Grade Level: Unit:



Time Frame: Approximately 1-2 weeks

Connections to Previous Learning:

Students use their understanding of data distribution or shape to determine more precise comparisons of data sets. Students have worked with a variety of units in the past and will bring that information to this unit of study. They will recall labeling graphs and units.

From the High School Statistics and Probability Progression Document p. 3:

Summarize, represent, and interpret data on a single count or measurement variable Students build on the understanding of key ideas describing distributions- shape, center, and spread described in the Grades 6-8 Statistics and Probability Progression. This enhanced understanding allows them to give more precise answers to deeper questions, often involving comparisons of data sets. Students use shape and the question(s) to be answered to decide on the median or mean as the more appropriate measure of center and to justify their choice through statistical reasoning. They also add a key measure of variation to their toolkits.

In connection with the mean as a measure of center, the standard deviation is introduced as a measure of variation. The standard deviation is based on the squared deviations from the mean, but involves much the same principal as the mean absolute deviation (MAD) that students learned about in Grades 6-8. 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.

Focus of the Unit:

Students use statistics to compare center and spread of two or more different data sets, including the use of scatter plots, histograms, box plots, and standard deviation. Students will interpret outliers and recognize associations and trends in the data.

The margin shows two ways of comparing height data for males and females in the 20-29 age group. Both involve plotting the data or data summaries (box plots or histograms) on the same scale, resulting in what are called parallel (or side-by-side) box plots and parallel histograms. The parallel box plots show an obvious difference in the medians and the IQRs for the two groups; the medians for males and females are, respectively, 71 inches and 65 inches, while the IQRs are 4 inches and 5 inches. Thus, male heights center at a higher value but are slightly more variable.

The parallel histograms show that distributions of heights to be mound shaped and fairly symmetrical (approximately normal) in shape. Therefore, the data can be succinctly described using the mean and standard deviation. Heights for males and females have means of 70.4 inches and 64.7 inches. Students should be able to sketch each distribution and answer questions about it just from knowledge of these three facts (shape, center, and spread). For either group, about 68% of the data values will be within one standard deviation of the mean. They should also observe that the two measures of center, median and mean, tend to be close to each other for symmetric distributions.

Connections to Subsequent Learning:

Students use comparison of data sets to explore the area under the normal curve. Students find the area under the normal curve using technology.

|Desired Outcomes |

|Standard(s): |

|Reason quantitatively and use units to solve problems. |

|N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data |

|displays. |

|N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. |

|N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. |

|Summarize, represent, and interpret data on a single count of measurement variable. |

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

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

|S.ID.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. |

|WIDA Standard: (English Language Learners) |

|English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. |

|English language learners benefit from: |

|Teachers guiding them in attending to the context of situations to assist with defining appropriate units. |

|Discussion about the connections between various representations of data and the situations they define. |

|Hands-on activities to collect and analyze data. |

|Understandings: Students will understand… |

|Units and quantities define the parameters of a given situation and are used to solve problems. |

|Data can be represented and interpreted in a variety of formats. |

|Extreme data points (outliers) can skew interpretations of a set of data. |

|Synthesizing information from multiple sets of data results in evidence-based interpretation. |

|Center and spread of a data set may be compared in multiple ways. |

|Data in a two –way frequency table can be summarized using relative frequencies in the context of the data. |

|Essential Questions: |

|How is attention to units and quantities meaningful in data analysis and problem solving? |

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

|When and how can extreme data points impact interpretation of data? |

|Why are multiple sets of data used? |

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

|How is a data set represented in a two-way frequency table summarized? |

|Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) |

|*1. Make sense of problems and persevere in solving them. Students make sense of the situations by attending to the units and quantities used to represent the context. They persevere in interpreting problem solving|

|contexts and data sets using accessible formats. |

|2. Reason abstractly and quantitatively. Students select and use the appropriate level of accuracy for measurement values to interpret data in the context of the given situation. |

|3. Construct viable arguments and critique the reasoning of others. Students articulate their interpretations of data as well as construct representations of data to communicate regarding trends and associations. |

|Students question each other to clarify or dispute interpretations and representations of data. |

|*4. Model with mathematics. Students use appropriate contexts to represent data sets in the real world. They also select appropriate mathematical representations to convey their messages in the context of |

|situations. |

|5. Use appropriate tools strategically. Students select and use appropriate mathematical and visual representations to convey messages in a given context. |

|*6. Attend to precision. Students recognize that the units used represent the context of the situation. They select and use appropriate labels given the context. |

|7. Look for and make use of structure. |

|8. Look for express regularity in repeated reasoning. |

|Prerequisite Skills/Concept: |Advanced Skills/Concepts: |

|Students should already be able to: |Some students may be ready to: |

|Construct and interpret scatter plots for bivariate measurement data to investigate patterns of |Defend choice of descriptor for a data set related to shape, center, and spread. |

|association between two quantities. Describe patterns such as clustering, outliers, positive or |Compare and contrast options for descriptors of a data set. |

|negative association, linear association, and nonlinear association. | |

|Know that straight lines are widely used to model relationships between two quantitative | |

|variables. For scatter plots that suggest a linear association, informally fit a straight line, | |

|and informally assess the model fit by judging the closeness of the data points to the line. | |

|Understand that patterns of association can also be seen in bivariate categorical data by | |

|displaying frequencies and relative frequencies in a two-way table. Construct and interpret a | |

|two-way table summarizing data on two categorical variables collected from the same subjects. Use| |

|relative frequencies calculated for rows or columns to describe possible association between the | |

|two variables. | |

|Knowledge: Students will know… |Skills: Students will be able to … |

| |Use and convert (as necessary) the appropriate unit when solving a multi-step real-world problem. |

|Joint, marginal, and conditional relative frequencies. |Interpret units used in formulas and real-world problems. |

| |Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. |

| |Choose and interpret the scale and origin in graphs and data displays. |

| |Define the appropriate quantities to describe the characteristics of interest for a population. |

| |Determine and interpret the appropriate quantities when communicating and using visual representations. |

| |Define variables in the context of a situation. |

| |Use and justify units to evaluate the appropriateness of a solution. |

| |Use correct numerical value based on context and tools used in measurement. |

| |Represent data visually in scatter plots, histograms, or box plots. |

| |Compute the measures of central tendencies of a data set (mean, median, and mode). |

| |Compute the range, max/min, quartiles and standard deviation of multiple data sets. |

| |Compare measures of center (mean, median) and spread (range, maximum, minimum, quartiles) from multiple data sets. |

| |Identify and describe possible outliers in a data set. |

| |Use measures of central tendencies, range, max/min, quartiles, and standard deviation to interpret differences between |

| |data sets. |

| |Create two-way frequency tables for categorical data. |

| |Identify joint, marginal, and conditional relative frequencies within two-way tables. |

| |Interpret relative frequencies in the context of the data. |

| |Recognized possible associations and trends in data represented in two-way tables. |

|Academic Vocabulary: |

| | | | |

|Critical Terms: | |Supplemental Terms: | |

|Joint relative frequency | |Association | |

|Marginal relative frequency | |Trend | |

|Conditional relative frequency | |Dot plot | |

|Outlier | |Histogram | |

|Skewed Distribution | |Box Plot | |

|Correlation Coefficient | |Scatter Plot | |

|Two-Way Frequency Table | |Measure of Center | |

|Standard deviation | |Normal Distribution | |

|Interquartile Range | |Categorical Data | |

| | |Accuracy | |

| | |Scale | |

| | |Quantity | |

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