151 terms
Cell Combining in Chi-squared Test
If any expected frequency is < 5, rows or columns with small expected frequencies should be combined before calculating
Chi-Squared Tests
Cohen's d
Cohen's d = (mean₁ - mean₂) / s_pooled, where s_pooled = √[((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)]. Interpretation: |d| < 0
Effect Size
Conditional Probability
The probability of event B occurring given that event A has already occurred, calculated as P(B|A) = P(A ∩ B) / P(A), wh
Probability
Conditional Probability Notation
The conditional probability P(A | B) is read 'the probability of A given B' and equals P(A ∩ B) / P(B), provided P(B) >
Probability
Conditions for Non-parametric Tests
Non-parametric tests require: random sampling, independent observations (for each test type), and that variables are mea
Non-Parametric Tests
Contingency Table Independence
A chi-squared test for independence uses χ² = Σ[(O - E)² / E] where O = observed frequency and E = expected frequency un
Chi-Squared Tests
Continuity Correction
When approximating a discrete random variable (taking integer values k) with a continuous distribution, the discrete val
Normal Distribution
Continuous Random Variable
A continuous random variable X can assume any real value within a specified interval. Probabilities are described by a p
Random Variables
Cumulative Distribution Function
The cumulative distribution function F(x) = P(X ≤ x) gives the total probability for all values up to and including x. F
Random Variables
Discrete Random Variable
A discrete random variable X assigns numerical values to outcomes of a random experiment and can only take on specific,
Random Variables
Distribution of Sample Mean
If X ~ N(μ, σ²), then the sample mean X̄ from n independent observations follows X̄ ~ N(μ, σ²/n). The standard error of
Normal Distribution
Paired t-test
The paired t-test compares the mean difference d̄ to zero using t = d̄ / (sₐ/√n), where sₐ is the standard deviation of
Paired t-Test
Power of Test
The power of a statistical test is P(reject H₀ | H₀ is false) = 1 - β, where β is the probability of Type II error. Powe
Hypothesis Testing
Probability Density Function
A probability density function f(x) for a continuous random variable X satisfies f(x) ≥ 0 for all x, and ∫f(x)dx = 1 ove
Random Variables
Sampling Bias
Sampling bias occurs when the sampling method inherently favors certain types of units. The sample is not a representati
Sampling Methods
SEC: Statistical Enquiry Cycle Initial Planning
Initial planning in the SEC involves: (1) identifying factors related to the problem, (2) defining a research question o
Statistical Enquiry Cycle
SEC: Statistical Enquiry Cycle Initial Planning
Data collection involves: (1) designing unbiased primary data collection methods, (2) researching secondary data sources
Statistical Enquiry Cycle
SEC: Statistical Enquiry Cycle Initial Planning
Data processing and presentation involves: (1) organising and processing data using technology, (2) making inferences ab
Statistical Enquiry Cycle
SEC: Statistical Enquiry Cycle Initial Planning
Interpretation of results involves: (1) analysing diagrams and calculations, (2) drawing conclusions related to original
Statistical Enquiry Cycle
SEC: Statistical Enquiry Cycle Initial Planning
Evaluation and review involves: (1) identifying weaknesses in data collection and display methods, (2) recognising limit
Statistical Enquiry Cycle
Wilcoxon Rank-sum Test
A non-parametric test for independent samples testing whether two populations have the same distribution, using ranks fr
Non-Parametric Tests
Wilcoxon Signed-Rank Procedure
The Wilcoxon signed-rank test: (1) calculates differences within pairs; (2) ignores signs, ranks absolute differences; (
Non-Parametric Tests
1.1: Mean
The arithmetic average of a set of values, calculated by summing all values and dividing by the number of values. Denote
Descriptive Statistics
1.1: Mean
The middle value of a dataset when arranged in ascending or descending order. For an even number of values, it is the av
Descriptive Statistics
1.1: Mean
The value that occurs with the highest frequency in a dataset. For grouped data, the modal class is the class with the h
Descriptive Statistics
1.1: Mean
The simplest measure of spread, calculated as the difference between the maximum and minimum values: Range = Max - Min.
Descriptive Statistics
1.1: Mean
A measure of spread calculated as IQR = Q₃ - Q₁, where Q₃ is the upper quartile and Q₁ is the lower quartile. It represe
Descriptive Statistics
1.1: Mean
A measure of dispersion representing the average distance of data values from the mean. Calculated as s = √(Σ(x - x̄)²/(
Descriptive Statistics
1.1: Mean
The average of the squared deviations from the mean, calculated as s² = Σ(x - x̄)²/(n-1) for a sample or σ² = Σ(x - μ)²/
Descriptive Statistics
1.2: Histogram
A graphical representation of grouped continuous data using rectangular bars where the area of each bar is proportional
Descriptive Statistics
1.2: Histogram
A graphical display consisting of a rectangular box showing Q₁, median, and Q₃, with lines (whiskers) extending to show
Descriptive Statistics
1.2: Histogram
A running total of frequencies, where the cumulative frequency for a class is the sum of the frequency of that class and
Descriptive Statistics
1.2: Histogram
A data display technique where values are split into a stem (all digits except the last) and a leaf (the last digit), ar
Descriptive Statistics
1.3: Skewness
A measure of the asymmetry of a distribution. Calculated using the formula: skewness = (mean - mode) / standard deviatio
Descriptive Statistics
1.3: Skewness
Values that are significantly different from the rest of the dataset. Formally, a value x is an outlier if x < Q₁ - 1.5×
Descriptive Statistics
10.1: Sign Test
A non-parametric test for paired data using the binomial distribution, testing H₀: median difference = 0 by comparing th
Non-Parametric Tests
10.1: Sign Test
A non-parametric test for paired data that ranks absolute differences and uses signed ranks, testing H₀: the distributio
Non-Parametric Tests
10.1: Sign Test
The Mann-Whitney U test compares two independent samples by ranking all observations together and calculating U = n₁n₂ +
Non-Parametric Tests
11.1: Bayes' Theorem
A formula for conditional probability: P(A|B) = [P(B|A) × P(A)] / P(B), allowing calculation of P(A|B) when P(B|A), P(A)
Bayes' Theorem
11.1: Bayes' Theorem
The conditional probability P(A|B) after observing event B, calculated using Bayes' theorem as the updated belief about
Bayes' Theorem
11.1: Bayes' Theorem
The probability P(A) before observing any evidence related to A, representing initial belief or background information.
Bayes' Theorem
12.1: Geometric Distribution
The geometric distribution models the number of independent Bernoulli trials X required until the first success, where e
Geometric Distribution
12.1: Geometric Distribution
The negative binomial distribution describes the number of trials X required to achieve r successes when each trial has
Geometric Distribution
12.1: Geometric Distribution
The continuous uniform distribution on interval [a, b] has constant probability density f(x) = 1/(b-a). It models situat
Geometric Distribution
12.1: Geometric Distribution
A Bernoulli trial is a random experiment with two mutually exclusive outcomes: success (with probability p) and failure
Geometric Distribution
13.1: Randomisation
Randomisation is the random allocation of experimental units to different treatment conditions. It ensures that differen
Experimental Design
13.1: Randomisation
Replication means applying each treatment to multiple independent experimental units rather than just once. This allows
Experimental Design
13.1: Randomisation
A control group is a set of experimental units that receive no treatment (or a standard/placebo treatment) and serves as
Experimental Design
13.1: Randomisation
In a blind (or single-blind) trial, participants do not know which group they're in or what treatment they're receiving.
Experimental Design
13.1: Randomisation
A confounding variable is an extraneous variable that is correlated with the treatment variable and affects the response
Experimental Design
13.1: Randomisation
Replication means each treatment is applied to multiple experimental units, not just one. Replication provides: (1) samp
Experimental Design
13.1: Randomisation
In a double blind trial, both participants and experimenters are unaware of treatment assignments. This eliminates bias
Experimental Design
13.2: Blocking
Blocking divides experimental units into homogeneous groups (blocks) based on some characteristic that is known to affec
Experimental Design
13.2: Blocking
Blocking divides experimental units into homogeneous groups (blocks) before randomization. Treatments are randomly assig
Experimental Design
13.3: Randomised Block Design
A randomised block design divides units into blocks based on variables expected to affect the response, then randomly as
Experimental Design
13.3: Randomised Block Design
In a completely randomised design, all units are randomly assigned to treatment groups without blocking. Each unit has e
Experimental Design
14.1: Parameter
A parameter is a numerical property of an entire population. Parameters are typically unknown and fixed for a given popu
Estimation
14.1: Parameter
A statistic is a numerical property of a sample, calculated from the observed data. Common statistics include x̄ (sample
Estimation
14.1: Parameter
A statistic is an unbiased estimator of a parameter if E(statistic) = parameter. For example, the sample mean x̄ is an u
Estimation
14.1: Parameter
The standard error (SE) is the standard deviation of a statistic's sampling distribution. For the sample mean, SE(x̄) =
Estimation
14.2: Central Limit Theorem
The Central Limit Theorem states that if X₁, X₂, ..., Xₙ are independent random variables from any distribution with mea
Estimation
14.2: Central Limit Theorem
The sampling distribution of a statistic is the probability distribution of that statistic calculated from all possible
Estimation
14.2: Central Limit Theorem
The bootstrap generates confidence intervals and estimates sampling distributions by repeatedly resampling from the obse
Estimation
15.1: Hypothesis Test
A hypothesis test is a formal procedure for evaluating evidence against a null hypothesis H₀ using a test statistic and
Hypothesis Testing
15.1: Hypothesis Test
The null hypothesis H₀ is a statement that specifies no change, no effect, or no difference between groups. It represent
Hypothesis Testing
15.1: Hypothesis Test
The alternative hypothesis H₁ (or Hₐ) is a statement that contradicts the null hypothesis, typically asserting that an e
Hypothesis Testing
15.1: Hypothesis Test
The p-value is P(test statistic as extreme as observed | H₀ is true). For a two-sided test with test statistic z, p-valu
Hypothesis Testing
15.1: Hypothesis Test
The significance level α is the predetermined probability threshold for making a Type I error (rejecting a true null hyp
Hypothesis Testing
15.1: Hypothesis Test
A test statistic is a function of the sample data that follows a known distribution (such as t, z, or χ²) under the null
Hypothesis Testing
15.1: Hypothesis Test
A critical value is a point on the distribution of the test statistic that separates the rejection region from the non-r
Hypothesis Testing
15.1: Hypothesis Test
A confidence interval for parameter θ is an interval [L, U] calculated from sample data such that P(L < θ < U) = confide
Hypothesis Testing
15.1: Hypothesis Test
The t-distribution is a family of distributions indexed by degrees of freedom (df). For a sample of size n, df = n - 1.
Hypothesis Testing
15.1: Hypothesis Test
Degrees of freedom (df) is the number of independent pieces of information available for estimation or hypothesis testin
Hypothesis Testing
15.1: Hypothesis Test
A 95% confidence interval means that if sampling and interval calculation are repeated many times, about 95% of the calc
Hypothesis Testing
15.5: Type I Error
A Type I error occurs when you reject H₀ even though H₀ is true. The probability of making a Type I error is α (the sign
Hypothesis Testing
15.5: Type I Error
A Type II error occurs when you fail to reject H₀ even though H₀ is false. The probability of a Type II error is denoted
Hypothesis Testing
15.7: One-sided Test
In a one-sided (or one-tailed) test, the alternative hypothesis is directional: H₁: θ > θ₀ (right-tailed) or H₁: θ < θ₀
Hypothesis Testing
15.7: One-sided Test
In a two-sided (or two-tailed) test, the alternative hypothesis is non-directional: H₁: θ ≠ θ₀. The critical region is s
Hypothesis Testing
18.1: Exponential Distribution
The exponential distribution with parameter λ > 0 has probability density function f(x) = λe^(-λx) for x ≥ 0, and cumula
Poisson and Exponential Distributions
18.1: Exponential Distribution
A discrete distribution with parameter λ > 0 where P(X = k) = e^(-λ) × λ^k / k! for k = 0, 1, 2, ... The mean is E(X) =
Poisson and Exponential Distributions
19.1: Goodness of Fit Test
A goodness of fit test uses χ² = Σ((O-E)²/E) to compare observed frequencies to those expected under a hypothesized dist
Goodness of Fit
19.1: Goodness of Fit Test
Goodness of fit tests compare observed frequencies O to expected frequencies E under a hypothesized distribution using χ
Goodness of Fit
2.1: Addition Rule of Probability
The rule that P(A ∪ B) = P(A) + P(B) - P(A ∩ B), which gives the probability that at least one of the events A or B occu
Probability
2.1: Addition Rule of Probability
The rule that P(A ∩ B) = P(A) × P(B|A), giving the probability that both events A and B occur. For independent events, t
Probability
2.1: Addition Rule of Probability
Set theory provides the formal language for probability, where the sample space S is represented as a set, events as sub
Probability
2.1: Addition Rule of Probability
The sample space S (or Ω) is the universal set containing every possible outcome of an experiment. Any event is a subset
Probability
2.2: Independent Events
Events A and B are independent if P(A ∩ B) = P(A) × P(B), or equivalently, if P(B|A) = P(B) and P(A|B) = P(A).
Probability
2.2: Independent Events
Events A and B are mutually exclusive if they cannot occur together, meaning P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
Probability
2.2: Independent Events
A two-way table (or contingency table precursor) presents frequencies or probabilities for two categorical variables, wi
Probability
2.2: Independent Events
A tree diagram represents probability experiments as a series of branches, with each branch labeled with a probability.
Probability
2.4: Venn Diagram
A graphical representation of sample spaces and events using overlapping circles or regions, where the area (or size) of
Probability
2.4: Venn Diagram
A visual representation of sequential events where branches show possible outcomes at each stage, with conditional proba
Probability
2.4: Venn Diagram
The addition law states P(A ∪ B) = P(A) + P(B) - P(A ∩ B); for mutually exclusive events, P(A ∪ B) = P(A) + P(B). The mu
Probability
21.1: Effect Size
Effect size quantifies the strength of an effect or the magnitude of a difference between groups. Common measures includ
Effect Size
21.1: Effect Size
Effect size quantifies the magnitude of an effect without depending on sample size. Cohen's d = (mean₁ - mean₂) / σ stan
Effect Size
3.1: Population
The entire collection of items, people, or observations of interest in a statistical investigation. A population can be
Sampling Methods
3.1: Population
A complete count of the entire population where data is collected from every individual member. Examples include the nat
Sampling Methods
3.1: Population
A subset of the population selected for investigation. The sample is studied to make inferences about the population. A
Sampling Methods
3.1: Population
When all outcomes in a sample space are equally likely, the probability of each outcome is 1/n where n is the total numb
Sampling Methods
3.2: Random Sampling
A sampling method where every member of the population has an equal probability of selection, and selections are indepen
Sampling Methods
3.2: Random Sampling
A sampling method where the population is divided into k equal intervals and every kth member is selected after a random
Sampling Methods
3.2: Random Sampling
A sampling method that divides the population into non-overlapping subgroups (strata) based on shared characteristics, t
Sampling Methods
3.2: Random Sampling
A sampling method that divides the population into naturally occurring clusters, randomly selects some clusters, and inc
Sampling Methods
3.2: Random Sampling
A non-random sampling method where the population is divided into strata and the researcher selects members from each st
Sampling Methods
3.2: Random Sampling
A convenience sampling method where the sample consists of individuals who are readily available and accessible, with no
Sampling Methods
3.3: Judgmental Sampling
Judgmental (purposive) sampling relies on the researcher's judgment to select a sample. Items are chosen because they're
Sampling Methods
3.3: Judgmental Sampling
In stratified sampling, strata are non-overlapping groups within the population, each defined by a characteristic (age,
Sampling Methods
4.1: Random Variable
A function that assigns numerical values to the outcomes of a random experiment. Discrete random variables take specific
Random Variables
4.1: Random Variable
For a discrete random variable, a table or formula showing P(X = x) for each possible value x. For a continuous random v
Random Variables
4.2: Expected Value
The mean or average value of a random variable, calculated as E(X) = ΣxP(X = x) for discrete variables or E(X) = ∫xf(x)d
Random Variables
4.2: Expected Value
Variance measures the spread of a random variable about its mean, calculated as Var(X) = E[(X - μ)²] = E(X²) - [E(X)]²,
Random Variables
5.1: Binomial Distribution
A discrete probability distribution for the number of successes X in n independent Bernoulli trials, each with success p
Binomial Distribution
5.1: Binomial Distribution
A binomial distribution B(n, p) requires: (1) a fixed number n of trials, (2) each trial has two mutually exclusive outc
Binomial Distribution
5.1: Binomial Distribution
In a binomial distribution X ~ B(n, p), n is the number of independent trials and p is the probability of success in eac
Binomial Distribution
6.1: Normal Distribution
A continuous probability distribution with probability density function determined by mean μ and standard deviation σ, d
Normal Distribution
6.1: Normal Distribution
The normal distribution N(μ, σ²) has: (1) a bell-shaped, symmetric probability density function; (2) mean μ, median, and
Normal Distribution
6.2: Standardisation
The process of converting a normal random variable X ~ N(μ, σ²) to a standard normal random variable Z ~ N(0, 1) using t
Normal Distribution
6.2: Standardisation
A standardized value representing how many standard deviations a value is from the mean. For X ~ N(μ, σ²), the z-score i
Normal Distribution
6.3: Normal Tables
Tables providing cumulative probabilities P(Z ≤ z) for the standard normal distribution, typically showing P(Z ≤ z) for
Normal Distribution
6.3: Normal Tables
The inverse normal process: given a probability p, find the z-score such that P(Z ≤ z) = p. This requires reverse lookup
Normal Distribution
7.1: Scatter Diagram
A graphical display of bivariate data using points (xi, yi) plotted on a coordinate plane, where the pattern of points r
Correlation and Regression
7.1: Scatter Diagram
Pearson's product-moment correlation coefficient r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² × Σ(y - ȳ)²]. Values range from -
Correlation and Regression
7.2: Pearson Correlation Coefficient
The product moment correlation coefficient (PMCC) calculated as r = Σ(x - x̄)(y - ȳ) / √[Σ(x - x̄)² × Σ(y - ȳ)²], rangin
Correlation and Regression
7.2: Pearson Correlation Coefficient
A rank-based correlation coefficient calculated as rs = 1 - (6Σd² / n(n² - 1)), where d is the difference in ranks for e
Correlation and Regression
7.3: Least Squares Regression
A regression method that finds the line y = a + bx minimizing Σ(observed y - predicted y)², where b = Σ(x - x̄)(y - ȳ) /
Correlation and Regression
7.3: Least Squares Regression
Using the regression equation to estimate y for an x-value within the range of x-values in the original data.
Correlation and Regression
7.3: Least Squares Regression
Using the regression equation to estimate y for an x-value outside the range of x-values in the original data.
Correlation and Regression
7.3: Least Squares Regression
The vertical distance from each data point to the regression line, calculated as ei = yi - ŷi, where ŷi = a + bxi is the
Correlation and Regression
7.4: Scatter Diagram with Regression Line
A scatter diagram plots pairs of observations as points on a graph. The least squares regression line y = a + bx minimiz
Correlation and Regression
7.4: Scatter Diagram with Regression Line
For a regression line ŷ = a + bx, the residual for observation (x, y) is e = y - ŷ = y - (a + bx). The sum of residuals
Correlation and Regression
7.4: Scatter Diagram with Regression Line
Regression diagnostics include: residual plots (checking for patterns, outliers, heteroscedasticity), normal probability
Correlation and Regression
8.1: Expectation Algebra
Mathematical rules for expected values: E(aX+b) = aE(X)+b, E(X+Y) = E(X)+E(Y), and E(XY) = E(X)E(Y) when X and Y are ind
Expectation Algebra
8.1: Expectation Algebra
Mathematical rules for variance: Var(aX+b) = a²Var(X), Var(X+Y) = Var(X)+Var(Y) when independent, and Var(X-Y) = Var(X)+
Expectation Algebra
8.4, 15.7: Acceptance Region
The acceptance region consists of test statistic values inside the critical value(s). If the test statistic falls in thi
Expectation Algebra
8.4, 15.7: Acceptance Region
The rejection region consists of test statistic values beyond the critical value(s). If the test statistic falls in this
Expectation Algebra
9.1: Chi-squared Test for Independence
A test statistic χ² = Σ[(observed - expected)² / expected] testing whether two variables in a contingency table are inde
Chi-Squared Tests
9.1: Chi-squared Test for Independence
A table displaying frequencies of observations categorized by two categorical variables, organized in rows and columns w
Chi-Squared Tests
9.1: Chi-squared Test for Independence
For a contingency table, expected frequency for a cell = (row total × column total) / grand total, calculated under the
Chi-Squared Tests
9.1: Chi-squared Test for Independence
A hypothesis test using the test statistic χ² = Σ((O-E)²/E) where O represents observed frequencies and E represents exp
Chi-Squared Tests
9.1: Chi-squared Test for Independence
In a contingency table, the expected frequency for a cell is calculated as E = (row total × column total) / grand total.
Chi-Squared Tests
9.1: Chi-squared Test for Independence
In hypothesis testing with categorical data, the observed frequency is the actual number of observations that fall into
Chi-Squared Tests