Statistics Chapter 5: z-scores and standardized distribution

A "z-score" specifies:

the precise location of each X value aka raw score within a distribution

•To evaluate the effect of the treatment, the researcher simply compares the treated sample with the original population. -If the individuals in the sample are noticeably different from the individuals in the original population, the researcher has evidence that the treatment has had an effect.

•On the other hand, if the sample is not noticeably different from the original population, it would appear that the treatment has no effect. •z-scores can tell us whether a sample is noticeably different

•It is possible to transform every X value in a distribution into a corresponding z-score. -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. •The z-score distribution will always have a mean of zero. •The distribution of z-scores will always have a standard deviation of 1. •Extreme scores: z=+2.00 or higher, Z=-2.00 or lower •Central scores: z scores in between 0 and 1, z scores in between 0 and -1

Looking Ahead to Inferential Statistics •A typical research study begins with a question about how a treatment will affect the individuals in a population.

-Because it is usually impossible to study an entire population, the researcher selects a sample and administers the treatment to the individuals in the sample.

A second purpose for z-scores

-Is to standardize an entire distribution. -For example: the distribution of IQ scores, they usually have a mean of 100 and a standard deviation of 15.

•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.

-Normally, if two scores come from different distributions, it is impossible to make any direct comparison between them. -Using z-scores makes such comparisons possible.

•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.

The purpose of the z-scores or standard scores is

-To identify and describe the exact location of each score in a distribution. -To find the location of your score, you must have information about the other scores in the distribution.

z-scores contain decimals and negative values

-it is common to standardize a distribution by transforming the scores into a new distribution with a predetermined mean and standard deviation that are whole round numbers. -standardized scores are frequently used in psychological or educational testing (SAT, IQ tests)

A population with μ=85 and σ=12 is transformed into z-scores, After the transformation, the population of z-scores will have a standard deviation of_____?

1.00 because it will be one standard deviation from one z-score to the next

•If all the scores in a sample are transformed into z-scores, the result is a sample 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.

1.The sample will have the same shape as the original sample. 2.The sample will have a mean of Mz = 0. The sample will have a standard deviation of sz = 1

•In a population with a mean of μ = 65, a score of X = 59 corresponds to z = −2.00. What is the standard deviation for the population?

A visual representation of the question in Example 5.4. If 2 standard deviations correspond to a 6-point distance, then 1 standard deviation must equal 3 points.

A z-score transformation

Following a z-score transformation, the X-axis is relabeled in z-score units. The distance that is equivalent to 1 standard deviation on the X-axis (σ = 10 points in this example) corresponds to 1 point on the z-score scale.

Last week Sarah had a score of X=43 on a Spanish exam and a score of X=75 on an English exam. For which exam should Sarah expect the better grade?

Impossible to determine without more information

a population ditribution

The relationship between z-score values and locations in a population distribution. Z score of -1.00 means the score is located one SD above the mean

If your exam scores is X=60, which set of parameters would give you the best grade?

Plug all the values into the z-score formula and the one with the highest standard deviation will give you the best grade. z=X-u/o=60-65/5=-1

For a distribution of exam scores with u=70, which value for the standard deviation would give the highest grade to a score of X=75?

Plug all the values into the z-zscore formula and the one with the lowest standard deviation will give you the highest grade. z=X-u/o= 75-70/1=5

A distribution with μ=35 and σ=8 is being standarized so that the new mean and standard deviation will be μ=50 and σ=10. In the new, standardized distribution your score is X=60. What was your score in the original distribution?

Step 1) Transform new score into z-score μ=50, σ=10, X=60 z= (x - μ)/σ=60-50/10=1 Step 2) Change z-score into old X value X= z*σ+μ=1*8+35=43

A distribution with μ=47 and σ=6 is being standarized so that the new mean and standard deviation will be μ=100 and σ=20. What is the standardized score for a person with X=56 in the original distribution?

Step 1) Transform original raw scores into z-scores μ=47, σ=6, X=56 z= (x - μ)/σ=56-47/6=+1.5 Step 2) Change z-score into new X value X= z*σ+μ=+1.5*20+100=130

•An instructor gives an exam to a psychology class. For this exam, the distribution of raw scores has a mean of μ = 57 with σ = 14. -The instructor would like to simplify the distribution by transforming all scores into a new, standardized distribution with μ = 50 and σ = 10. -To demonstrate this process, we will consider what happens to two specific students: Maria, who has a raw score of X = 64 in the original distribution; and Joe,whose original raw score is X = 43

Step 1) Transform original raw scores into z-scores using the formula: z = (x - μ)/σ Step 2) Change each z-score into an X value in the new standardized distribution using the formula: X= z*σ+μ

•In a population distribution, a score of X = 54 corresponds to z = +2.00 and a score of X = 42 corresponds to z = −1.00. What are the values for the mean and the standard deviation for the population?

The 12-point distance from 42-54 corresponds to 3 standard deviations. Therefore, the standard deviation must be σ = 4. Also, the score X = 42 is below the mean by one standard deviation, so the mean must be μ = 46.

Although z-scores are most commonly used in the context of a population, the same principles can be used to identify individual locations within a sample

The definition 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

A Research Study

The goal of the study is to evaluate the effect of a treatment. A sample is selected from the population and the treatment is administered to the sample. If, after treatment, the individuals in the sample are noticeably different from the individuals in the original population, then we have evidence that the treatment does have an effect.

two distributions of exam scores

Two distributions of exam scores. For both distributions, μ = 70, but for one distribution, σ = 3, and for the other, σ = 12. The relative position of X = 76 is very different for the two distributions.

Example: •A researcher is evaluating the effect of a new growth hormone. It is known that regular adult rats weigh an average of μ = 400 g. The weights vary from rat to rat, and the distribution of weights is normal with a standard deviation of σ = 20 g.

Z-scores and interpreting results in research: Control group: no hormone is injected to the rat(X= 418) Treatment group: hormone is injected to the rat(X= 450) **If the individuals who receive the treatment in a research study have extreme z-scores compared to those who do not receive the treatment, we can conclude that the treatment does appear to have an effect.**

In N=25 games last season, the college basketball team averaged μ=74 points with a standard deviation of σ=6. In their final game of the season, the team scored 90 points. Based on this information, the number of points scored in the final game was ____.

far above average

A population has μ=50 and σ=10. If these scores are transformed into z-scores, the population of z-scores will have a mean of ____ and a standard deviation of ____.

in a transformation the mean is always 0 and the standard deviation is 1.00

Which of the following is an advantage of transofrming X values into z-scores?

none of the option is an advantage

Of the following z-score values, which one represents the location closet to the mean?

the median for a z-score is 0 therefore the location closet to zero will represent this location +0.50

The numerical value of the z-score specifies

the distance from the mean by counting the number of standard deviations between X and μ.

Using z-scores, a population with μ=37 and σ=6 is standardized so that the new mean is μ=50 and σ=10. How does an individual's z-score in the new distribution compare with his/her z-score in the original population?

the z-scores do not change: new z = old z (only the x values change)

A population of scores has σ=4. In this population, an X valus of 58 corresponds to z=2.00. What is the population mean?

use the formula μ=X-z*σ =58-2.00*4=50

A population of scores has μ=44. In this population, an X value of 40 corresponds to z=-0.50. What is the population standard deviation?

use the formula σ=X-μ/z =40-44/-0.50=8

Determining a Raw Score (X) from a z-Score

use the formula X=μ+z*σ

a z-score establishes a relationship between the score, mean, and standard deviation.

we simply transform scores (X values) into z-scores, or change z-scores back into X values

Under what circumstances would a score that is 15 points above the mean be considered to be near the center of the distribution?

when the population standard deviation is much larger than 15

Under what circumstances would a score that is 20 points above the mean be considered to be an extreme, unrepresented value?

when the population standard deviation is much smaller than 20

The sign of the z-score (+ or −) signifies

whether the score is above the mean (positive) or below the mean (negative).

For a population with μ=100 and σ=20, what is the z-score corresponding to X=105?

z = (x - μ)/σ =105-100/20 =+0.25

z score formula

z = (x - μ)/σ x=score μ=mean σ=standard deviation