Statistics Ch:5 Z-scores

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*. Easy way to remember is it's the only one that doesn't include the mean.

Z score rules

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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.

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.

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.

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.

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=mew + z sigma

Standardized Score

The number of standard deviations that a piece of data lies above or below the mean. Z = (X - μ) / σ

What is the formula to determine the x-value from z-score?

X = mew + z times sigma (X = u + zo). (Mean plus (2 multiplied by standard deviation)

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.

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.

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.

Deviation score

score minus the mean = how much the score deviates from the mean.

What is the purpose of z-scores?

to describe the exact location of each score in a distribution; -always refers to population (must use a different formula for samples).

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 does the z-score number represent?

the number of standard deviations from the mean. Aka standardized scores.

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.

z-score transformation

statistical technique that uses the mean and standard deviation to transform each raw score into a standard score

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*. Because the body column includes the mean & the tail.