Unit 4

Statistics

 

Algebra I 

Unit Length and Description:

 

25 days

 

Experience with descriptive statistics began as early as Grade 6. Students were expected to display numerical data and summarize it using measures of center and variability. By the end of middle school they were creating scatter-plots and recognizing linear trends in data. This unit builds upon that prior experience, providing students with more formal means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.

 (Mathematics Appendix A, p.22)

 

·        Summarize, represent, and interpret data on a single count or measurement variable; choose a summary statistic appropriate to the characteristics of data distribution.

·        Summarize, represent, and interpret data on two categorical quantitative variables; use linear functions to model the relationship between two numerical variables and assess how well the line of best fit models the data using residuals.

·        Interpret linear models; focus on computation and interpretation of the correlation coefficient as a measure of the of how well the data fit the relationship

·        Distinction between a statistical relationship and a cause-and-effect relationship.

 

Standards:

 

S-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

 

S-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

 

S-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

 

S-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

 

S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

  1. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
  2. Informally assess the fit of a function by plotting and analyzing residuals.
  3. Fit a linear function for a scatter plot that suggests a linear association.
  • Standard Clarifications: i) Assessment tasks have a real-world context. ii) Exponential functions are limited to those with domains in the integers.

 

S-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

 

S-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

 

S-ID.C.9 Distinguish between correlation and causation.

 

Focus Standards of Mathematical Practice:

 

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

S-ID.A.1

·        I can represent data with plots on the real number line using various display types by creating dot plots, histograms and box plots

S-ID.A.2

·        I can choose the appropriate measure for center (mean, median) and spread (interquartile range, standard deviation) based on the shape of a data distribution

·        I can use appropriate statistics for center and spread to compare two or more data sets

S-ID.A.3

·        I can define “the context of data sets” as measuring the specific nature of the attributes under investigation

·        I can interpret differences in shape, center and spread in the context of data sets

·        I can describe the possible effects the presence of outliers in a set of data can have on shape, center, and spread in the context of the data sets

S-ID.B.5

·        I can recognize the differences between joint, marginal and conditional relative frequencies

·        I can calculate relative frequencies including joint, marginal and conditional relative frequencies

·        I can summarize categorical data for two categories in two-way frequency tables

·        I can interpret relative frequencies in the context of the data

·        I can recognize possible associations and trends in the data

S-ID.B.6

·        I can represent data on a scatter plot (2 quantitative variables)

·        I can fit a given function class (e.g. linear, exponential) to data

·        I can, using given scatter plot data represented on the coordinate plane, informally describe how the two quantitative variables are related

·        I can determine which function best models scatter plot data represented on the coordinate plane, and describe how the two quantitative variables are related

·        I can use functions fitted to data to solve problems in the context of the data

·        I can represent the residuals from a function and the data set it models numerically and graphically

·        I can informally assess the fit of a function by analyzing residuals from the residual plot

·        I can fit a linear function for a scatter plot that suggests a linear association

S-ID.C.7

·        I can interpret the slope (rate of change) and model the intercept (constant term) of a linear model in the context of the data

S-ID.C.8

·        I can compute (using technology) the correlation coefficient of a linear fit

·        I can define the correlation coefficient

·        I can interpret the correlation coefficient of a linear fit as a measure of how well the data fit the relationship

S-ID.C.9

·        I can define positive, negative, and no correlation and explain why correlation does not imply causation

·        I can define causation

·        I can distinguish between correlation and causation

Enduring Understandings:

 

·        The way that data is collected, organized, and displayed influences interpretation.

·        Describing center, spread, and shape is essential analysis of both univariate and bivariate data.

·        Mathematical models are used to predict and make inferences about data.

·        Mathematics can be used to solve real world problems and can be used to communicate solutions.

 

Essential Questions:

 

·        How do various representations of data lead to different interpretations of the data?

·        How are center and spread of data sets described and compared?

·        What information is appropriate to interpret from a data-based model?

·        How can mathematical representations be used to communicate information effectively?