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

• Outcomei = modeli + errori• Most statistical methods are based on a linear

modeloutcomei = (slope)xi + 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

Day 1 Day 20

10

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Fahrenheit

The Same Temperatures on Another Interval Scale

Day 1 Day 2-5

0

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Celsius

The Same Temperatures on a Ratio Scale (Rankine = F + 459.6)

Day 1 Day 20

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Rankine

The Same Temperatures on a Ratio Scale (Kelvin = C + 273.15)

Day 1 Day 20

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Kelvin

DESCRIPTIVE STATISTICS

• Central Tendency• Variability• Frequency Distributions• z-Scores

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• Take into account what assumptions you can

reasonably make about the data• Central tendency and variability go together