Psych Statistics-Chap 5

Because each individual score stays in its same position within the distribution, the overall shape of the distribution:

Does not change

Using z-scores to describe the position of X = 76:

-In Figure 5.1(a), with a standard deviation of σ = 3, the score X = 76 corresponds to a z-score of z = +2.00. That is, the score is located above the mean by exactly 2 standard deviations. -In Figure 5.1(b), with σ = 12, the score X = 76 corresponds to a z-score of z = +0.50. In this distribution, the score is located above the mean by exactly 1/2 standard deviation.

Give an example of how a z-score describes the exact location of a score within a distribution:

-In a distribution of IQ scores with μ= 100 and σ = 15, a score of X = 130 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 (15 + 15 = 30 points)

How do you evaluate the effect of a treatment?

-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

The original population mean is transformed into a value of what in the z-score distribution?

0

Note that a z-score of z = +1.00 corresponds to a position exactly __ standard deviation above the mean

1

The z-score always consists of what 2 parts?

1. A sign (+ or −) 2. A magnitude (the numerical value)

Transforming X-scores into z-scores serves what 2 purposes?

1. Each z-score tells the exact location of the original X value within the distribution 2. The z-scores form a standardized distribution that can be directly compared to other distributions that also have been transformed into z-scores

If every X value is transformed into a z-score, then the distribution of z-scores will have the following 3 properties:

1. Shape-The distribution of z-scores will have exactly the same shape as the original distribution of scores (ex. if the original distribution is negatively skewed, for example, then the z-score distribution will also be negatively skewed) 2. Mean-The z-score distribution will ALWAYS have a mean of zero 3. Standard Deviation-The distribution of z-scores will ALWAYS have a standard deviation of 1

The procedure for standardizing a distribution to create new values for the mean and standard deviation is a 2-step process that can be used either with a population or a sample:

1. The original scores are transformed into z-scores 2. The z-scores are then transformed into new X values so that the specific mean and standard deviation are attained

For a SAMPLE, each X value is transformed into a z-score so that (2 purposes):

1. The sign of the z-score indicates whether the X value is above (+) or below (−) the sample mean 2. 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)

For intelligence tests, raw scores are frequently converted to standard scores that have a mean of __ and a standard deviation of __

100, 15

A z-score of z = +2.00 is always located exactly __ standard deviations above the mean

2

The deviation from the mean (zσ) determines:

Both the direction and the size of the distance from the mean

The fact that the z-score distribution has a mean of zero makes the mean:

A convenient reference point

A typical research study begins with:

A question about how a treatment will affect the individuals in a population

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

A sample distribution of z-scores

In this chapter, we introduce 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

Notice that we cannot compare Dave's two exam scores because the scores come from different distributions with different means and standard deviations. However, we can compare the two z-scores because:

All distributions of z-scores have the same mean and the same standard deviation

A z-score is a relative value, not:

An absolute value (ex. a z-score of z = -2.00 does not necessarily suggest a very low raw score—it simply means that the raw score is among the lowest within that specific group)

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

Why is the deviation score divided by σ?

Because we want the z-score to measure distance in terms of standard deviation units

Note that Dave's z-score for biology is +2.0, which means that his test score is 2 standard deviations above the class mean. On the other hand, his z-score is +1.0 for psychology, or 1 standard deviation above the mean. In terms of relative class standing, Dave is doing much better in which class?

Biology

Because most IQ tests are standardized so that they have the same mean and standard deviation, it is possible to:

Compare IQ scores even though they may come from different tests

Instead of drawing the two distributions to determine where Dave's two scores are located, we simply can:

Compute the two z-scores to find the two locations

A standardized distribution is composed of scores that have been transformed to:

Create predetermined values for μ and σ

In inferential statistics, z-scores provide an objective method for:

Determining how well a specific score represents its population

This 2-step process ensures that:

Each individual has exactly the same z-score location in the new distribution as in the original distribution

A z-score beyond +2.00 (or −2.00) indicates that the score is:

Extreme and is noticeably different from the other scores in the distribution

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.

A common example of a standardized distribution is the distribution of:

IQ scores

What is the easiest way to transform z-scores into X values?

In general it is easier to use the definition of a z-score, rather than a formula, when you are changing z-scores into X values

What is a solution to this disadvantage of z-scores?

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

What is an advantage of standardizing?

It is possible to compare distributions even though they may have been quite different before standardization

What is the result of standardizing IQ tests?

It is possible to understand and compare IQ scores even though they come from different tests (ex. we all understand that an IQ score of 95 is a little below average, no matter which IQ test was used. Similarly, an IQ of 145 is extremely high, no matter which IQ test was used.)

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

A score by itself does not necessarily provide much information about:

Its position within a distribution

Normally, if two scores come from different distributions, it is impossible to:

Make any direct comparison between them (ex. suppose, for example, Dave received a score of X = 60 on a psychology exam and a score of X = 56 on a biology test. For which course should Dave expect the better grade?)

What is the deviation score?

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 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.

Standardized scores of this type (with whole numbers) are frequently used in:

Psychological or educational testing (ex. raw scores of the Scholastic Aptitude Test (SAT) are transformed to a standardized distribution that has μ = 500 and σ = 100)

Transforming a distribution from X values to z values does not move scores from one position to another; the procedure simply:

Relabels each score

A z-score near 0 indicates that the score is close to the population mean and therefore is:

Representative

Because all z-score distributions have the same mean and the same standard deviation, the z-score distribution is called a:

STANDARDIZED distribution

The definition and the purpose of a z-score is the same or different for a sample as for a population?

Same (provided that you use the sample mean and the sample standard deviation to specify each z-score location)

The transformed distribution of z-scores will have the same or different properties that exist when a population of X value is transformed into z-scores?

Same (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, and the sample of z-scores will have a standard deviation of sz = 1)

The locations identified by z-scores are the same or different for all distributions?

Same-no matter what mean or standard deviation the distributions may have

Note that the set of z-scores is still considered to be a sample (just like the set of X values) and the _____ formulas must be used to compute variance and standard deviation

Sample

A second purpose for z-scores is to:

Standardize an entire distribution

An entire population of scores is transformed into z-scores:

The transformation does not change the shape of the distribution but the mean is transformed into a value of 0 and the standard deviation is transformed to a value of 1.

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

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 (ex. a z-score of 1.50 is exactly 1.50 standard deviations from the mean)

To find the location of your score, you must have information about:

The other scores in the distribution

In general terms, what does the process of standardizing do?

The process of standardizing takes different distributions and makes them equivalent

A z-score establishes a relationship between:

The score, the mean, and the standard deviation

When you use the z-score formula, it can be useful to pay attention to:

The definition of a z-score as well (helpful to find inconsistencies in calculations)

Following a z-score transformation, the X-axis is simple 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

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

A diagram of 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.

The numerical value of the z-score tells you:

The number of standard deviations from the mean

Before you can begin to make comparisons, you must know:

The values for the mean and standard deviation for each distribution (ex. suppose the biology scores had μ = 48 and σ = 4, and the psychology scores had μ = 50 and σ = 10. With this new information, you could sketch the two distributions, locate Dave's score in each distribution, and compare the two locations)

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 (PROBABILITY helps determine this)

Although z-score distributions have distinct advantages, many people find them cumbersome because:

They contain negative values and decimals

Although there are several different tests for measuring IQ, the tests usually are standardized so that:

They have a mean of 100 and a standard deviation of 15

What can this relationship between z-scores and the raw scores, the mean, and standard deviation be used for?

This relationship can be used to answer a variety of different questions about scores and the distributions in which they are located

The goal of changing the mean and standard deviation into positive whole numbers 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

What is the purpose of z-scores/standard scores?

To identify and describe the exact location of each score in a distribution

One technique for deciding whether a sample is noticeably different is:

To use z-scores (ex. an individual with a z-score near 0 is located in the centre 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)

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 -The NUMBER tells the distance between the score and the mean in terms of the number of standard deviations

What do we do to make raw scores more meaningful?

We transform X values into z-scores so that the resulting z-scores tell exactly where within a distribution the original scores are located (new values that contain more information)

When do we use the z-score formula instead of the definition?

When the numbers are more difficult to organize the calculations

Notice that the interpretation of the research results depends on:

Whether the sample is NOTICEABLY DIFFERENT from the population

There is no need to create a whole new distribution when transforming X values into z-scores, instead:

You can think of the z-score transformation as simply relabeling the values along the X-axis; that is, after a z-score transformation, you still have the same distribution, but now each individual is labeled with a z-score instead of an X value

Thus, we can use _____ to help decide whether the treatment has caused a change

Z-scores (if z-scores are extreme, they caused a change, if they are close to 0, they did not cause a change)

A z-score distribution is an example of a standardized distribution with:

μ = 0 and σ = 1

That is, when any distribution (with any mean or standard deviation) is transformed into Z-SCORES, the transformed distribution will always have:

μ = 0 and σ = 1