Review of Basics

REVIEW OF BASICS PART I Measurement Descriptive Statistics Frequency Distributions

MEASUREMENT CONCEPTS Measured vs. True Scores Statistical Models Measurement Scales

Measured Scores Any measured score represents: True underlying score Measurement error Lower measurement error means higher reliability

Statistical Models A statistical model is a way to represent the data Outcome i = model i + error i Most statistical methods are based on a linear model outcome i = (slope)x i + y-intercept

MEASUREMENT SCALES What assumptions can you make about a score? Many statistics require a certain measurement scale. The measurement scale is a property of the data.

1. Nominal Scale Numbers classify into groups. Math, other than counting, is not meaningful.

2. Ordinal Scale Numbers are rank orders. Math, other than counting, is not meaningful.

3. Interval Scale Numbers represent amounts, with equal intervals between numbers. Math, other than ratio comparisons, is meaningful.

4. Ratio Scale Numbers represent amounts, with equal intervals and a true zero true zero: score of zero represents a complete absence Math, including ratios, is meaningful.

Why You Can’t do Ratios on an Interval Scale

The Same Temperatures on Another Interval Scale

The Same Temperatures on a Ratio Scale (Rankine = F )

The Same Temperatures on a Ratio Scale (Kelvin = C )

DESCRIPTIVE STATISTICS Central Tendency Variability Frequency Distributions

Central Tendency – Typical Score mean: arithmetic average median: middle score mode: most frequent score

Variability – Spread of Scores deviation: difference between observed score and model (e.g., mean) sum of squares(SS): sum of squared differences from the mean

Variability variance: average squared difference from the mean standard deviation: average unsquared difference from the mean

FREQUENCY DISTRIBUTIONS frequency: number of times a score occurs in a distribution frequency distribution: list of scores with the frequency of each score indicated

Normal Distributions symmetrical equal mean, median, and mode bell-shaped

Why Be Normal?  Many variables are affected by many random factors.  Effects of random factors tend to balance out.

Skewness Extent to which scores are piled more on one end of the distribution than the other positive skew negative skew

Skewness Skewness = 0 for a normal distribution Skewness < 0 for a negatively skewed distribution Skewness > 0 for a positively skewed distribution

Kurtosis Measure of the steepness of the curve Platykurtic: flat Leptokurtic: steep

Kurtosis  Kurtosis = 0 for a normal distribution  Kurtosis < 0 when the distribution is flatter than a normal  Kurtosis > 0 when the distribution is steeper than a normal

Take-Home Points  Measurement is always open to error.  Think about what assumptions you can reasonably make about the data.  Central tendency and variability go together.