Secondary Mathematics Benchmarks Progressions, Grades 7–12: Probability and Statistics (PS)

Probability and statistics allow us to make sense of the enormous mass of data that come from measurements of the natural and constructed worlds. Statistical data from opinion polls and market research are integral to informing business decisions and governmental policies. Many jobs require workers who are able to analyze, interpret, and describe data and who can create visual representations of data—charts, graphs, diagrams—to help get a point across succinctly and accurately. Moreover, a free society depends on its citizens to understand information, evaluate claims presented as facts, detect misrepresentations and distortions, and make sound judgments based on available data. Gathering, organizing, representing, summarizing, transforming, analyzing, and interpreting data are essential mathematical skills incorporated in the study of statistics that help fulfill these needs. Statistical reasoning rests on a foundation of probability. The study of probability quantifies the likelihood of an event and includes the analysis of odds and risk and the rigorous prediction of future events.

PS.A.1 Simple probability

a. Represent probabilities using ratios and percents.

Task related to this benchmark: How Odd . . .

b. Compare probabilities of two or more events and recognize when certain events are equally likely.

Task related to this benchmark: Rock, Paper, Scissors

c. Use sample spaces to determine the (theoretical) probabilities of events.

A sample space consists of all of the disjoint (mutually exclusive) outcomes possible in a given situation involving chance.

• Calculate theoretical probabilities in simple models (e.g., dice, coins, spinners).

Task related to this benchmark: Rolling for the Big One

d. Know and use the relationship between probability and odds.

The odds of an event occurring is the ratio of the number of favorable outcomes to the number of unfavorable outcomes, whereas the probability is the ratio of favorable outcomes to the total number of possible outcomes.

Task related to this benchmark: How Odd . . .

PS.A.2 Relative frequency and probability

a. Describe the relationship between probability and relative frequency.

If an action is repeated n times and a certain event occurs b times, the ratio is called the relative frequency of the event occurring.

• Recognize and use relative frequency as an estimate for probability.
• Use theoretical probability to determine the most likely result if an experiment is repeated a large number of times.

b. Identify, create, and describe the key characteristics of frequency distributions of both discrete and continuous data.

A frequency distribution shows the number of observations falling into each of several ranges of values; if the percentage of observations is shown, the distribution is called a relative frequency distribution. Both frequency and relative frequency distributions are portrayed through tables, histograms, or broken-line graphs.

• Describe key characteristics (e.g., shape, symmetry/skewness, typical value, spread) of a frequency distribution
• Use a probability distribution to assess the likelihood of the occurrence of an event

c. Analyze and interpret actual data to estimate probabilities and predict outcomes.

Example: In a sample of 100 randomly selected students, 37 of them could identify the difference in two brands of soft drinks. Based on these data, what is the best estimate of how many of the 2,352 students in the school could distinguish between the soft drinks?

d. Compare theoretical probabilities with the results of simple experiments (e.g., tossing dice, flipping coins, spinning spinners).

• Explain how the Law of Large Numbers explains the relationship between experimental and theoretical probabilities.

  The law of large numbers indicates that if an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions approaches p as the number of repetitions becomes arbitrarily large.

• Use simulations to estimate probabilities.

Task related to this benchmark: Rock, Paper, Scissors

e. Compute and graph cumulative frequencies.

PS.A.3 Question formulation and data collection

a. Formulate questions about a phenomenon of interest that can be answered with data.

• Recognize the need for data.
• Understand that data are numbers in context (with units) and identify units.
• Define measurements that are relevant to the questions posed.

Task related to this benchmark: Click It

b. Design a plan to collect appropriate data.

• Understand the differing roles of a census, a sample survey, an experiment, and an observational study.
• Select a design appropriate to the questions posed.
• Begin the use of random sampling in sample surveys and the role of random assignment in experiments, introducing random sampling as a "fair" way to select an unbiased sample.

Task related to this benchmark: Click It

c. Collect and record data.

• Organize written or computerized data records, making use of computerized spreadsheets.
• Display data on tables, charts, or graphs.
• Evaluate the accuracy of the data.

Task related to this benchmark: Click It

PS.A.4 Linear trends

a. Determine whether a scatter plot suggests a linear trend.

Tasks related to this benchmark: Equal Salaries for Equal Work?, Television and Test Grades

b. Visually determine a line of good fit to estimate the relationship in bivariate data that suggests a linear trend.

• Identify criteria that might be used to assess how good the fit is.

Task related to this benchmark: Equal Salaries for Equal Work?

PS.B.1 Compound probability

a. Calculate probabilities of compound events.

• Employ Venn diagrams to summarize information concerning compound events.
• Distinguish between dependent and independent events.

Tasks related to this benchmark: Gamers, Rolling for the Big One

b. Use probability to interpret odds and risks and recognize common misconceptions.

Examples: After a fair coin has come up heads four times in a row, explain why the probability of tails is still 50% in the next toss; analyze the risks associated with a particular accident, illness, or course of treatment; assess the odds of winning the lottery or being selected in a random drawing.

Task related to this benchmark: How Odd . . .

c. Show how a two-way frequency table can be used effectively to calculate and study relationships among probabilities for two events.

Task related to this benchmark: Gamers

d. Recognize probability problems that can be represented by geometric diagrams, the number line, or in the coordinate plane; represent such situations geometrically and apply geometric properties of length or area to calculate the probabilities.

PS.B.2 Analysis and interpretion of categorical and quantitative data

a. Represent both univariate and bivariate categorical data accurately and effectively.

• For univariate data, make use of frequency and relative frequency tables and bar graphs.
• For bivariate data, make use of two-way frequency and relative frequency tables and two-dimensional bar graphs.

Task related to this benchmark: Equal Salaries for Equal Work?

b. Represent both univariate and bivariate quantitative (measurement) data accurately and effectively.

• For univariate data, make use of line plots (dot plots), stem-and-leaf plots, and histograms.
• For bivariate data, make use of scatter plots.
• Describe the shape, center, and spread of data distributions. (For bivariate data, a scatter plot may have a linear shape with a trend line marking its center and distances between the data points and the line showing spread.)

c. Summarize and compare data sets by using a variety of statistics.

• For univariate categorical data, use percentages and proportions (relative frequencies).
• For bivariate categorical data, use conditional (row or column) percentages or proportions.
• For univariate quantitative data, use measures of center (mean and median) and measures of spread (percentiles, quartiles, and interquartile range).
• For bivariate quantitative data, use trend lines (linear approximations or best-fit line).
• Graphically represent measures of center and spread (variability) for quantitative data.
• Interpret the slope of a linear trend line in terms of the data being studied.
• Use box plots to compare key features of data distributions.

Task related to this benchmark: Equal Salaries for Equal Work?

d. Read, interpret, interpolate, and judiciously extrapolate from graphs and tables.

Extrapolation depends on the questionable assumption that the trend indicated continues beyond the known data.

Task related to this benchmark: Television and Test Grades

e. Judge accuracy, reasonableness, and potential for misrepresentation.

• Identify and explain misleading uses of data by considering the completeness and source of the data, the design of the study, and the way the data are analyzed and displayed.

  Examples: Determine whether the height or area of a bar graph is being used to represent the data; evaluate whether the scales of a graph are consistent and appropriate or whether they are being adjusted to alter the visual information conveyed.

f. Interpret data and communicate conclusions.

• State conclusions in terms of the question(s) being investigated.
• Use appropriate statistical language when reporting on plausible answers that go beyond the data actually observed.
• Use oral, written, graphic, pictorial, or multi-media methods to create and present manuals and reports.

PS.C.1 Probability distributions

a. Identify and distinguish between discrete and continuous probability distributions.

• Reason from empirical distributions of data to make assumptions about their underlying theoretical distributions.

b. Know and use the chief characteristics of the normal distribution.

The normal (or Gaussian) distribution is actually a family of mathematical functions that are symmetric in shape with scores more concentrated in the middle than in the tails. They are sometimes described as bell shaped. Normal distributions may have differing centers (means) and scale (standard deviation). The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. In normal distributions, approximately 68% of the data lie within one standard deviation of the mean and 95% within two.

• Demonstrate that the mean and standard deviation of a normal distribution can vary independently of each other (e.g., that two normal distributions with the same mean can have different standard deviations).
• Identify common examples that fit the normal distribution (height, weight) and examples that do not (salaries, housing prices, size of cities) and explain the distinguishing characteristics of each.

Task related to this benchmark: Is Your Score Normal?

c. Calculate and use the mean and standard deviation to describe the characteristics of a distribution.

Task related to this benchmark: Is Your Score Normal?

d. Understand how to calculate and interpret the expected value of a random variable having a discrete probability distribution.

PS.C.2 Correlation and regression

a. Determine a line of good fit for a scatter plot.

• Identify and evaluate methods of determining the goodness of fit of a linear model (e.g., pass through the most points, minimize the sum of the absolute deviations, minimize the sum of the square of the deviations).
• Use a computer or a graphing calculator to determine a linear regression equation (least-squares line) as a model for data that suggest a linear trend, and understand the criteria it satisfies for goodness of fit.

  The linear regression equation for a set of data minimizes the sum of the squared vertical deviations of the points from the line and passes through the point (, ) where is the mean of the x-coordinates of the data points and is the mean of the y-coordinates of the data points. The linear regression line is often called the line of best fit.

• Identify the effect of outliers on the position and slope of the regression line.
• Interpret the slope of the regression line in the context of the relationship being modeled.
• Construct and interpret residual plots to assess the goodness of fit of a regression line.

Tasks related to this benchmark: Equal Salaries for Equal Work?, Television and Test Grades

b. Determine and interpret correlation coefficients.

Be alert to the risk of confusing correlation with causation.

• Recognize correlation as a number between –1 and +1 that measures the strength of linear association between two variables.
• Use the relationship among the standard deviations, correlation coefficients, and slope of the regression line to assess the strength of association suggested by the underlying scatter plot.

PS.D.1 Sample surveys, experiments, and observational studies

a. Describe the nature and purpose of sample surveys, experiments, and observational studies, relating each to the types of research questions they are best suited to address.

• Identify specific research questions that can be addressed by different techniques for collecting data.
• Critique various methods of data collection used in real-world problems, such as a clinical trial in medicine, an opinion poll, or a report on the effect of smoking on health.

Task related to this benchmark: Click It

b. Recognize and explain the rationale for using randomness in research designs.

• Distinguish between random sampling from a population in sample surveys and random assignment of treatments to experimental units in an experiment.

  Random sampling is how items are selected from a population so that the sample data can be used to estimate characteristics of the population; random assignment is how treatments are assigned to experimental units so that comparisons among the treatment groups can allow cause-and-effect conclusions to be made.

Task related to this benchmark: Click It

c. Use simulations to analyze and interpret key concepts of statistical inference.

• Analyze and interpret the notion of margin of error and how it relates to the design of a study and to sample size.
• Analyze and interpret the basic notion of confidence interval and how it relates to margin of error.
• Analyze and interpret the notion of p-value and how it relates to the interpretation of results from a randomized experiment.

d. Plan and conduct sample surveys to estimate population characteristics and experiments to compare treatments.

Task related to this benchmark: Click It

e. Explain why observational studies generally do not lead to good estimates of population characteristics or cause-and-effect conclusions regarding treatments.

PS.D.2 Risks and decisions

a. Apply probability to practical situations to make informed decisions.

• Communicate an understanding of the inverse relation of risk and return.
• Explain the benefits of diversifying risk.

PS.E.1 Transformations of data

a. Explore transformations of data for the purpose of "linearizing" a scatter plot that exhibits curvature.

• Use squaring, square root, reciprocal, and logarithmic functions to transform data.
• Interpret the results of specific transformations in terms of what they indicate about the trend of the original data.

b. Estimate the rate of exponential growth or decay by fitting a regression model to appropriate data transformed by logarithms.

c. Estimate the exponent in a power model by fitting a regression model to appropriate data transformed by logarithms.

d. Analyze how linear transformations of data affect measures of center and spread, the slope of a regression line, and the correlation coefficient.

PS.E.2 Advanced probability

a. Interpret and use the Central Limit Theorem: The distribution of the average of independent samples approaches a normal distribution.

• Use computer simulations to demonstrate the Central Limit Theorem.

b. Know and use equations for the binomial and normal distributions.

c. Use and interpret the normal approximation to the binomial distribution.

d. Calculate and apply expected value.

PS.E.3 Cross-classified data

a. Recognize problems that call for the use of conditional probability and calculate conditional probability in such cases.

• Analyze conditional probabilities using two-way and three-way tables.
• Use and interpret Boolean (and, or, not) operators in the context of two-way and three-way tables.

  Example: Student participation on sports teams in relation to Title IX gender equity goals.

b. Use contingency tables to analyze categorical data.

• Understand and illustrate Simpson's Paradox and other data-based paradoxes.

c. Use Χ² tests to evaluate significance of conditional probabilities.

PS.E.4 Statistical reasoning

a. Explain the protocol for hypothesis testing and apply it in problem situations.

b. Explain statistical estimation and error.

c. Design, conduct, and interpret a simple comparative experiment.

• Formulate questions and identify quantitative measures that may be used to provide answers.
• Draw appropriate conclusions from the collected data.

PS.E.5 Statistical inference

a. Estimate population parameters (point estimators and confidence intervals).

b. Know common tests of significance and use them to test hypotheses.

c. Know and explain the difference between mathematical and statistical inference.

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