PSY 360 - Chapter 5: Z-Scores
What are three other standardized distributions based on z-scores?
1) Select a mean and standard deviation you want to use. 2) Transform original X values into z-scores. 3) Transform z-scores into new X values.
During the process of changing an X value into a z-score involves creating a (+ or -) number, what are the two things that it can tell us?
1) The sign of the z score (+ or -) tells whether the score is located above (+) or below (-) the mean. 2) The number (magnitude) tells the distance between the score and the mean in terms of the number of standard deviations.
What are the three things to be mindful of when standardizing z-scores in a distribution?
1. Shape 2. The Mean 3. The Standard Deviation
In Figure 5.7, compare and contrast both the distributions.
Both distribution have exactly the same shape. In the z-score distribution, you could calculate the mean and the standard deviation - just like we've done with X distribution. The mean would be 0 and the SD would be 1.
What can be inferred about a distribution if every X value can be transformed into a corresponding z-score?
Every X value can be transformed into a corresponding z-score. Thus, an entire distribution of x values can be transformed into a distribution of z-scores.
What is the formula to transform z-scores back to an x value?
X = M + zs
In Figure 5.8, explain the example of other standardized distribution based on z-scores?
Joe's position relative to the other students hasn't change. IQ tests are often standardized so they have the same mean and standard deviation. This makes it possible to compare IQ scores even when they come from different tests.
What is the formula to calculate z-scores for samples?
The definition of a z-score is the same for both samples and populations, and the formulas are also the same, except the SAMPLE mean and SAMPLE standard deviation is used instead of POPULATION mean and POPULATION standard deviation. z = (X - M)/s
What affect does the z-score have on mean?
The z-score distribution will ALWAYS have a mean of zero. Ex: The original distribution has a mean of mu = 100. When that value of X = 100 is transformed into a z-score, the result is: z = (X - mu)/lowercase sigma = (100 - 100)/10 = 0 - Figure 5.5 - The mean is transformed into a value of 0 and the standard deviation is transformed into a value of 1.
What affect does the z-score have on standard deviation?
The z-score distribution will ALWAYS have a standard deviation of 1. Ex: The distribution of X values has a mean of mu = 100 and lowercase sigma = 10. The score of X = 110 is 10 points above the mean, or 1 standard deviation. When X = 110 is transformed into a z-score, it becomes z = +1.00, which is above the mean by exactly 1 point in the z-score distribution. The SD of 10 points in the X distribution was transformed into a SD of 1 point in the Z distribution.
What formula can be used to transform a z-score into a raw score?
X = mu + (z)(lowercase sigma)
What formula can be used to transform a raw score into a z-score aka z-score formula?
Z = ( x - mu)/ lowercase sigma *lowercase sigma = standard deviation * mu = mean of population
What is another example of the relationship between z, X, mu, and lowercase sigma?
A population has a standard deviation of lowercase sigma of 4, and a score of X =33 corresponds to z = +1.50. What is the mean? - Again, start by looking at the z-score (1.5 above mean). - SD is 4, so the score is located (lowercase sigma)(z) = (4)(1.5) = 6 points above the mean - mu = 33 - 6 = 27
Explain an example of a z-score.
A score that is located two standard deviations above the mean will have a z-score of +2.00. And, a z-score of +2.00 always indicates a location above the mean by two standard deviations.
What does the term standardized distribution mean?
Because all z-score distributions have the same mean and the same standard deviation, the z-score distribution is called a standardized distribution. By standardizing distributions, we can then make comparisons. *In the book: A standardized distribution is composed of scores that have been transformed to create predetermined values for mean (mu) and standard deviation (lowercase sigma). Standardized distributions are used to make dissimilar distributions comparable.
In Figure 5.2, which distribution of exam scores are better?
Both distributions have a mean of 70, but the position of the score x = 76 are very different. It would be better off with the top scores.
By itself, a raw score (an X value) doesn't do what?
By itself, a raw score (an x value) doesn't provide much info about how that particular score compares with other values in the distribution. Ex: A score of x = 70 could be a low score, an average score, or an extremely high score depending on the mean and the standard deviation of the distribution?
What is an example of transforming a z-score into a raw score?
Ex: A distribution with a mean of mu = 60 and lowercase sigma = 5, what X value corresponds to a z-score of z = -3.00? X = mu + (z)(lowercase sigma) X = 60 + (-3.00)(5) X = 60 - 15 = 45
What is an example of the relationship between z, X, mu, and lowercase sigma?
Ex: A population has a mean of mu = 65, and a a score of X =59 corresponds to z = -2.00. What is the standard deviation for the population? - Start by looking at the z=score (located 2 SD below the mean). - Compare the score and mean (X = 59 is located 6 points below the mean of mu = 65). - Thus, 2 SD corresponds to a distance of 6 points, so 1 SD must be lowercase sigma = 3.
According to Figure 5.3, what relationship is described?
Figure 5.3 shows a population distribution with various positions identified by their z-scores values. --> All z-scores above the mean are positive and all the z-scores below the mean are negative. --> The sign of a z-score tells you immediately whether the score is located above or below the mean. Figure 5.3 does not give any specific values for the population mean or the standard deviation. The locations identified by z-scores are the same for all distributions, no matter what mean or standard deviation the distribution may have been. *The numerical value tells you how many standard deviations the score is away from the mean, and the sign tells you if the score is located above or below the mean.
What can be a raw score be transformed into?
If a raw score is transformed into a z-score, the value of the z-score tells us exactly where the score is located relative to the other scores.
What are the three properties of the z-score distribution to be mindful of when standardizing a sample distribution?
If all of the scores in a sample are transformed into a distribution of z-scores, then the new distribution will have all the same properties as when a population of X values is transformed into Z-scores. 1) The sample of z-scores will have the same shape as the original sample of score. 2) The sample of z-scores will have a mean of M = 0. 3) The sample of z-scores will have a standard deviation of s = 1.
Why do most people find z-scores cumbersome? What is more convenient?
Most people find z-scores cumbersome because they consist of decimal values and negative numbers. It is often convenient to standardize the distribution of z-scores into numerical values.
What is an example of using z-scores to make comparisons? How would you make a comparison?
Suppose that you received a 89 on your last exam in Statistics, and you received on a Biology exam. These two distribution have different means and standard deviations, so you can't make a comparison on how you did on each exam relative to the rest of the class. To make comparisons between distributions, we need mean and SD. Statistics: mu = 74, sigma = 14 Biology: mu = 48, sigma = 4 We can now compute the two z-scores to find the location of the grades. Statistics: z = (89-74)/14 = +1.07 Biology: z = (70-74)/4 = -1.00
What is an example of other standardized distributions based on z-scores?
Suppose you want to standardize a distribution that has a mean of 57 and a standard deviation of 14. We want to simplify this distribution by transforming it into a new, standardized distribution with a mean of 50 and a standard deviation of 10. - Transform each raw score into a z-score (using original mean and SD). - Transform each z-score into an X value in the new distribution that has a mean of 50 and a standard deviation of 10. Table 5.1/ Example 5.7 Original Scores mu = 57 and sigma = 14 Maria X =64 Joe X = 43 Z-Score Location z = +0.50 z = -1.00 Standardized Scores mu = 50 and sigma = 10 x =55 x = 40
In Figure 5.10, explain the distribution of weights for the population of adults rats.
The distribution of weights for the population of adult rats. Note that individuals with z-scores near 0 are typical or representative. However, individuals with z-scores beyond +2.00 or -2.00 are extreme and noticeably different from most of the others in the distribution.
What affect does the z-score distribution have on shape?
The z-score distribution will have exactly the same shape as the original distribution of a score. -> If the original distribution is negatively skewed, z-score distribution will be negatively skewed. -> For example, any raw score above the mean by 1 SD will be transformed into a z-score of +1.00, which is still above the mean by 1 SD. - Figure 5.5 - The mean is transformed into a value of 0 and the standard deviation is transformed into a value of 1.
What is the bottom line when any distribution is transformed into z-scores?
When any distribution is transformed into z-scores, the z-score distribution will ALWAYS have a mean of 0 and a standard deviation of 1.