Instructional programs from prekindergarten through grade 12 should enable each and every student to—

- Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
- Select and use appropriate statistical methods to analyze data
- Develop and evaluate inferences and predictions that are based on data
- Understand and apply basic concepts of probability

**Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them **

**Pre-K–2 Expectations: In pre-K through grade 2 each and every student should–**

- pose questions and gather data about themselves and their surroundings;
- sort and classify objects according to their attributes and organize data about the objects;
- represent data using concrete objects, pictures, and graphs.

**Grades 3–5 Expectations: In grades 3–5 ****each and every ** student should–

- design investigations to address a question and consider how data-collection methods affect the nature of the data set;
- collect data using observations, surveys, and experiments;
- represent data using tables and graphs such as line plots, bar graphs, and line graphs;
- recognize the differences in representing categorical and numerical data.

**Grades 6–8 Expectations: In grades 6–8 ****each and every **student should–

- formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population;
- select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

**Grades 9–12 Expectations: In grades 9–12 ****each and every ** student should–

- understand the differences among various kinds of studies and which types of inferences can legitimately be drawn from each;
- know the characteristics of well-designed studies, including the role of randomization in surveys and experiments;
- understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable;
- understand histograms, parallel box plots, and scatterplots and use them to display data;
- compute basic statistics and understand the distinction between a statistic and a parameter.

**Select and use appropriate statistical methods to analyze data **

**Pre-K–2 Expectations: In pre-K through grade 2 ****each and every ** student should–

- describe parts of the data and the set of data as a whole to determine what the data show.

**Grades 3–5 Expectations: In grades 3–5 ****each and every ** student should–

- describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed;
- use measures of center, focusing on the median, and understand what each does and does not indicate about the data set;
- compare different representations of the same data and evaluate how well each representation shows important aspects of the data.

**Grades 6–8 Expectations: In grades 6–8 ****each and every ** student should–

- find, use, and interpret measures of center and spread, including mean and interquartile range;
- discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

**Grades 9–12 Expectations: In grades 9–12 ****each and every ** student should–

- for univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics;
- for bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools;
- display and discuss bivariate data where at least one variable is categorical;
- recognize how linear transformations of univariate data affect shape, center, and spread;
- identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.

**Develop and evaluate inferences and predictions that are based on data **

**Pre-K–2 Expectations: In pre-K through grade 2 ****each and every ** student should–

- discuss events related to students' experiences as likely or unlikely.

**Grades 3–5 Expectations: In grades 3–5 ****each and every **student should–

- propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions.

**Grades 6–8 Expectations: In grades 6–8 ****each and every **student should–

- use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken;
- make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit;
- use conjectures to formulate new questions and plan new studies to answer them.

**Grades 9–12 Expectations: In grades 9–12 ****each and every ** student should–

- use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions;
- understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference;
- evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions;
- understand how basic statistical techniques are used to monitor process characteristics in the workplace.

**Understand and apply basic concepts of probability **

**Grades 3–5 Expectations: In grades 3–5 ****each and every **student should–

- describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible;
- predict the probability of outcomes of simple experiments and test the predictions;
- understand that the measure of the likelihood of an event can be represented by a number from 0 to 1.

**Grades 6–8 Expectations: In grades 6–8 ****each and every **student should–

- understand and use appropriate terminology to describe complementary and mutually exclusive events;
- use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations;
- compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

**Grades 9–12 Expectations: In grades 9–12 ****each and every ** student should–

- understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases;
- use simulations to construct empirical probability distributions;
- compute and interpret the expected value of random variables in simple cases;
- understand the concepts of conditional probability and independent events;
- understand how to compute the probability of a compound event.