Unit 4 — Variability, Distributions, and Relationships Between Quantities
Description
Unit 4 introduces formal statistical thinking and develops understanding of quantitative relationships. Students distinguish statistical questions from non-statistical questions by recognizing that statistical questions anticipate variability in data. They analyze data distributions by identifying center, spread, and shape. Students calculate and interpret measures of center (mean and median) and measures of variability (interquartile range and mean absolute deviation) in context. They display numerical data using dot plots, histograms, and box plots on number lines, and they choose appropriate measures based on the shape of the distribution and context. The unit also focuses on relationships between quantities by having students write equations in two variables to represent dependent and independent variables. Students analyze these relationships using graphs, tables, and equations, understanding how changes in one quantity affect another. The unit culminates with students applying ratio and rate reasoning to solve problems involving unit rates and percents.
Essential Questions
- What is the best way to organize a set of data?
- How can I describe the center and spread of a data set?
- How can I decide which measure of center or variability best describes the data?
- How can I use data to compare different groups?
- What conclusions can be drawn from data?
- How can two quantities change in relationship to one another?
Learning Objectives
- Distinguish questions that are statistical from those that are not by recognizing variability in data.
- Understand that data distributions can be described by center, spread, and overall shape.
- Recognize that measures of center and variability each summarize data in different ways.
- Display numerical data in dot plots, histograms, and box plots on number lines.
- Calculate measures of center: mean and median for a numerical data set.
- Calculate measures of spread: interquartile range and mean absolute deviation for a data set.
- Describe the overall shape of a distribution and identify striking deviations (outliers).
- Choose and justify measures of center and variability appropriate to the shape of distribution and context.
- Write equations in two variables to represent relationships between dependent and independent variables in real-world situations.
- Analyze relationships between quantities using graphs, tables, and equations.
- Solve real-world problems by graphing points in all four quadrants of the coordinate plane.
- Use ratio and rate reasoning to solve problems involving unit rates, unit pricing, percentages, and measurement conversions.
Supplemental Resources
- Printed data sets and frequency tables for students to analyze and display
- Graphic organizers for calculating mean, median, and comparing data sets
- Coordinate plane grids for graphing relationships between variables
- Sticky notes for identifying outliers and describing distributions
- Chart paper for displaying different data displays (dot plots, histograms, box plots) for comparison
Expressions and Equations
The Number System
Ratios and Proportional Relationships
Statistics and Probability
Standards for Mathematical Practice
Students use close-reading skills to understand and solve complex word problems and write mathematical reflections after each unit. Students utilize reading comprehension skills by acting out or drawing the order of important events in story problems. Students read and write stories to represent mathematical concepts.
Students apply mathematical reasoning to understand scientific phenomena including temperatures, data analysis, and quantitative relationships in science investigations.
Students understand how to read dates properly, interpret geographic and economic data, and use quantitative evidence to support historical and civic arguments.
Formative Assessments
- Homework practice calculating mean, median, and measures of variability
- Exit tickets assessing understanding of statistical vs. non-statistical questions
- Journal writing explaining choice of data display and interpretation of distributions
- Task cards for calculating measures of center and creating appropriate graphs
- Self-assessments on understanding independent and dependent variables
Summative Assessment
Chapter tests on distributions, measures of center and variability, and relationships between quantities; performance tasks requiring students to collect, organize, and analyze data sets, choosing appropriate displays and measures; extended projects designing surveys or analyzing real-world data from multiple sources
Benchmark Assessment
Renaissance/STAR assessments for data analysis; MAP Testing; built-in assessments in adopted programs measuring proficiency with statistical concepts
Alternative Assessment
Students may demonstrate understanding through verbal explanation of statistical concepts, recorded responses, or teacher-led interviews in place of written assessments. Simplified data sets, pre-made charts, number lines with labeled intervals, and sentence frames for describing center, spread, and relationships between quantities may be provided as needed.
IEP (Individualized Education Program)
Students with IEPs may benefit from graphic organizers that visually distinguish statistical from non-statistical questions and anchor key vocabulary such as mean, median, interquartile range, and mean absolute deviation. Providing a reference card with step-by-step calculation procedures and labeled diagram examples of dot plots, histograms, and box plots supports processing during both instruction and independent practice. Teachers should allow alternate output modes—such as oral explanation, guided note completion, or scribing—when students are asked to interpret distributions or justify their choice of measures in context. Assignments and assessments may be chunked into smaller sections, and calculator access should be provided so that computational demands do not interfere with demonstrating conceptual understanding of statistical reasoning.
Section 504
Students with 504 plans should be provided extended time on chapter tests and performance tasks that require multi-step data analysis and graphing across all four quadrants. Preferential seating and a low-distraction environment are especially important during tasks that require sustained attention to data sets and equation writing. Printed copies of any graphs, tables, or coordinate planes displayed on the board should be made available so students can annotate directly on the material rather than copying from a distance.
ELL / MLL
Multilingual learners benefit from a visual word wall or illustrated vocabulary reference covering statistical terms—such as variability, distribution, outlier, dependent, and independent—that remain accessible throughout the unit. Directions for data collection tasks, graph construction, and equation writing should be given in short, clear steps, and teachers should ask students to restate instructions in their own words before beginning. Where possible, real-world data contexts used in this unit should reflect familiar, culturally relevant scenarios, and students should be encouraged to discuss patterns and relationships in their home language before expressing ideas in English.
At Risk (RTI)
Students who need additional support should be connected to concrete, familiar contexts—such as sports statistics or school-related data—when first exploring measures of center and variability, helping them build meaning before moving to abstract calculation. Reducing the number of data points in an initial data set allows students to develop procedural fluency with mean, median, and interquartile range without being overwhelmed, and frequent brief check-ins at the start of each practice session help identify and address misconceptions early. Scaffolded templates for constructing dot plots, histograms, and box plots, as well as partially completed tables for exploring relationships between variables, give students a structured entry point into the unit's more complex tasks.
Gifted & Talented
Students who demonstrate early mastery of center, variability, and graphical displays should be invited to investigate how the same data set tells different stories depending on the measure and display chosen, connecting statistical decision-making to real fields such as public health, economics, or sports analytics. Extending the relationships-between-quantities work to include analysis of non-proportional linear relationships, exploration of how changes in rate affect the shape of a graph, or introduction to two-variable data displays such as scatter plots provides meaningful depth beyond grade-level expectations. These students may also design and carry out an original statistical investigation—formulating a statistical question, collecting data, selecting and defending appropriate measures and displays, and communicating findings—serving as an authentic culminating extension of the unit's core ideas.