stats - textbook

Continuous variable

a variable that can take any value in a range

Discrete variable

a variable that takes only separate, distinct values

Probability distribution

a smooth curve showing how probability is spread across values

Area under curve

the probability of a value falling in that region

Normal distribution

symmetric, bell‑shaped distribution defined by mean and SD

Standard normal distribution

normal distribution with mean 0 and SD 1

z‑score

distance from the mean measured in SD units

z = (X − μ) / σ

formula to convert X to z

X = μ + zσ

formula to convert z to X

Population mean (μ)

average of all values in the population

Population SD (σ)

spread of values in the population

Sample mean (M)

average of values in a sample

Sample SD (s)

spread of values in a sample

Sampling variability

natural variation in statistics across repeated samples

Sampling distribution

distribution formed by the means of many samples

Standard error (SE)

SD of the sampling distribution of the mean

SE = σ / √N

formula for standard error

Dance of the means

visualization of sample means varying from sample to sample

Mean heap

histogram of many sample means showing the sampling distribution

Central Limit Theorem

sample means become normally distributed as N increases

Tail area

probability in the extreme ends of a distribution

95% rule

about 95% of values lie within ±2 SD of the mean

68% rule

about 68% of values lie within ±1 SD of the mean

99.7% rule

about 99.7% of values lie within ±3 SD of the mean

1.96 rule

z = ±1.96 captures 95% of a normal distribution

Random sampling

every population member has equal and independent chance of selection

Probability density

height of the curve showing how probability is concentrated

Point estimate (M)

the sample mean used to estimate the population mean

Estimation error

the difference between the sample mean and the true mean (M − μ)

Margin of error (MoE)

the largest likely estimation error, usually based on 95%

Error bars

line segments extending MoE on each side of the sample mean

Confidence interval (CI)

the interval [M − MoE, M + MoE] likely to contain μ

95% CI

interval that captures μ in 95% of repeated samples

Confidence level (C)

the percentage of CIs expected to capture μ

Plausibility curve

curve showing which values are most plausible as the true mean

MoE formula (σ known)

1.96 × SE for a 95% CI

MoE formula (general)

z_{C/100} × SE

Dance of the CIs

visualization of CIs from repeated samples

Red CI

a CI that fails to include μ (about 5% of 95% CIs)

CI slogan

"It might be red!" meaning any single CI might miss μ

z‑value

critical value from the normal distribution

t‑value

critical value from the t distribution when σ is unknown

t formula

(M − μ) / (s / √N)

t distribution

family of distributions used when σ is unknown

Degrees of freedom (df)

number of independent pieces of information (N − 1)

As df increases

t distribution approaches the normal distribution

CI formula (σ unknown)

M ± t × (s / √N)

MoE (σ unknown)

t × (s / √N)

Higher confidence level

wider CI

Lower confidence level

narrower CI

Larger sample size

smaller SE and narrower CI

CI coverage

long‑run proportion of CIs that include μ

Independent groups design

each participant is in only one condition

Group independence

scores in one group do not influence the other

Effect size (independent groups)

difference between group means (M₂ − M₁)

Difference axis

scale showing the mean difference and its CI

Plausibility curve (difference)

curve showing plausible differences

Normal populations

both groups' scores come from normal distributions

Homogeneity of variance

population variances assumed equal

CI on difference

interval estimating the true mean difference

MoE for difference

t × sₚ × √(1/n₁ + 1/n₂)

Pooled SD (sₚ)

weighted average of group SDs

df for independent groups

(n₁ − 1) + (n₂ − 1)

Difference (M₂ − M₁)

observed effect size

Pooled SD formula

√[((n₁−1)s₁² + (n₂−1)s₂²) / (n₁ + n₂ − 2)]

CI formula (difference)

(M₂ − M₁) ± MoE

Wide CI

indicates high uncertainty

Narrow CI

indicates higher precision

Welch method

CI calculation not assuming equal variances

One‑way independent groups design

one IV with three or more levels

Level of an IV

one condition or category

Extended design

more than two groups but one IV

Comparison

difference between two group means

Planned comparison

specified before data collection

Exploratory comparison

chosen after seeing data

Contrast

combination of means answering a research question

Subset contrast

difference between subsets of groups

Overlap rule

eyeballing evidence using independent CIs

Gap or touching CIs

moderate evidence (≈ p < .01)

Overlap ≤ half MoE

small evidence (≈ p < .05)

Pooled SD (three groups)

weighted SD across groups

df for pooled SD

N − 3

MoE for single mean

t × sp × (1/√n)

MoE for comparison

t × sp × √(1/n₁ + 1/n₂)

Two‑way independent groups design

two IVs, independent groups

Factorial design

includes all combinations of IV levels

2 × 2 design

two IVs with two levels each

Cell

one group representing a combination of IV levels

Main effect

overall effect of one IV

Interaction

effect of one IV depends on the level of the other

Difference of differences

numerical expression of interaction

Interaction contrast

difference between simple differences

Parallel lines

no interaction

Non‑parallel lines

interaction present

Crossing lines

strong interaction

Correlation

measure of linear relationship between two variables

Pearson's r

statistic from −1 to 1 measuring linear association

Positive correlation

higher X with higher Y

Negative correlation

higher X with lower Y

Zero correlation

no linear relationship