Week 3 Chapter 5 Flashcards
equation used to transform z-scores back into X values, for a population
X = μ + zσ
equation used to transform z-scores back into X values, for a sample
X = M + zs
formula used to transform X values into z-scores, for a sample
z = (X - M)/s
formula used to transform X values into z-scores, for a population
z = (X - μ)/σ
The numerator of the equation, X - μ, is a deviation score (Chapter 4). 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. The deviation score is then divided by σ because we want the z-score to measure distance in terms of standard deviation units. The formula performs exactly the same arithmetic that is used with the z-score definition, and it provides a structured equation to organize the calculations when the numbers are more difficult.
raw score
original, unchanged datum that is the direct result of measurement
z-score
specification of the precise location of each X value within a distribution
z-score transformation
relabeling of X values in a population into precise X-value locations within a distribution
standardized distribution
composition of data used to make dissimilar distributions comparable
standardized score
result from relabeling data into new table with positive, whole-number predetermined mean and standard deviation
the process of transforming X values into z-scores serves two useful purposes
- Each z-score tells the exact location of the original X value within the distribution.
- The z-scores form a standardized distribution that can be directly compared to other distributions that also have been transformed into z-scores.
One of the primary purposes of a z-score is to describe the exact location of a score within a distribution. The z-score accomplishes this goal by transforming each X value into a signed number (+ or −) so that
- 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.
Thus, in a distribution of IQ scores with μ = 100 and σ = 15, a score of would be transformed into z = +2.00. The z-score value indicates that the score is located above the mean (+) by a distance of 2 standard deviations (30 points).
The locations identified by z-scores are
the same for all distributions, no matter what mean or standard deviation the distributions may have.
It is possible to transform every X value in a population into a corresponding
z-score. The result of this process is that the entire distribution of X values is transformed into a distribution of z-scores
The new distribution of z-scores has characteristics that make the z-score transformation a very useful tool. Specifically, if every X value is transformed into a z-score, then the distribution of z-scores will have the following properties (part 1):
- Shape: The distribution of z-scores will have exactly the same shape as the original distribution of scores. If the original distribution is negatively skewed, for example, then the z-score distribution will also be negatively skewed. If the original distribution is normal, the distribution of z-scores will also be normal. Transforming raw scores into z-scores does not change anyone’s position in the distribution. For example, any raw score that is above the mean by 1 standard deviation will be transformed to a z-score of +1.00, which is still above the mean by 1 standard deviation. Transforming a distribution from X values to z values does not move scores from one position to another; the procedure simply relabels each score (see Figure 5.5). Because each individual score stays in its same position within the distribution, the overall shape of the distribution does not change.
The new distribution of z-scores has characteristics that make the z-score transformation a very useful tool. Specifically, if every X value is transformed into a z-score, then the distribution of z-scores will have the following properties (part 2):
- The Mean: The z-score distribution will always have a mean of zero. In Figure 5.5, the original distribution of X values has a mean of μ = 100. When this value, X = 100, is transformed into a z-score, the result is
Z = (X - μ)/σ = (100 - 100)/100 = 0
Thus, the original population mean is transformed into a value of zero in the z-score distribution. The fact that the z-score distribution has a mean of zero makes the mean a convenient reference point. Recall from the definition of z-scores that all positive z-scores are above the mean and all negative z-scores are below the mean. In other words, for z-scores, μ = 0
The new distribution of z-scores has characteristics that make the z-score transformation a very useful tool. Specifically, if every X value is transformed into a z-score, then the distribution of z-scores will have the following properties (part 3):
The Standard Deviation: The distribution of z-scores will always have a standard deviation of 1. In Figure 5.5, the original distribution of X values has μ = 100 and σ = 10. In this distribution, a value of X = 110 is above the mean by exactly 10 points or 1 standard deviation. When X = 110 is transformed, it becomes z = +1.00 , which is above the mean by exactly 1 point in the z-score distribution. Thus, the standard deviation corresponds to a 10-point distance in the X distribution and is transformed into a 1-point distance in the z-score distribution. The advantage of having a standard deviation of 1 is that the numerical value of a z-score is exactly the same as the number of standard deviations from the mean. For example, a z-score of z = 1.50 is exactly 1.50 standard deviations from the mean.
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 value is transformed into z-scores. Specifically,
- 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 Mz = 0
- the sample of z-scores will have a standard deviation of sz = 1
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.
This process ensures that each individual has exactly the same z-score location in the new distribution as in the original distribution. The following example demonstrates the standardization procedure for a population.
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. For example, an individual with a z-score near 0 is located in the center of the population and would be considered to be a fairly typical or representative individual. However, an individual with an extreme z-score, beyond +2.00 or −2.00 for example, would be considered “noticeably different” from most of the individuals in the population. Thus, we can use z-scores to help decide whether the treatment has caused a change. 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.
Although 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,” you should realize that these z-score boundaries were not determined by any mathematical rule.
