Grade 12 · secular · Common Core-aligned
This is the non-calculus path to a real senior-year math credit, statistics and data science instead of Calculus. Your child learns to summarize data, spot when a relationship between two things is probably real versus coincidence, design a study that can actually support a "this causes that" claim, and use probability to say how confident anyone should be in a result. Every technique gets done by hand on a tiny dataset first (8-10 numbers), then redone in a free spreadsheet on real government and survey data, Census, CDC, World Bank, that kind of thing. It ends with your child running their own actual study from scratch: pick a question, collect or pull real data, analyze it, and defend in writing and out loud what the results do and don't prove.
The whole year, in plain English. Tap any unit to see every skill inside, nothing is hidden.
This is where all the vocabulary for the rest of the year gets built: distribution, shape, center, spread, standard deviation. Your child moves from just describing a plot in words to computing mean, median, mode, range, IQR, and standard deviation by hand, then z-scores and boxplots, then finally the same statistics in a spreadsheet on a bigger, real dataset. The big idea underneath all of it: every summary number throws away information, and choosing which one to report is a choice about what to hide.
Given an ordered list of 8-10 numeric values, compute the mean, median, and mode by hand and state which, if any, are equal.
Given a raw dataset of 8-10 values, compute the range, interquartile range, and standard deviation by hand, showing the deviation-from-mean step explicitly.
Explain why the mean and median diverge in a skewed distribution and which one is being pulled by which feature of the data.
Classify a given value in a dataset as an outlier or not, using a stated numeric or contextual criterion, and justify whether it should be reported or removed for a specific analytic purpose.
Compute a z-score for a given value and interpret what it says about that value's position relative to the rest of the distribution, in the units of the original variable.
Given two datasets with identical mean and range but different shape, determine which additional statistic (median, IQR, or SD) reveals the difference and justify why the matching statistics failed to.
Given a real public-dataset variable never encountered in class (e.g., a CDC BRFSS or ACS variable), select and compute an appropriate set of summary statistics, describe the distribution's shape/center/spread in writing, and defend the choice of statistics used.
Classify a given histogram, dotplot, or boxplot by shape (symmetric, skewed left, skewed right, uniform, bimodal) using visual features alone.
Construct a boxplot by hand from a five-number summary computed from raw data, correctly placing whiskers and the box relative to Q1, median, and Q3.
Now your child moves from describing one variable to describing how two variables move together, scatterplots, the correlation coefficient r, and a regression line, all computed by hand first. The unit's real teaching moment is a set of datasets (Anscombe-style) that all have the identical r but look completely different when plotted, proof that a single number can hide a lot.
Given a scatterplot of paired numeric data, describe the association using direction, form, and strength in words.
Compute the correlation coefficient r by hand on a small paired dataset (n=8-10) using the deviation-product formula.
Explain why two datasets can share an identical correlation coefficient while representing very different underlying relationships (e.g., one linear and consistent, one driven by a single outlier, one curved).
Fit a least-squares regression line by hand to a small paired dataset and interpret the slope and intercept in the context of the variables.
Given a regression equation and a new x-value, calculate and evaluate a prediction, classifying it as interpolation or extrapolation and stating the confidence warranted by that classification.
Construct a residual plot from a regression line's residuals and use its pattern (or lack of pattern) to judge whether a linear model is appropriate for the data.
Interpret the coefficient of determination (r-squared) as the proportion of variability in the response variable accounted for by the linear model, in the context of a specific pair of variables.
Given a completely unfamiliar pair of real-world variables and a scatterplot with no accompanying r or regression statistics, judge whether a linear model would be appropriate before any computation is performed, and justify the judgment from the shape of the cloud of points alone.
Using a real, previously unseen paired dataset (e.g., a World Bank indicator pair not used in class) accessed via spreadsheet software, compute r, the regression line, and r-squared, and write an interpretation that names at least one thing the summary statistics conceal about the relationship.
This unit is a turn in the road: instead of asking 'how do I compute this,' it asks 'am I even allowed to conclude this, given how the data was collected.' It covers population vs. sample, the four sampling methods and the specific bias each one causes, observational studies vs. experiments, confounding variables, and why random assignment is the one thing that actually licenses a cause-and-effect claim.
Given a written research scenario, state whether the group described is the population or a sample, and identify the corresponding number described as a parameter or a statistic.
Given a sampling scenario that uses the same structural cue words practiced in the Day 3-4 worked examples (e.g., 'names drawn from a hat' for SRS, 'divided by grade level first, then randomly selected within each' for stratified, 'entire classrooms selected' for cluster, 'surveys whoever happens to walk by' for convenience), correctly apply the taught classification routine to label the method as simple random, stratified, cluster, or convenience.
Classify a described sampling procedure as simple random, stratified, cluster, or convenience, and name the specific bias (if any) that procedure introduces.
Given two study summaries reporting the same correlation but differing in whether treatment was randomly assigned, explain why only one licenses a causal claim, connecting the presence or absence of random assignment to the possibility of confounding.
For a described observational study, generate a specific confounding variable (not previously discussed in class) and justify why it plausibly affects both the explanatory and response variable.
Compare two survey vignettes that ask 'the same' question with different wording and identify the specific wording feature responsible for the differing response patterns.
Given a real, previously unseen news article reporting a causal claim from data, determine from the body text whether the underlying study was observational or experimental, even when the headline's language implies causation regardless of design.
Write a critique of a real news article's causal claim that states precisely what claim the study's actual design licenses, distinct from what the headline claims, integrating sampling method, study design, and at least one specific unresolved confounder into a single coherent judgment.
Given a scenario describing both the sampling method and the assignment method of a study, evaluate separately what the sampling method licenses (generalizability to the population) and what the assignment method licenses (a causal claim), rather than forming one combined judgment about whether the study is 'good.'
Probability here isn't formulas to memorize, it's a model for describing patterns in random processes over the long run. Your child moves from listing outcomes and applying addition/multiplication rules, to conditional probability and independence on real two-way tables, to random variables and their expected value, ending with simulation as a way to check theoretical probability against what actually happens when you run something many times.
Given a description of a random process (e.g., drawing two cards, rolling two dice), list the sample space and identify a specified event as a subset of it.
Compute P(A or B) using the addition rule, correctly determining first whether A and B are mutually exclusive from a given scenario.
Determine whether two events described by a real two-way table are independent by comparing P(A) to P(A|B), and explain in writing what the comparison shows.
Given a probability distribution table for a discrete random variable, compute the expected value E(X) using summation notation and interpret it as a long-run average.
Compute the standard deviation of a discrete random variable by hand and explain what a large versus small SD(X) implies about the variable's real-world behavior.
Design and run a hand or spreadsheet simulation (using random digits or RAND) to estimate the probability of a compound event, and compare the simulated relative frequency to the theoretical probability computed by rule.
Given a completely novel real-world scenario involving three or more categorical variables (not two-way, not modeled in class), determine what additional information would be required to assess whether any pair of variables is independent, without being told which variables to compare.
Critique a media or textbook claim that treats a single trial's outcome as evidence against (or for) a stated long-run probability (e.g., 'the model said 70% chance and it didn't happen, so the model was wrong'), using the distinction between single-trial outcomes and long-run relative frequency.
Recall the definition of independent events (P(A and B) = P(A)·P(B), equivalently P(A|B) = P(A)) from memory without reference to a specific scenario.
Two specific, reusable models, plus the judgment to know which one (if either) fits a given situation. The first third builds the binomial distribution by hand for small numbers of trials. The rest builds the normal distribution as a model for continuous measurements, reusing Unit 1's z-scores and standard deviation, working up through the 68-95-99.7 rule to spreadsheet tools (NORM.DIST) for any proportion, and ending with the harder skill of checking whether a normal model even fits a given dataset in the first place.
Given a description of a random process, state whether it meets the four binomial conditions (fixed number of trials, two outcomes, constant probability of success, independent trials) and identify which condition fails when it does not.
Compute the probability of exactly k successes in n binomial trials by hand using the binomial probability formula, for n ≤ 6.
Calculate the mean (np) and standard deviation (√(np(1-p))) of a binomial random variable and explain what each measures in terms of repeated trials of the process.
Explain, using the shape of overlaid binomial histograms for increasing n, why a sum of many small independent random effects tends toward a bell shape regardless of the shape of the individual trials.
Apply the 68-95-99.7 rule to estimate the percentage of a normally distributed population falling within 1, 2, or 3 standard deviations of the mean, or between two given values that fall exactly on those boundaries.
Standardize a raw score from an approximately normal distribution into a z-score and interpret it as a distance in standard deviations from the mean, including for values that do not fall on the 68-95-99.7 boundaries.
Use a standard normal table or the spreadsheet function NORM.DIST to find the proportion of a normal distribution above, below, or between arbitrary values, and use NORM.INV (or table lookup in reverse) to find a value corresponding to a given proportion.
Given a real, unfamiliar dataset's histogram and summary statistics, judge whether a normal model is an appropriate simplification, and justify the judgment by citing specific features of the shape (symmetry, tails, number of modes, outliers) that support or undermine the fit.
Given a short scenario, decide whether a binomial model, a normal model, or neither is the appropriate model, and justify the choice by naming which defining feature of the scenario drove the decision.
Given a novel measurement context outside any taught example (e.g., a manufacturing tolerance, a biological measurement), design a plan for whether and how a normal model would be used to answer a specific question, and state what additional information would be needed before trusting that plan.
This is the unit everything else in the second half of the course depends on. Your child first generates, by hand and in a spreadsheet, the fact that a sample statistic (like an average) changes from sample to sample and that this variation has its own shape, before the Central Limit Theorem is even named. Only after they've seen it happen is the rule stated. The unit also uses a side-by-side comparison of a genuinely random simulation and a deliberately biased one to nail down that more data reduces spread but never fixes bias.
Given a small finite population (4-6 values) and a fixed sample size, list every possible simple random sample of that size, compute the sample mean for each, and state how many distinct samples produce each possible mean value.
Explain why a sample statistic computed from a random sample is itself a random variable with its own distribution, distinguishing this distribution from the distribution of the original population data.
Predict, before instruction, what will happen to the shape, center, and spread of an accumulating dot-plot of sample means as the number of simulation trials and the sample size per trial both increase, and revise the prediction using evidence from the simulation.
State the Central Limit Theorem in words, specifying the two conditions (large enough n; independent random samples) under which the sampling distribution of a sample mean is approximately normal regardless of population shape.
Compute the standard error of a sample mean given a population standard deviation and sample size, and of a sample proportion given a population proportion and sample size, using the square-root-of-n formulas.
Given a description of a real sampling scenario, judge whether a reported statistic's sampling distribution can be assumed approximately normal, and identify what additional information (n, population shape, independence) would be needed to justify that judgment.
Explain, using a specific biased-versus-unbiased sampling comparison, why increasing sample size reduces the spread of a sampling distribution but does not correct a sampling method's bias.
Given a novel real-world scenario (e.g., a nonresponse-heavy online poll, or a convenience-sample health study reported in the news) never discussed in class, identify whether the described flaw would primarily inflate the sampling distribution's spread, shift its center, or both, and justify the classification.
Design and run a spreadsheet simulation that draws repeated random samples of a specified size from a real dataset treated as a population, computes the sample statistic each time, and produces an empirical sampling distribution to compare against the CLT's shape/center/spread predictions.
This is where sampling variability (Unit 6) becomes the actual machinery of drawing conclusions. Your child builds confidence intervals and hypothesis tests for one and two groups, by hand on small data first, then by spreadsheet. The core idea running through the whole unit: a p-value or confidence interval tells you how surprising your data would be under an assumed model, it is not a verdict on truth, and mixing those two up is an error professionals make constantly, not just students.
State, from a written research question, the correct null and alternative hypotheses in symbols and in words.
Compute a one-sample confidence interval for a proportion or a mean by hand from raw data (n=8-12), then reproduce the same interval using spreadsheet functions.
Explain how each of confidence level, sample size, and sample variability changes the margin of error, holding the other two fixed.
Given a confidence interval and its stated confidence level, distinguish a correct long-run interpretation of the interval from a common incorrect probability-of-the-parameter interpretation.
Carry out a complete one-sample or two-sample hypothesis test (t-test or proportion z-test) on small raw data, including stating hypotheses, checking conditions, computing the test statistic and p-value, and stating a conclusion in context.
Given a new research context not matched to any lesson by date or label, decide whether the appropriate procedure is a one- or two-sample, proportion or mean test, and justify the choice from the structure of the variables and design.
Explain, in a specific scenario, what a Type I error and a Type II error would each look like in context, and why lowering one generally raises the other for a fixed sample size.
Given a student-written or media statement interpreting a p-value or confidence interval, identify which specific documented misinterpretation it commits and rewrite it correctly.
Analyze a real two-group dataset (e.g., a GSS attitude comparison) to determine whether an observed difference is statistically significant, whether it is practically important, and what the result does and does not establish about causation.
Recall the defining formula elements (test statistic = (statistic - null value) / standard error) common to every hypothesis test in the unit.
The last new technique of the course, the chi-square test for categorical data, plus a framework for choosing among every inference tool met since Unit 6. Then, for the last two and a half weeks, the capstone: your child designs an original study or pulls together real public data, collects or retrieves it, picks and justifies a procedure, computes part of it by hand and the rest by spreadsheet, and writes and defends a report on what the results do and don't prove.
Given a two-way frequency table of observed counts, compute expected counts under independence and a chi-square test statistic by hand, matching the row-total-times-column-total-over-grand-total formula.
Explain why the chi-square test for independence and the chi-square goodness-of-fit test answer different questions (association between two categorical variables vs. fit of one categorical variable to a hypothesized distribution), using the same test statistic formula for both.
Given an unfamiliar research scenario with a data structure (one sample vs. two, categorical vs. quantitative, one variable vs. two), select and justify the single correct inference procedure from the full set met in Units 6-8 using a decision framework.
Given a novel, real public-health or social-science report (not used in instruction) that presents a statistically significant result, identify what the reported statistic does and does not license the reader to claim about causation, generalizability, and practical importance.
Formulate an original, testable research question and a data-collection or dataset-selection plan that names the sampling or experimental design and at least one specific safeguard against a bias named in Unit 3 (e.g., confounding, non-response, voluntary-response sampling).
Execute the full inference procedure appropriate to their own capstone data (compute by hand for a representative subset, then reproduce for the complete dataset using spreadsheet functions), matching the hand and spreadsheet results within rounding error.
Write an explicit, specific statement of the limitations of their own capstone conclusion, identifying at least one threat to the validity of their causal or generalizability claim that is particular to their own study design (not a generic disclaimer).
Given a two-way table already set up with observed and expected counts, recall which row of a chi-square distribution table (which degrees of freedom) applies, using the (rows-1)(columns-1) rule.
From the parent guide
This is the non-calculus path to a real senior-year math credit, statistics and data science instead of Calculus. Your child learns to summarize data, spot when a relationship between two things is probably real versus coincidence, design a study that can actually support a "this causes that" claim, and use probability to say how confident anyone should be in a result. Every technique gets done by hand on a tiny dataset first (8-10 numbers), then redone in a free spreadsheet on real government and survey data, Census, CDC, World Bank, that kind of thing. It ends with your child running their own actual study from scratch: pick a question, collect or pull real data, analyze it, and defend in writing and out loud what the results do and don't prove.
Unit 1 · what to expect
This is where all the vocabulary for the rest of the year gets built: distribution, shape, center, spread, standard deviation. Your child moves from just describing a plot in words to computing mean, median, mode, range, IQR, and standard deviation by hand, then z-scores and boxplots, then finally the same statistics in a spreadsheet on a bigger, real dataset. The big idea underneath all of it: every summary number throws away information, and choosing which one to report is a choice about what to hide.
The full guide covers all 8 units: where kids get stuck, what to say, and how to tell it's working. Included with the course.
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