Essential Understandings
6-8 Statistics
Big Idea 1:
Distributions describe variability in data.
EU 1a: Graphs and tables are useful for displaying distributions of categorical data.
EU 1b: Numerical summaries of categorical data are useful for describing particular features of
a distribution.
EU 1c: Graphs and tables are useful for displaying distributions of quantitative data.
EU 1d: Numerical summaries of quantitative data are useful for measuring the center of a
distribution.
EU 1e: Numerical summaries of quantitative data are useful for measuring the amount of
variability within a distribution.
EU 1f: Graphs and tables based on grouped data are useful for displaying distributions of
quantitative data.
EU 1g: The shape of a distribution influences which summary measure is most appropriate for
describing the center of a distribution for quantitative data.
EU 1h: Graphs and tables based on a division of the ordered data into equal-sized groups are
useful for displaying distributions of quantitative data.
EU 1i: Some numerical summaries of quantitative data are more resistant than others to
extreme data values, called outliers.
Big Idea 2:
Statistics can be used to compare two or more groups of data.
EU 2a: The focus of comparisons between two or more groups of data is on similarities and
differences between the distributions.
EU 2b: The amount of separation between two or more distributions is related to the amount
of variability within them.
Big Idea 3:
Bivariate distributions describe patterns or trends in the covariability in
data on two variables.
EU 3a: Graphs and tables are useful for displaying bivariate distributions of data on two
categorical variables.
EU 3b: Conditional relative frequency distributions are useful for establishing an association
between two categorical variables.
EU 3c: Graphs and tables are useful for displaying bivariate distributions of data on two
quantitative variables.
EU 3d: A correlation coefficient is a numerical summary of bivariate data that measures the
strength of the relationship between two variables.
EU 3e: When the trend in bivariate data on two quantitative variables is generally linear, a
centrally located line can be useful for making predictions.
Big Idea 4:
Inferential statistics uses data in a sample selected from a population to
describe features of the population.
EU 4a: The sampling distribution of a statistic describes the sample-to-sample variability in
avalues of the statistic from multiple samples of the same size selected from the same
population.
EU 4b: Selecting a simple random sample from a population is a fair way to select a sample.
EU 4c: The predictable pattern for the sampling distribution of a statistic based on random
sampling provides a way for making inferences about the population.
9-12 Statistics
Big Idea 1: Data consist of structure and variability.
EU 1a: Mathematical models describe structure.
EU 1b: Statistical models extend mathematical models by describing variability around the
structure.
EU 1c: Statistical models are evaluated by how well they describe data and whether they are
useful.
Big Idea 2: Distributions describe variability.
EU2a: A population distribution describes variability in the values that make up a population.
EU2b: The population distribution is often unknown but can be approximated by a sample
distribution.
EU2c: The sampling distribution of a sample statistic describes how the value of the statistic
varies from sample to sample.
EU2d: Simulation can be used to approximate sampling distributions.
Big Idea 3: Hypothesis tests answer the question, “Do I think that this could have
happened by chance?”
EU3a: A hypothesis test involves choosing between two competing hypotheses—the null
hypothesis and the alternative hypothesis.
EU3b: The alternative hypothesis is determined by the statistical question of interest.
EU3c: The null hypothesis is rejected in favor of the alternative hypothesis if the sample data
provide convincing evidence against the null hypothesis.
EU3d: The p-value measures surprise.
EU3e: Hypothesis tests do not always lead to a correct decision.
Big Idea 4: The way in which data are collected matters.
EU4a: Observational studies, including surveys, provide information about the characteristics
of a population or sample, whereas controlled experiments provide information about
treatment effects.
EU4b: Random assignment in an experiment permits drawing causal conclusions about
treatment effects and quantifying the uncertainty associated with these conclusions.
EU4c: Random selection tends to produce samples that are representative of the population,
permitting generalization from the sample to the larger population and also allowing the
uncertainty in estimates to be quantified.
EU4d: Random selection and random assignment are different things, and the type and scope
of conclusions that can be drawn from data depend on the role of random selection and
random assignment in the study design.
Big Idea 5: Evaluating an estimator involves considering bias, precision, and the sampling
method.
EU5a:
EU5b:
EU5c:
EU5d:
EU5e:
Estimators are evaluated on the basis of their performance in repeated sampling.
Some estimators are biased.
The standard error describes the precision of an estimator.
Confidence intervals are estimators that convey information about precision.
The precision of estimators depends both on the way in which the sample was selected
and on the size of the sample.
EU5f: If the sampling method is good, a larger sample is always more useful that a smaller
sample.
EU5g: A small sample selected by using a good method can yield better results than a large
sample selected by using a poor method.
EU5h: The size of the sample relative to the population size is not an important factor in
determining the accuracy of estimators.