Statistics Chapter 5: Z-scores Flashcards

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} ).

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.