Chapter 5

When an entire distribution of X values is transformed into a Z- scores

The resulting distribution of z-scores will always have mean=0 and standard deviation=1

Using Z-scores to standardizing a sample also has

The same affect as standardizing a population

A score that is located two standard deviations above the mean will always have a

z-score of +2.00. And a z-score of +2.00 always indicates a location above the mean by 2 standard deviation

Z=0 is in the center (at the mean), and the extreme tails corresponding

to z-scores of approximately -2.00 on the left and +2.00 on the right

Although more extreme z-scores are possible

most of the distribution is contained z=-2.00 and z=+2.00

Z scores

-2,-1,0,+1,+2

Specifically, the mean of the Z scores will be

0 and a standard deviation of the z-scores will equal to 1 provided the standard deviation is computed using the sample formula

Half a standard deviations above the mean

Average

The basic z-score definition is usually sufficient to

Complete most z-score transformations

The process of changing an X value into a z-score involves

Creating a signed number, called a z-score

Transforming X values into z-scores creates a standardized distribution, many people find that z-scores burdensome because they consist of many..

Decimal values and negative numbers

As descriptive statistics, z scores

Describe exactly where each individual is located

Let's say the population mean is 70

If the standard deviation is 3, your score of 76 is above the mean by 6 points since the mean is 70

because z-scores distribution all have the same mean and standard deviation,

Individual scores from different distributions can be directly compared

Two standard deviations above the mean

Is superior

In a z-score distribution the following are always true

Mean for sample or population is =0 Standard deviation for a sample or a population= 1

The numerical value of the z-score corresponds to the

Number of standard deviations between X and the mean of the distribution

It is more convenient to standardize a distribution into

Numerical values that are simpler than z-scores

If the standard deviation is 12, your score..

Of 76 is above the mean by 6 points

The formulas are also the same except that the

Sample mean and standard deviation are used in place of the population mean and standard deviation

The transformation does not change the

Shape of the original distribution and it does not change the location of any individuals or relative to others in the distribution

A score of X= 76, for example, may be a relatively low, or an average score, or an extremely high score depending on..

The mean and standard deviation for the distribution from which the score was obtained

An entire population of scores is transformed into z scores

The transformation does not change the shape of the population, but the mean is 0 and standard deviation is 1 since its z scores

In addition to knowing the basic definition of a z score and the formula for a z score, it is useful to be able

To visualize z-scores as locations in a distinction

The advantage of standardizing distributions is that

Two or more different distributions can be made the same

The fact that z-scores identify exact locations within a distribution means that z-scores can be

Used as descriptive statistics and as inferential statistics

If the raw score is transformed into a z score the ..

Value of the z-score tells exactly where the score is located relative to all the other scores in he distribution

A raw scores( the original, unchanged scores) or X value provides..

Very little information about how that particular scores compares with other value in the distribution

As inferential statistics, z scores determine

Whether a specific sample is representative of its population, or is extreme and unrepresentative

The sign of the z- score (+ or -) identifies

Whether the X value is located a love the mean ( positive ) or below the mean ( negative)

Formula to convert z-scores back to a raw score

X = population mean + z population standard deviation

It is also possible to calculate

Z-scores for samples

The definition can be written in mathematical notation to create a formula for computing the z-score for an value of X

Z= X- population mean / Population standard deviation

Formula for sample z-score

Z=X-M/S

Definition of a Z score is the same for either

a sample or population