What is the purpose of z-scores?
describe exact location of each score in a distribution;
-always refers to population (must use a different formula for samples).
Z-scores are turned into
a standard score. The purpose of z-scores is to identify and describe the exact location of each score in a distribution & to standardize an entire distribution to understand & compare scores from different tests.
To describe the exact position of a score within a distribution, z-score must transform each x-value into a signed number; positive or negative.
all z-scores above the mean are positive and all z-scores below
the mean are negative. The number tells the distance between the score and the mean in terms of the number of standard deviations.
What does the z-score number represent?
the number of standard deviations from the mean. Aka standardized scores.
What is the formula for the z-score?
z = x value - mean or mew/ divided by standard deviation or sigma. The numerator X - mew is a
deviation score
. The denominator expresses deviation in standard deviation units.
What is the formula to determine the x-value from z-score?
X = mew + z times sigma. (Mean plus (2 multiplied by standard deviation)
If every x value is transformed into a z-score, then the distribution of z-scores will have what following properties regarding shape, mean, and standard deviation?
distribution of z-scores will have exactly the same shape as original distribution of scores; z-score mean will always have mean of 0 & z-scores will always have standard deviation of 1.
Raw score
original, unchanged scores that are the direct result of measurement. A test score that has not been transformed or converted in any way.
z-score
Describes the exact location of a score in a distribution relative to the mean. Aka Standard Score; how many standard deviations you are away from the norm. Used to make different distributions, or metric scales, comparable.
Deviation score
score minus the mean
z-score transformation
statistical technique that uses the mean and standard deviation to transform each raw score into a standard score
Standardized Distribution
Composed of scores that have been transformed to create predetermined values for mean standard deviation. They are used to make dissimilar distributions comparable.
Standardized Score
The number of standard deviations that a piece of data lies above or below the mean.
Z = (X - ?) / ?
Standardizing a distribution has two steps:
1. Original raw scores transformed to z-scores.
2. The z-scores are transformed to new X values so that the specific mew or mean & sigma/standard deviation are attained.
3 Properties of Standard Scores
1. The mean of a set of z-scores is always 0.
2. The standard distribution of a set of standardized scores is always 1.
3. The distribution of a set of standardized scores has the same shape as the original scores, the scaling is just different.
Z score rules
...
1. When you need to find a proportion between a negative (-) & positive (+) z-score:
Go to
mean-to-z column
for each Z.; Find proportions and add together.
2. When you need to find a proportion between 2 positive OR 2 negative z-scores, you:
consult the
mean to z column
for both. Find proportions & subtract the smaller from the larger.
3. When you need to find the P that is
greater
than a positive Z or a negative Z you will go to the:
tail column
.
4. When you need to find the P for an area
greater than
a negative Z or
Less than
a positive Z use:
the
Body column
.
5. When you need to find the z-score that forms the boundary between 2 areas under the bell curve i.e. between top 20% & bottom 80% use:
The
Tail column
& find the proportion closest to the percentage e.g. the proportion closest to .2000; the z-score in that row is the z-score that forms that boundary.
6. When you need to compute a raw score, that represents the minimum or maximum score needed to answer a question, look for the percentage in the question e.g. "What raw scores form the boundaries of the middle 60% of the distribution:
The middle 60% straddles the mean & can be divided into 2 = percentages; 30% & 30%. You look for the value closest to .3000 in the
mean to z column
& locate the z-score in that row. Then you use that z-score in the formula we use to compute raw score: X=m