Stats 100 Flashcards

For my Goddess :)

What makes two events mutually exclusive

If

P(A ∩ B) = 0

Experimental Study

A study where researchers actively manipulate the independent variable and use random assignment to place subjects into treatment groups. Allows causal conclusions.

Observational Study

A study where researchers observe without intervention; subjects self-select or are naturally grouped. Can show correlation but NOT causation.

Random Assignment

Using chance to assign subjects to treatment groups in an experiment. Enables causal conclusions by balancing confounding variables.

Random Sampling

Selecting subjects from a population using chance, where each member has equal probability of selection. Ensures representative sample.

Population

The entire group of individuals about which we want information. Described by parameters (µ, σ, p)

Sample

A subset of the population from which we collect data.

Parameter

A numerical characteristic of a population (µ, σ, p). Usually unknown - we estimate using sample statistics.

Statistic

A numerical characteristic of a sample

Independent Events

Two events where occurrence of one doesn't affect probability of the other. P(A and B) = P(A) × P(B). Example: Two coin flips.

Mutually Exclusive Events

Two events that cannot both occur at the same time. P(A and B) = 0. Example: Getting heads AND tails on one coin flip.

Conditional Probability

Probability of event A given that B has occurred. Notation: P(A|B). Formula: P(A|B) = P(A and B) / P(B)

Law of Total Probability

P(A) = P(A and B) + P(A and B')

Bayes' Theorem

Formula for reversing conditional probabilities. P(A|B) = [P(B|A) × P(A)] / P(B). Used in medical testing, diagnostics.

Sensitivity

In medical testing: P(positive test | disease). True positive rate; ability to detect disease.

Specificity

In medical testing: P(negative test | no disease). True negative rate; ability to rule out disease.

Binomial Distribution

Distribution for number of successes in n independent trials with constant probability p. Notation: X ~ Binomial(n, p). Requires BINS: Binary, Independent, n fixed, Same p.

Normal Distribution

Continuous, symmetric, bell-shaped distribution. X ~ N(µ, σ). Properties: Mean=Median=Mode=µ, symmetric, 68-95-99.7 rule applies

Standard Normal

Normal distribution with µ=0 and σ=1. Z ~ N(0,1). All normal distributions can be standardized using z-scores.

t-Distribution

Symmetric, bell-shaped like normal but heavier tails. Shape depends on df. Used for inference about µ when σ is unknown.

Sampling Distribution

The probability distribution of a sample statistic, based on all possible samples of size n.

Mean

The Average

Median

Middle value when data is ordered; 50th percentile. Resistant to outliers (robust).

Mode

Most frequently occurring value. Can have multiple modes or no mode. Best for categorical data.

Standard Deviation

Typical distance of values from mean

Variance

Average of squared deviations from mean, Relationship: Variance = (SD)². Units are squared.

Quartiles

Values dividing data into 4 parts

IQR (Interquartile Range)

Range of middle 50% of data.

Outlier

Observation far from rest of data. 1.5×IQR rule: outlier

Skewness

Measure of asymmetry. Right-skewed: long tail right, mean>median. Left-skewed: long tail left, mean

Percentile

Value below which a given percentage of observations fall. Example: 90th percentile = 90% of values below this point.

z-score

Number of standard deviations a value is from mean. z = (x-µ)/σ. Example: z=2 means 2 SD above mean.

68-95-99.7 Rule

For normal distributions: ≈68% within µ±1σ, ≈95% within µ±2σ, ≈99.7% within µ±3σ. Also called Empirical Rule

Central Limit Theorem

As n increases, sampling distribution of x■ approaches normal regardless of population shape. Rule: n≥30 usually sufficient.

Standard Error (SE)

Standard deviation of sampling distribution of x, SE = σ/√n. Measures variability of sample means, NOT individuals.

Point Estimate

Single value estimating a parameter. Examples: x estimates µ, s estimates σ. Gives no uncertainty info.

Confidence Interval

Interval estimate with confidence level. General form: Estimate ± (Critical Value)(SE). For µ: x ± t*(s/√n)

Confidence Level

Percentage of time the method produces interval containing true parameter (if repeated many times). Common: 90%, 95%, 99%

Margin of Error (ME)

Maximum expected difference between estimate and true parameter. ME = (Critical Value)(SE). For µ: ME = t*(s/√n)

Critical Value

Multiplier from t or z distribution determining CI width. Example: 95% CI has t*≈2 (depends on df)

Degrees of Freedom (df)

Number of independent pieces of info for estimation. For one sample: df = n-1. Affects t-distribution shape.

α (Alpha Level)

Probability CI doesn't contain true parameter. α = 1 - Confidence Level. Example: 95% CI has α=0.05.

Independence Assumption

Observations don't influence each other. Check: random sampling, 10% condition (n<10% of population).

Normality Assumption

Population or sampling distribution is approximately normal. Check: n≥30 (CLT), histogram, no strong skew/outliers.

10% Condition

When sampling without replacement, n should be <10% of population size (n<0.10N). Ensures approximate independence.

Sample Space

Set of all possible outcomes of random experiment. Example: coin flip {H,T}, die roll {1,2,3,4,5,6}.

Event

Subset of sample space; collection of outcomes. Example: rolling even on die = {2,4,6}.

Probability

Number between 0 and 1 representing likelihood. Properties: 0≤P(A)≤1, P(sure)=1, P(impossible)=0.

Variable

Characteristic taking different values for individuals. Types: Categorical (labels) or Quantitative (numerical).

Continuous Variable

Quantitative variable taking any value in interval (infinitely many values). Examples: height, weight, time.

Discrete Variable

Quantitative variable taking specific countable values. Examples: number of students, dice rolls, errors.

Probability Distribution

Function/table giving probability for each value of random variable. Properties: all probs 0-1, sum/area=1.

Expected Value E(X)

Long-run average if experiment repeated many times. For discrete: E(X)=Σ[x·P(X=x)]. Also called expectation or µ.

Robust Procedure

Statistical method performing well when assumptions violated. Example: t-procedures robust to normality violations when n large. Median robust to outliers.

Statistical Inference

Using sample data to make conclusions about population. Two types: Estimation (CIs) and Hypothesis testing.

Union (A or B)

Event that A, B, or both occur. Notation: A∪B. Formula: P(A or B) = P(A) + P(B) - P(A and B)

Intersection (A and B)

Event that both A and B occur. Notation: A∩B. For independent: P(A and B) = P(A)×P(B)