Statistics: Measurement Scales, Descriptive & Inferential Methods, Regression, Probability, and Hypothesis Testing

What is the difference between a numeral and a number?

A numeral is a symbol or label; a number has quantitative meaning.

Name the four scales of measurement

Nominal, Ordinal, Interval, Ratio.

Which scale has a meaningful zero and allows ratios?

Ratio scale.

When is it acceptable to use the mean?

For interval or ratio data (and when distribution not severely skewed).

What is sampling error?

The natural discrepancy between a sample statistic and the population parameter.

Descriptive vs inferential statistics — main difference.

Descriptive summarizes data; inferential uses sample data to make conclusions about a population.

Define mode.

The most frequent score or category in a distribution.

When is the mode the only appropriate measure?

For nominal data.

Define median.

The middle value in an ordered dataset (50th percentile).

When is the median preferred over the mean?

For ordinal data or when interval/ratio data are severely skewed or have outliers.

Define mean (sample and population notation).

Sample mean = Xˉ; population mean = μ; arithmetic average of values.

Define range.

Max − Min; distance covered by scores.

What is the semi-interquartile range (SIQR)?

Half the IQR; distance of a typical value from the median (IQR/2).

How do you compute median absolute deviation (MAD)?

Median of absolute deviations from the median.

Why square deviations when computing variance?

To remove signs and produce a minimum-sum measure (SS) that weights larger deviations more.

Relationship between variance and standard deviation.

Variance = average squared deviation; SD = square root of variance.

Formula for population variance and sample variance (conceptual).

Population: divide SS by N; Sample estimate uses SS divided by (N−1).

Which variability measures are robust to outliers?

SIQR and MAD are less affected by outliers than SD.

In a positively skewed distribution, order the mean, median, mode.

Mode < Median < Mean.

What does Pearson coefficient of skew use to indicate direction?

Relationship between mean and median divided by SD; sign indicates skew direction.

What does Pearson's r measure?

Strength and direction of linear relationship between two interval/ratio variables.

What is the regression equation for predicting Y from X?

Y^=a+bX where b is the slope, and a is the intercept.

Interpret the slope b

Expected change in Y for a one-unit increase in X.

How is the intercept a calculated from means and slope?

a=Yˉ−bXˉ

What is the meaning of R^2 in regression?

Proportion of variance in Y explained by X (explained variability).

What is the standard error of estimate (SD Y−Y^)?

Standard deviation of residuals; average prediction error around regression line.

What is a residual?

Y−Y^, the difference between observed and predicted Y.

What does heteroscedasticity mean for regression?

Non-constant variance of residuals across levels of X; affects slope reliability.

When do Y' and X' regression lines overlap?

When r = ±1 (perfect linear relationship).

Probability range and notation

Between 0 and 1; p(A) denotes probability of event A.

Addition rule for mutually exclusive events.

p(A or B)= p(A) + p(B) if events cannot co-occur.

Multiplication rule for independent events.

P(A and B)=P(A)*P(B)

Combination rule for order-not-important events.

Sum of probabilities of each order (e.g., p(A&B) + p(B&A)).

Difference between a priori and empirical probability.

A priori uses logic/counting; empirical uses observed frequencies.

What is subjective (Bayesian) probability?

Probability as degree of belief; updates priors with data to get posterior.

What is the sampling distribution of the mean?

Distribution of all possible sample means for samples of size N from a population.

Rule for mean of sampling distribution.

μXˉ= μ (sampling distribution mean equals population mean).

Central Limit Theorem in one sentence.

As N increases, the sampling distribution of the mean approaches normal regardless of population shape.

Practical N thresholds for CLT when population skewed.

Mild skew: N ≥ 10-12; moderate: N > 18-20; severe: N > 25-30.

State the null and alternative hypotheses conceptually.

H0: no effect/difference; H1: there is an effect/difference.

What is alpha (α)?

Pre-set probability of Type I error (commonly .05).

Type I vs Type II error.

Type I: reject true H0; Type II: fail to reject false H0

Two-tailed vs one-tailed tests — main difference.

Two-tailed tests check for difference in either direction; one-tailed checks a specific direction.

When can you use a single-sample z-test?

Population mean and population SD known; compare sample mean to population.

When do you use a single-sample t-test instead of z?

Population SD unknown; estimate using sample SD and use t-distribution.

How are degrees of freedom defined for single-sample t-test?

df= N - 1.

Why divide SS by (N−1) when estimating variance from a sample?

To correct bias (Bessel's correction); gives an unbiased estimate of population variance.