Statistics Chapter 5: Z-scores Flashcards

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Questions and Answers

What is the purpose of z-scores?

To describe the exact location of each score in a distribution; always refers to population.

Z-scores are turned into?

A standard score.

What does a z-score indicate about a score's position?

All z-scores above the mean are positive, and all z-scores below the mean are negative.

What does the z-score number represent?

The number of standard deviations from the mean. Signup and view all the answers

What is the formula for the z-score?

z = (x value - mean) / standard deviation. Signup and view all the answers

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

X = mean + z * standard deviation. Signup and view all the answers

If every x value is transformed into a z-score, what properties does the distribution of z-scores have?

It has the same shape as the original distribution, a mean of 0, and a standard deviation of 1. Signup and view all the answers

What are raw scores?

Original, unchanged scores that are the direct result of measurement. Signup and view all the answers

What is a z-score?

Describes the exact location of a score in a distribution relative to the mean. Signup and view all the answers

What is a deviation score?

Score minus the mean. Signup and view all the answers

What does z-score transformation do?

Uses the mean and standard deviation to transform each raw score into a standard score. Signup and view all the answers

What is a standardized distribution?

Composed of scores transformed to create predetermined values for mean and standard deviation. Signup and view all the answers

What is a standardized score?

The number of standard deviations a piece of data lies above or below the mean. Signup and view all the answers

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 or mean & are attained.

mean, standard deviation Signup and view all the answers

What are the three properties of standard scores?

<ol> <li>The mean of z-scores is always 0. 2. The standard deviation of standardized scores is always 1. 3. The distribution shape remains the same as the original scores.</li> </ol> Signup and view all the answers

When finding a proportion between a negative and positive z-score, where do you go?

Mean-to-z column for each z-score. Signup and view all the answers

How do you find a proportion between two positive or two negative z-scores?

Consult the mean to z column for both and subtract the smaller from the larger. Signup and view all the answers

To find the probability greater than a positive z or negative z, you go to the?

Tail column. Signup and view all the answers

To find the probability for an area greater than a negative z or less than a positive z, you use?

Body column. Signup and view all the answers

To find the z-score that forms the boundary between two areas under the bell curve, you use?

The tail column and the proportion closest to the percentage. Signup and view all the answers

Study Notes

Z-scores Overview

Z-scores indicate the exact position of each score in a distribution, essential for understanding data characteristics.
They are always based on population data; a different formula applies to sample data.

Purpose of Z-scores

Serve as standard scores, allowing comparison across different tests by standardizing distributions.

Positive and Negative Z-scores

Z-scores are signed numbers: positive indicates a score above the mean, while negative signifies below the mean.
The value represents the distance from the mean in standard deviation units.

Z-score Calculation

The formula for calculating a z-score is:
- ( z = \frac{x - \mu}{\sigma} )
- This involves subtracting the mean from the x-value to produce a deviation score, divided by the standard deviation.

Back Calculation from Z-score

To find the original x-value from a z-score, use:
- ( X = \mu + z \cdot \sigma )

Properties of Z-score Distribution

The distribution of z-scores maintains the original distribution's shape.
The mean of z-scores is always 0, and the standard deviation is always 1.

Raw and Standardized Scores

Raw scores are unaltered results measured directly from tests.
Standardized scores reflect how many standard deviations a data point is from the mean.

Standardized Distribution

A standardized distribution consists of scores adjusted to specified mean and standard deviation for comparability.

Steps for Standardizing a Distribution

Transform original raw scores into z-scores.
Convert z-scores to obtain new values for defined mean and standard deviation.

Properties of Standard Scores

Mean of z-scores is 0, standard deviation is 1, and the shape remains consistent with the original score distribution.

Z-score Calculation Rules

To find proportions involving z-scores:
- For a mix of negative and positive z-scores, sum proportions from the mean-to-z column for both.
- For two positive or two negative z-scores, find proportions and subtract the smaller from the larger.
- For area greater than a z-score, use the tail column; this excludes the mean.
- For area greater than a negative or less than a positive z-score, reference the body column, which includes the mean.
- To find a z-score marking boundaries between two areas under the curve (e.g., top 20% and bottom 80%), check tail column proportions against desired percentages.

Deviation Scores and Transformation

A deviation score expresses how far a score is from the mean.
Z-score transformation is a key technique using mean and standard deviation for standardization.

Standardized Score Definition

Standardized scores are expressed in terms of standard deviations from the mean, calculated using ( Z = \frac{X - \mu}{\sigma} ).

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Description

Explore the fundamental concepts of z-scores in statistics with these flashcards. Learn the purpose of z-scores and how they standardize scores to help describe their exact location within a distribution. Perfect for mastering Chapter 5 of your statistics course.