🌍 Worldwide · Mathematics

Statistics Syllabus

Mathematical study and application. The full Statistics syllabus: 10 chapters and 50 topics, with 50 flashcards written against it.

10Chapters

50Topics

50Flashcards

Statistics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Statistics, not a summary of it.

Introduction to Statistics
5 topics
- Definition and Scope
- Types of Data
- Levels of Measurement
- Sampling Methods
- Descriptive vs Inferential Statistics
Descriptive Statistics
5 topics
- Measures of Central Tendency
- Measures of Dispersion
- Skewness and Kurtosis
- Data Visualization
- Frequency Distributions
Probability
5 topics
- Basic Probability Concepts
- Conditional Probability
- Probability Distributions
- Bayes' Theorem
- Random Variables
Probability Distributions
5 topics
- Discrete Probability Distributions
- Binomial Distribution
- Poisson Distribution
- Continuous Probability Distributions
- Normal Distribution
Inferential Statistics
5 topics
- Estimation
- Confidence Intervals
- Hypothesis Testing
- Types of Errors
- Power of a Test
Correlation and Regression
5 topics
- Correlation Coefficients
- Simple Linear Regression
- Multiple Regression
- Assumptions of Regression Analysis
- Model Fit and Diagnostics
Analysis of Variance (ANOVA)
5 topics
- One-Way ANOVA
- Two-Way ANOVA
- Assumptions of ANOVA
- Post Hoc Tests
- Repeated Measures ANOVA
Non-Parametric Tests
5 topics
- Chi-Square Test
- Mann-Whitney U Test
- Wilcoxon Signed-Rank Test
- Kruskal-Wallis Test
- Spearman Rank Correlation
Time Series Analysis
5 topics
- Components of Time Series
- Trend Analysis
- Seasonal Variations
- Moving Averages
- ARIMA Models
Multivariate Statistics
5 topics
- Principal Component Analysis (PCA)
- Factor Analysis
- Cluster Analysis
- Discriminant Analysis
- Multivariate Analysis of Variance (MANOVA)

Statistics flashcards

The complete Statistics deck: 50 questions with worked answers.

What is the definition of Statistics as a field of study?
Statistics is the science of collecting, organizing, summarizing, analyzing, interpreting, and presenting data to make decisions and draw conclusions under uncertainty.
Distinguish between a population and a sample.
A population is the entire set of all elements of interest; a sample is a subset of the population actually observed or measured. A numerical summary of a population is a parameter; of a sample, a statistic.
What is the difference between qualitative (categorical) and quantitative (numerical) data?
Qualitative data describe categories or attributes (e.g., color, gender); quantitative data are numerical measurements or counts (e.g., height, number of cars) on which arithmetic is meaningful.
Distinguish discrete from continuous quantitative data.
Discrete data take countable, separate values (often integers, e.g., number of students); continuous data can take any value within an interval (e.g., weight, time) limited only by measurement precision.
What are the four levels (scales) of measurement, in order of increasing information?
Nominal, ordinal, interval, and ratio.
Compare the nominal and ordinal levels of measurement.
Nominal data are labels/categories with no order (e.g., blood type). Ordinal data have a meaningful order or ranking but unequal or undefined intervals between values (e.g., satisfaction: low/medium/high).
Compare the interval and ratio levels of measurement.
Interval data have ordered values with equal intervals but no true zero, so ratios are meaningless (e.g., Celsius temperature). Ratio data have equal intervals and a true zero, so ratios are meaningful (e.g., weight, height).
What is the difference between descriptive and inferential statistics?
Descriptive statistics summarize and describe features of a data set (tables, graphs, measures like mean and SD). Inferential statistics use sample data to draw conclusions, estimate parameters, and test hypotheses about a population.
What is the difference between probability sampling and non-probability sampling?
In probability sampling every element has a known, nonzero chance of selection (enabling valid inference); in non-probability sampling selection is subjective and probabilities are unknown (e.g., convenience, quota sampling).
Describe simple random sampling.
A method in which every possible sample of size $n$ has an equal chance of being chosen, so each individual is selected independently with equal probability.
Describe stratified sampling and when it is used.
The population is divided into homogeneous subgroups (strata) and random samples are drawn from each stratum. Used to ensure representation of key subgroups and to increase precision.
Describe cluster sampling and how it differs from stratified sampling.
The population is divided into clusters (often geographic); some clusters are randomly selected and all (or a sample of) members within them are surveyed. Unlike stratified sampling, only some groups are sampled and clusters are ideally internally heterogeneous.
Describe systematic sampling and define the sampling interval.
Every $k$-th element is selected from an ordered list after a random start, where the sampling interval is $k = \frac{N}{n}$ ($N$ = population size, $n$ = sample size).
What are the three common measures of central tendency?
The mean (arithmetic average), the median (middle value), and the mode (most frequent value).
State the formula for the arithmetic mean of a sample.
$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$$
How is the median found, and why is it preferred over the mean for skewed data?
Order the data; the median is the middle value if $n$ is odd, or the average of the two middle values if $n$ is even. It is preferred for skewed data or outliers because it is resistant (robust) to extreme values.
Define the mode and note how many modes a distribution can have.
The mode is the value that occurs most frequently. A distribution may have no mode, one mode (unimodal), two modes (bimodal), or more (multimodal).
For a moderately skewed unimodal distribution, what empirical relationship links mean, median, and mode?
$$\text{Mean} - \text{Mode} \approx 3(\text{Mean} - \text{Median})$$
What are the main measures of dispersion (spread)?
Range, interquartile range (IQR), variance, standard deviation, and coefficient of variation.
Define the range and the interquartile range (IQR).
Range = maximum − minimum. IQR = $Q_3 - Q_1$, the spread of the middle 50% of the data, which is resistant to outliers.

Show the other 30 Statistics flashcards

State the formula for the sample variance.
$$s^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^{2}$$
Why does the sample variance divide by $n-1$ instead of $n$?
Dividing by $n-1$ (Bessel's correction, the degrees of freedom) makes $s^{2}$ an unbiased estimator of the population variance $\sigma^{2}$; dividing by $n$ would underestimate it.
State the formula for the population standard deviation.
$$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^{2}}$$
Define the coefficient of variation (CV) and state its use.
$$CV = \frac{s}{\bar{x}} \times 100\%$$ It is a unitless relative measure of dispersion used to compare variability between data sets with different units or means.
What does skewness measure, and what do positive and negative skew indicate?
Skewness measures the asymmetry of a distribution. Positive (right) skew has a long right tail (mean > median); negative (left) skew has a long left tail (mean < median); zero skew is symmetric.
What does kurtosis measure, and name the three classifications?
Kurtosis measures the 'tailedness'/peakedness of a distribution. Mesokurtic (normal, excess kurtosis $=0$), leptokurtic (heavy tails/peaked, excess $>0$), and platykurtic (light tails/flat, excess $<0$).
What is excess kurtosis and what is its value for the normal distribution?
Excess kurtosis = kurtosis $-\ 3$. For the normal distribution kurtosis $=3$, so its excess kurtosis is $0$.
Which graphs are appropriate for categorical data versus quantitative data?
Categorical data: bar charts and pie charts. Quantitative data: histograms, stem-and-leaf plots, box plots, dot plots, and (for two variables) scatter plots.
What is a histogram and how does it differ from a bar chart?
A histogram displays the distribution of continuous quantitative data using adjacent bars over class intervals, where area/height represents frequency. A bar chart shows categorical data with separated bars whose order is arbitrary.
What five values make up a box plot (five-number summary)?
Minimum, first quartile $Q_1$, median $Q_2$, third quartile $Q_3$, and maximum. The box spans $Q_1$ to $Q_3$ with the median inside; whiskers extend to the extremes (or fences).
How are outliers commonly identified using the IQR rule?
A value is an outlier if it lies below $Q_1 - 1.5\times IQR$ or above $Q_3 + 1.5\times IQR$.
What is a frequency distribution, and what are relative and cumulative frequency?
A frequency distribution is a table pairing classes/values with their counts. Relative frequency = $\frac{\text{class frequency}}{n}$ (a proportion); cumulative frequency is the running total of frequencies up to a class.
How do you compute the class width for a grouped frequency distribution?
$$\text{Class width} \approx \frac{\text{Range}}{\text{Number of classes}} = \frac{\text{max} - \text{min}}{k}$$ rounded up to a convenient value.
Define the classical (theoretical) probability of an event with equally likely outcomes.
$$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}}$$
State the three axioms of probability (Kolmogorov).
1) $0 \le P(E) \le 1$ for any event $E$; 2) $P(S) = 1$ for the sample space $S$; 3) for mutually exclusive events, $P(A \cup B) = P(A) + P(B)$.
State the general addition rule for two events.
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ For mutually exclusive events $P(A\cap B)=0$.
What does it mean for two events to be mutually exclusive versus independent?
Mutually exclusive events cannot occur together: $P(A \cap B) = 0$. Independent events have $P(A \cap B) = P(A)P(B)$; the occurrence of one does not affect the other.
Define conditional probability and give its formula.
The probability of $A$ given that $B$ has occurred: $$P(A\mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$$
State the general multiplication rule for the intersection of two events.
$$P(A \cap B) = P(A)\,P(B\mid A) = P(B)\,P(A\mid B)$$
State the law of total probability for a partition $B_1,\dots,B_n$.
$$P(A) = \sum_{i=1}^{n} P(A\mid B_i)\,P(B_i)$$
State Bayes' Theorem.
$$P(A\mid B) = \frac{P(B\mid A)\,P(A)}{P(B)} = \frac{P(B\mid A)\,P(A)}{\sum_i P(B\mid A_i)\,P(A_i)}$$
In Bayes' Theorem, identify the prior, likelihood, and posterior.
For $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$: $P(A)$ is the prior, $P(B\mid A)$ is the likelihood, $P(B)$ is the evidence/marginal, and $P(A\mid B)$ is the posterior.
What is a random variable, and how do discrete and continuous random variables differ?
A random variable assigns a numerical value to each outcome of a random experiment. A discrete random variable takes countable values (described by a PMF); a continuous one takes values over an interval (described by a PDF).
Define the expected value (mean) of a discrete random variable.
$$E(X) = \mu = \sum_{i} x_i\,P(X = x_i)$$
State the formula for the variance of a discrete random variable.
$$\operatorname{Var}(X) = E[(X-\mu)^2] = \sum_i (x_i-\mu)^2 P(x_i) = E(X^2) - [E(X)]^2$$
What two conditions must a discrete probability distribution (PMF) satisfy?
1) $0 \le P(x_i) \le 1$ for every value; 2) the probabilities sum to one: $\sum_i P(x_i) = 1$.
What are the four conditions defining a binomial experiment?
1) A fixed number $n$ of trials; 2) each trial has two outcomes (success/failure); 3) constant success probability $p$; 4) trials are independent.
State the binomial probability mass function.
$$P(X=k) = \binom{n}{k} p^{k}(1-p)^{\,n-k}, \quad k = 0,1,\dots,n$$
State the mean and variance of a binomial distribution.
$$\mu = np, \qquad \sigma^{2} = np(1-p)$$
State the Poisson probability mass function and its mean/variance.
$$P(X=k) = \frac{\lambda^{k} e^{-\lambda}}{k!}, \quad k=0,1,2,\dots$$ with mean and variance both equal to $\lambda$. It models the number of events in a fixed interval, and approximates the binomial when $n$ is large and $p$ is small ($\lambda = np$).

Statistics syllabus FAQ

What is in the Statistics syllabus?

The Statistics syllabus is split into 10 chapters: Introduction to Statistics, Descriptive Statistics, Probability, Probability Distributions, Inferential Statistics, Correlation and Regression, Analysis of Variance (ANOVA), Non-Parametric Tests, Time Series Analysis and Multivariate Statistics. Together they contain 50 topics.

How many chapters and topics are in the Statistics syllabus?

10 chapters and 50 topics. The largest chapter is Introduction to Statistics, with 5 topics.

Are there Statistics flashcards?

Yes — 50 question-and-answer flashcards, printed in full on this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.

Mathematics syllabus

Mathematics · full syllabus breakdown

7 subjects · 63 chapters · 351 flashcards Calculus syllabus