Statistics Chapter 6

Standard Normal Distribution

A normal distribution with mean = 0 and standard deviation and variance = 1. See also normal distribution.

Galton Board

A physical device that demonstrates the distribution of outcomes for Bernoulli trials (binary chance decisions). Also called a quincunx.

Quincunx or Galton Board

A physical device that demonstrates the distribution of outcomes for Bernoulli trials (binary chance decisions). Also called a quincunx.

positively skewed

An asymmetric distribution that has a longer tail at the high (or positive) end of the distribution is said to be positively skewed.

negatively skewed

An asymmetric distribution that has a longer tail at the low (or negative) end of the distribution is said to be negatively skewed.

Unit Free

Scores are unit free (also called standardized) when they have been converted into units that have a mean of 0 and a standard deviation of 1. When individual X scores such as height in inches are converted to z scores, they become unit free.

Skewness

Skewness is asymmetry. If you divide the normal distribution at its mean, asymmetry means that the left- and right-hand sides are not mirror images of each other. This can be evaluated by visually examining a histogram; SPSS also provides a skewness statistic. Also see positively skewed and negatively skewed.

Standard Score or Standardized Score

The distance of an individual score from the mean of a distribution expressed in unit-free terms (i.e., in terms of the number of standard deviations from the mean). If μ and σ are known, the z score is given by z = (X - μ)/σ. When μ and σ are not known, a distance from the mean can be computed using the corresponding sample statistics, M and s (or SD). If the distribution of scores has a normal shape, a table of the standard normal distribution can be used to assess how distance from the mean (given in z-score units) corresponds to proportions of area or proportions of cases in the sample that correspond to distances of z units above or below the mean.

z score

The formula to calculate a standard score or z score is z = (X - M)/SD. A distribution of z scores has M = 0 and SD = 1. See also standard score and standardized scores.

Normal Distribution

The mathematical definition of a normal distribution is given in Appendix 6A. Analysts typically call an empirical distribution seen in a histogram "approximately normal" if its shape approximates that of a bell curve. Also called the Gaussian distribution.

Standardization

The term standardization has two different meanings in this book. In data analysis, standardization refers to the conversion of scores in original units of measurement (e.g., pounds, degrees, inches) into unit-free z scores (see Chapter 6). In experimental design and measurement, standardization means keeping data collection procedures as similar as possible across all participants or cases (see Chapter 2). See experimental control over other situational factors or extraneous variables.

Standardized Score

These are scores expressed in z-score units, that is, as unit-free distances from the mean. For example, the standardized score version of X, zX, is obtained as zX = (M - MX)/sX.

Floor Effect

When scores have a fixed lower limit, such as 0 on an exam, and when many scores are close to that minimum possible value, there is a floor effect. If this distribution occurs for examination scores, it suggests that the examination was too difficult. A floor effect is undesirable.

Ceiling Effect

When there is a fixed upper limit for scores (such as 100% on an exam), and most scores are close to top of that range, a ceiling effect is present. A ceiling effect in a histogram of exam scores suggests that an exam is too easy.

Gaussian distribution

normal distribution