Chapter 4: Variability

Definition of Variability

• Variability indicates that scores are not all the same.
• It provides a quantitative measure of the difference between scores in a distribution.
• Describes the degree to which scores are spread out or clustered.
• Defined in terms of distance, showing how far apart scores are.
• Measures how well an individual score represents the entire distribution.
• Purpose: Obtain an objective measure of score spread in a distribution.

When Variability is Zero

• If scores are all the same, there is no variability.

Measures of Variability

Range

• First step in defining and measuring variability.
• Distance between the smallest score and the largest score.
• Formula: Range = Xmax – Xmin
• For continuous variables, Range = URL – LRL
• Disadvantage: Affected by extreme values; does not accurately describe variability for the entire distribution.

Standard Deviation

• Most commonly used and important measure of variability.
• Uses the mean as a reference point.
• Measures variability by considering the distance between each score and the mean.
• Formula: SD = √variance
• Deviation: Distance from the mean for each score, deviation score = X - µ.
• Variance: Mean of the squared deviations (average squared distance from the mean).

Variance

• Also known as mean squared deviation.
• Provides a measure of variability based on squared distances.

Example Calculation

• Scores: 12, 0, 1, 7, 4, 6; Mean = 5
• Calculate deviation, squared deviation, variance (16), and SD (4).

Sum of Squares (SS)

• Sum of squared deviation scores.
• Formulas:
  • Definitional: SS = ∑(x-µ)²
  • Computational: SS = ∑X² - (mean²)N

Population Variance and Standard Deviation

• Population Variance (σ²): Mean squared distance from the mean.
• Population Standard Deviation (σ): Square root of population variance.

Inferential Statistics

• Use sample information to draw conclusions about populations.
• High Variability: Obscures patterns.
• Low Variability: Makes patterns clear.

Sample Variance and Standard Deviation

• Sample Variance (s²): Mean squared distance from the mean, divide SS by n-1.
• Sample Standard Deviation (s): Square root of sample variance.

Degrees of Freedom

• Number of scores in a sample that are independent and free to vary.
• Formula: df = n-1
• Produces an unbiased estimate of the population variance.

Transformation of Scales

• Adding a Constant: Does not change SD.
• Multiplying by a Constant: Multiplies SD by the same constant.

Functions of Standard Deviation

1. Describe entire distribution (mean and SD).
2. Describe location of individual scores.

Error Variance

• Indicates unexplained, uncontrolled score differences.
• High variance complicates detection of patterns or differences in research data.