The sign tells whether the score is located above (+) or below (−) the mean
The number tells the distance between the score and the mean in terms of the number of standard deviations.
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 locations identified by z-scores are the same for all distributions, no matter what mean or standard deviation the distributions may have.
z-Score Formula for a Population
z = (X - μ)/σ
X - μ , is a
deviation score
Computing z-Scores for Samples
The sign of the z-score indicates whether the X value is above (+) or below (−) the sample mean
The numerical value of the z-score identifies the distance from the sample mean by measuring the number of sample standard deviations between the score (X) and the sample mean (M)
Other Relationships between z, X, the Mean, and the Standard Deviation
z-Score establishes a relationship between the score, the mean, and the standard deviation
Using z-Scores to Standardize a Distribution
Population distributions
If every X value is transformed into a z-score, then the distribution of z-scores will have the following properties:
Shape
- The distribution of z-scores will have exactly the same shape as the original distribution of scores.
Mean
- The z-score distribution will
always
have a mean of zero.
Standard Deviation
- The distribution of z-scores will always have a standard deviation of σ .
z-score transformation
- transforming every X value in a population into a corresponding z-score
Sample distributions
If all the scores in a sample are transformed into z-scores, the result is a sample distribution of z-scores. The transformed distribution of z-scores will have the same properties that exist when a population of X values is transformed into z-scores.
the distribution for the sample of z-scores will have the same shape as the original sample of scores.
the sample of z-scores will have a mean of M = 0.
the sample of z-scores will have a standard deviation of s = 1
Note: the set of z-scores is still considered to be a sample (just like the set of X values) and the sample formulas must be used to compute variance and standard deviation.
Using z-Scores for making comparisons
When any distribution (with any mean or standard deviation) is transformed into z-scores, the resulting distribution will always have a mean of μ and a standard deviation of σ .
Because all z-score distributions have the same mean and the same standard deviation, the z-score distribution is called a standardized distribution .
standardized distribution
: composed of scores that have been transformed to create predetermined values for μ and σ . Standardized distributions are used to make dissimilar distributions comparable.
One advantage of standardizing distributions is that it makes it possible to compare different scores or different individuals even though they come from completely different distributions.
Other Standardized Distribution Based on z-Scores
Although z-score distributions have distinct advantages, many people find them cumbersome because they contain negative values and decimals. For this reason, it is common to standardize a distribution by transforming the scores into a new distribution with a predetermined mean and standard deviation that are positive whole numbers.
The goal is to create a new (standardized) distribution that has “simple” values for the mean and standard deviation but does not change any individual’s location within the distribution.
The procedure for standardizing a distribution to create new values for the mean and standard deviation is a two-step process that can be used either with a population or a sample:
The original scores are transformed into z-scores.
The z-scores are then transformed into new X values so that the specific mean and standard deviation are attained.
Looking Ahead to Inferential Statistics
inferential statistics are techniques that use the information from samples to answer questions about populations.
Notice that the interpretation of the research results depends on whether the sample is noticeably different from the population. One technique for deciding whether a sample is noticeably different is to use z-scores.
Specifically, if the individuals who receive the treatment finish the research study with extreme z-scores, we can conclude that the treatment does appear to have an effect.
It is reasonable to describe individuals with z-scores near 0 as “highly representative” of the population, and individuals with z-scores beyond ±2.00 as “extreme,”
The purpose of z-scores (standard scores) is to identify the exact location of each score in a distribution (Each z-score tells the exact location of the original X value within the distribution.)
Another purpose for z-scores is to standardize an entire distribution. The z-scores form a standardized distribution that can be directly compared to other distributions that also have been transformed into z-scores.