Statistics & Probability

Understand the distinction between mathematical models and statistical models.

• S.Q.1.1(Objective)

  Distinguish among different sources of variability, including measurement, natural, induced, and sampling variability.

• S.Q.1.2(Objective)

  Formulate meaningful statistical questions to clarify the problem at hand.

Distinguish between the distribution of a population, a distribution of sample data, and a sampling distribution.

• S.Q.2.1(Objective)

  Distinguish between sample statistics and population parameters.

• S.Q.2.2(Objective)

  Recognize a population distribution has fixed values of its parameters and that these parameter values are typically unknown.

• S.Q.2.3(Objective)

  Recognize that a sample data distribution is taken from a population distribution, and the data distribution is what is seen in practice.

• S.Q.2.4(Objective)

  Recognize a sampling distribution is the distribution of a sample statistic (e.g., sample mean, sample proportion) obtained from repeated samples.

Identify differences between categorical and quantitative data.

• S.Q.3.1(Objective)

  Determine whether categorical or quantitative data is appropriate to answer a statistical question.

• S.Q.3.2(Objective)

  Compare and contrast different potential graphical or visual representations given the same data set.

Distinguish among different types of study designs for collecting data, and know the scope of inference for each design type.

• S.DC.1.1(Objective)

  Distinguish among sample surveys, experiments, and observational studies.

• S.DC.1.2(Objective)

  Compare and contrast the benefits of different sampling techniques.

• S.DC.1.3(Objective)

  Determine the appropriate scope of inference for generalizing results.

• S.DC.1.4(Objective)

  Explain how sample size impacts the precision with which generalizations can be made.

• S.DC.1.5(Objective)

  Determine when a cause-and-effect inference can be drawn from an association, based on how the data were collected.

Identify common sources of bias and the role of randomization in study design.

• S.DC.2.1(Objective)

  Explain how randomization and sources of bias impact the results of a study.

• S.DC.2.2(Objective)

  Understand the different roles of random selection and random assignment in study design.

Use distributions of quantitative and categorical data to identify the key features of the data collected in context.

• S.DA.1.1(Objective)

  Summarize and represent the distribution for univariate quantitative data by describing and analyzing the shape of the distribution, the measures of center for the distribution, the patterns in variability for the distribution, and any outliers, gaps, or other unusual features in the distribution.

• S.DA.1.2(Objective)

  Select and create an appropriate display (e.g., dot plots, histograms, box plots) for univariate data.

• S.DA.1.3(Objective)

  Use statistics appropriate to the shape of the data distribution to compare center and variability of two or more different data sets.

• S.DA.1.4(Objective)

  Describe and analyze the distribution of univariate categorical data.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.

• S.DA.2.1(Objective)

  Use calculators, computers, or tables to estimate areas under the normal curve. Recognize that there are data sets for which such a procedure is not appropriate.

Compare two or more groups by analyzing distributions.

• S.DA.3.1(Objective)

  Construct appropriate parallel graphical displays of distributions.

• S.DA.3.2(Objective)

  Use numerical attributes of distributions to make comparisons between distributions.

Analyze associations between two variables.

• S.DA.4.1(Objective)

  Create two-way tables for bivariate categorical data and analyze for possible associations between the two categories using marginal, joint, and conditional frequencies.

• S.DA.4.2(Objective)

  Make predictions and draw conclusions from regression models (linear, exponential, quadratic) from two-variable quantitative data.

• S.DA.4.3(Objective)

  Analyze scatter plots for patterns, linearity, outliers, and influential points.

• S.DA.4.4(Objective)

  Using technology, compute and interpret the correlation coefficient.

• S.DA.4.5(Objective)

  Understand the implications of extrapolating data to make predictions.

Make statistical inferences and evaluate claims from studies.

• S.DA.5.1(Objective)

  Construct and interpret confidence intervals for the mean of a normally distributed population and for a population proportion.

• S.DA.5.2(Objective)

  Explain how a sample statistic and a confidence level are used in the construction of a confidence interval.

• S.DA.5.3(Objective)

  Explain how changes in the sample size, confidence level, and standard error affect the margin of error of a confidence interval.

• S.DA.5.4(Objective)

  Construct a confidence interval for the mean of a normally distributed population (with a known standard deviation) and for a population proportion. Use confidence intervals to evaluate claims.

• S.DA.5.5(Objective)

  Use confidence intervals to evaluate claims for a single population parameter.

Interpret and communicate the results of a statistical analysis in context.

• S.IR.1.1(Objective)

  Recognize when the difference between two sample proportions or two sample means is due to random variation or if the difference is statistically significant.

• S.IR.1.2(Objective)

  Understand the concept of a confidence interval, including the interpretation of confidence level, margin of error, and statistical significance.

• S.IR.1.3(Objective)

  Develop inferences or predictions to construct resulting decisions or recommendations.

• S.IR.1.4(Objective)

  Create and evaluate recommendations for areas of future research.

Evaluate practical implications of statistical significance or lack thereof.

• S.IR.2.1(Objective)

  Develop and critique arguments for practical implications based on statistical significance.

• S.IR.2.2(Objective)

  Identify potential lurking variables which may explain an association between two variables.

Evaluate real-world claims and conclusions.

• S.IR.3.1(Objective)

  Evaluate strengths and weaknesses in the studies or methods used to generate data.

• S.IR.3.2(Objective)

  Evaluate the statistical validity of claims made.

Connect basic probability concepts to statistical analysis.

• S.P.1.1(Objective)

  Describe events as subsets of a sample space.

• S.P.1.2(Objective)

  Describe the relationship between theoretical and empirical probabilities using the Law of Large Numbers.

• S.P.1.3(Objective)

  Use counting techniques (e.g., permutations and combinations) to solve mathematical and real-world problems, including determining probabilities of compound events.

Determine probabilities, including joint probabilities, conditional probabilities, probabilities of independent events, and probabilities of dependent events. Interpret the results.

• S.P.2.1(Objective)

  Understand that two events, A and B, are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if two events are independent.

• S.P.2.2(Objective)

  Understand and calculate the conditional probability of A given B as P(A and B)/P(B).

• S.P.2.3(Objective)

  Interpret independence of A and B as saying that the conditional probability of A, given B, is the same as the probability of A.

Use probability to make decisions.

• S.P.3.1(Objective)

  Analyze decisions and strategies using probability concepts and expected values.

• S.P.3.2(Objective)

  Analyze decisions about statistical significance based on reported p-values.