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