Colorado Academic Standards Online

Colorado Essential Skills and Mathematical Practices:

1. Understand statistical descriptors of data and interpret and be critical of the use of statistics outside of school. (Professional Skills: Information Literacy)
2. Create, analyze, and synthesize visual representations of statistical data. (Entrepreneurial Skills: Literacy/Reading)
3. Reason about the context of the data separate from the numbers involved and about the numbers separate from the context; move fluidly between contextualized reasoning and decontextualized reasoning. (MP2)
4. Use statistics and statistical reasoning to make sense of, interpret, and generalize about real-world situations. (MP4)
5. Use technology to reason about, model, and compute statistical problems and to interpret results within the context. Additionally, students sketch a distribution and answer questions about it just from knowledge of these three facts: shape, center, and spread. (MP5)

Inquiry Questions:

1. How would you describe the difference between the distributions of two data sets with the same measure of center but different measures of spread?
2. Why do we have multiple measures of center? Why wouldn’t we always just use the mean?
3. What questions might a statistician ask about extreme data points? How do they/should they affect the interpretation of the data?

Coherence Connections:

1. This expectation is in addition to the major work of high school.
2. In Grade 6, students study data displays, measures of center, and measures of variability. Standard deviation, introduced in high school, involves much the same principle as the mean absolute deviation (MAD) that students use beginning in Grade 6. Students should see that the standard deviation is the appropriate measure of spread for data distributions that are approximately normal in shape, as the standard deviation then has a clear interpretation related to relative frequency.
3. At this level, students are not expected to fit normal curves to data. Instead, the aim is to look for broad approximations, with application of the rather rough “empirical rule” (also called the 68%–95% Rule) for distributions that are somewhat bell-shaped. The better the bell, the better the approximation.

Colorado Essential Skills and Mathematical Practices:

1. Create, interpret and demonstrate statistical understanding using technology. (Professional Skills: Use Information and Communications Technologies)
2. Analyze, synthesize, and interpret information from scatter plots and residual plots, and construct explanations for these interpretations. (Entrepreneurial Skills: Literacy/Reading and Writing)
3. Use calculators or computer software to compute with large data sets then interpret and make statistical use of the results. (MP5)
4. Look for patterns in tables and on scatter plots. (MP7)

Inquiry Questions:

1. Does a high correlation (close to \(\pm1\)) in the data of two quantitative variables mean that one causes a response in the other? Why or why not?
2. In what way(s) does a plot of the residuals help us consider the best model for a data set?

Coherence Connections:

1. This expectation supports the major work of high school.
2. In Grade 8, students explore scatter plots with linear associations and create equations for informal “lines of best fit” in support of their in-depth study of linear equations. In high school, this statistical topic is formalized and includes fitting quadratic or exponential functions (where appropriate) in addition to linear. Additionally, students use graphing calculators or software to analyze the residuals and interpret the meaning of this analysis in terms of the correctness of fit.
3. It is important that students understand the foundational concept that “correlation does not equal causation” within their study of curve/line-fitting and the associated numerical calculations. This presents a launching point for discussions about the design and analysis of randomized experiments, also included in high school statistics.
4. The mathematics of summarizing, representing, and interpreting data on two categorical or quantitative variables lays the foundation for more advanced statistical topics, such as inference.

Colorado Essential Skills and Mathematical Practices:

1. Critically interpret the use of statistics in their lives outside of school. (Professional Skills: Information Literacy)
2. Use technology and interpret results as they relate to life outside of school. (Professional Skills: Use Information and Communications Technologies)
3. Reason quantitatively about the contextual meaning of slope and intercept of linear models of real-world data, and when the numerical value has no meaning within the context. (MP2)
4. Use technology to compute, model, and reason about linear representations of bivariate data, and interpret the meaning of the calculated values. (MP5)

Inquiry Questions:

1. How is it possible for the intercept of a linear model to not have meaning in the context of the data?
2. What does the correlation coefficient of a linear model tell us? What actions, recommendations, or interpretations might we have about the correlation coefficient?

Coherence Connections:

1. This expectation is in addition to the major work of high school.
2. The comprehensive study of linear functions in Grade 8 allows the high school focus to shift from computation to interpretation of the components of a linear function. Whereas in Grade 8 the slope/rate of change is described mathematically, the work here focuses on the contextual meaning of the rate of change and its applicability to the linear function as a model to predict unknown values of the real-world scenario.
3. The statistics concepts in high school lend themselves to application in other content areas, such as science (e.g., the relationship between cricket chirps and temperature), sports (e.g., the relationship between the year and the average number of home runs in major league baseball), and social studies (e.g., the relationship between returns from buying Treasury bills and from buying common stocks).

Colorado Essential Skills and Mathematical Practices:

1. Students understand how statistics serves to make inferences about a population. (Professional Skills: Information Literacy)
2. Use a variety of statistical tools to construct and defend logical arguments based on data. (MP3)
3. Understand and describe the differences between statistics (derived from samples) and parameters (characteristic of the population). (MP6)

Inquiry Questions:

1. What is the difference between a statistic and a parameter? Why do we need both?
2. Why is it important that random sampling be used to make inferences about population parameters?

Coherence Connections:

1. This expectation represents major work of high school.
2. In Grade 7, students approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
3. The concepts of this expectation are foundational for advanced study of statistical inference.

Colorado Essential Skills and Mathematical Practices:

1. Apply statistical methods to interpret information and draw conclusions in real-world contexts. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Evaluate reports based on data and explain the practical and statistical significance of the results. (Entrepreneurial Skills: Literacy/Reading and Writing)
3. Use sampling, design, and results of sample surveys, experiments, and observational studies and justify reasonable responses and misleading or inaccurate results. (MP3)
4. Use sampling, randomization, and simulations to model, describe, and interpret real-world situations, and use margin of error, p-values and effect size to describe the meaning of the results. (MP4)
5. Observe regular patterns in distributions of sample statistics and use them to make generalizations about the population parameter. (MP8)

Inquiry Questions:

1. How can the results of a statistical investigation be used to support or critique a hypothesis?
2. What happens to sample-to-sample variability when you increase the sample size?
3. How does randomization minimize bias?
4. Can the practical significance of a given study matter more than statistical significance? Why is it important to know the difference?
5. Why is the margin of error in a study important?

Coherence Connections:

1. This expectation represents major work of high school.
2. In Grades 6–8, students engage with statistics to: (a) draw informal comparative inferences about two populations; (b) informally assess degree of visual overlap of two numerical data distributions; (c) use measures of center and measure of variability for numerical data from random samples to draw comparative inferences; and (d) generate or simulate multiple samples to gauge variation in estimates and predictions. These concepts are extended and formalized in high school.
3. Students’ understanding of random sampling is the key that allows the computation of margins of error in estimating a population parameter and can be extended to the random assignment of treatments in an experiment.

Colorado Essential Skills and Mathematical Practices:

1. Apply probability concepts and interpret their real-world meaning. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Use probability values to describe how multiple random events are related, including the ideas of independence and conditional probability as they have meaning in the real world. (MP3)
3. Use probability to support the independence of two random events, or to make sense to conditional probabilities. (MP4)
4. Use clear definitions and accurate notation to express probability concepts. (MP6)

Inquiry Questions:

1. How can you describe the formula for determining independence in everyday language? Why does this make sense?
2. How can you describe the formula for conditional probability in everyday language? Why does this make sense?
3. How can a careful and clear display of categorical data in a table help in interpreting relationships between the values expressed?

Coherence Connections:

1. This expectation is in addition to the major work of high school.
2. In Grade 7, students encounter the development of basic probability, including chance processes, probability models, and sample spaces.
3. In high school, the relative frequency approach to probability is extended to conditional probability and independence, rules of probability and their use in finding probabilities of compound events, and the use of probability distributions to solve problems involving expected value. As seen in the expectations for Making Inferences & Justifying Conclusions, there is a strong connection between statistics and probability.

Colorado Essential Skills and Mathematical Practices:

1. Understand and apply probability to the real world. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Consider multiple approaches and representations for representing and understanding probabilities of random events. (MP1)
3. Consider probability concepts in context and mathematically and make connections between both types of reasoning. (MP2)
4. Use probability models to represent and make sense of real-world phenomena. (MP4)

Inquiry Questions:

1. What is an everyday situation that helps explain the Addition Rule? How does the context help you understand the subtraction of \(P \left( A \text{ and } B \right)\) from \(P \left(A \right) + P \left( B \right)\)?

Coherence Connections:

1. This expectation is in addition to the major work of high school and includes advanced (+) outcomes.
2. Studying and understanding probability, which is always in a context, provides high school students with a mathematical structure for dealing with the many changes they will experience as part of life.

Colorado Essential Skills and Mathematical Practices:

1. Understand and apply probability to the real world. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Consider probability concepts in context and mathematically, and make connections between both types of reasoning. (MP2)
3. Apply probability models to real-world situations, calculate appropriately, and interpret the results. (MP4)

Inquiry Questions:

1. What is a random variable?
2. Create a context which can be used to describe a random variable.

Coherence Connections:

1. This expectation represents advanced (+) work of high school.

Colorado Essential Skills and Mathematical Practices:

1. Students understand and apply probability to the real world. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Apply probability models to real-world situations, calculate appropriately, and interpret the results. (MP4)

Inquiry Questions:

1. How does probability help in the decision-making process?
2. Why does expected value require the weighted average of all possible values?

Coherence Connections:

1. The expectation represents advanced (+) work of high school.