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 Instructional Focus DocumentMathematical Models with Applications
 TITLE : Unit 10: Statistical Data Analysis SUGGESTED DURATION : 12 days

#### Unit Overview

This unit bundles student expectations that address interpreting univariate or bivariate data from various graphs to draw conclusions and analyze numerical data using measures of central tendency and variability in order to make inferences. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Grade 2, bar graphs were introduced and were revisited throughout elementary and middle school mathematics. In Grades 3–7, students were introduced to and used dot plots to organize and analyze data. In Grade 5, students were introduced to scatterplots. In Grade 6, students were introduced to box and whisker plots, histograms, and stem-and-leaf plots. In Grade 7, students were introduced to circle graphs. Middle School mathematics addressed mean, median, mode and variance, including range and interquartile ranges. In Grade 8, students were introduced to absolute mean to describe data.

During this unit, students use prior knowledge and experience with various graphs, including line graphs, bar graphs, circle graphs, histograms, scatterplots, dot plots, stem-and-leaf plots, and box and whisker plots. Students use various sources (e.g., internet, print media, etc.) to find various types of graphs and interpret the graphs to draw conclusions and determine the strengths or weaknesses of the data and the graphs. Students analyze numerical data using measures of central tendency (mean, median and mode) and variability (range, interquartile range or IQR, and standard deviation) in order to make inferences with normal distributions. Mathematical Models with Applications introduces examining data using standard deviation. Throughout the unit, students examine various data sets and analyze various measures to describe the center, spread, and shape of the data distributions in order to make inferences and predictions regarding real world situations.

After this unit, in the last six weeks, students will apply statistical data representations, formulating a question, determining information needed, and collecting and analyzing data in order to make conclusions and draw inferences.

This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1, C1; II. Algebraic Reasoning D1; IV. Measurement Reasoning A1; VI. Statistical Reasoning B1, B2, B2, B3, B4, C1, C2, C3; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the Connections Standard for Grades 9-12 from the National Council of Teachers of Mathematics (NCTM), “Instructional programs from pre-kindergarten through grade 12 should enable students to:

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

When students can see the connections across different mathematical content areas, they develop a view of mathematics as an integrated whole. As they build on their previous mathematical understandings while learning new concepts, students become increasingly aware of the connections among various mathematical topics. As students' knowledge of mathematics, their ability to use a wide range of mathematical representations, and their access to sophisticated technology and software increase, the connections they make with other academic disciplines, especially the sciences and social sciences, give them greater mathematical power” (NCTM, 2000, p. 354).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: Connections standard for grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Education Policy Improvement Center (2009), Texas College and Career Standards, Austin, TX, University of Texas Printing.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Statistical data are collected, analyzed graphically and numerically, and interpreted to determine the reliability of the data, make predictions, and draw conclusions.

• What is the purpose of analyzing statistical data?
• Why is it important to understand the analysis and interpretation of statistical data?
• How does the type of data determine the type of graphical analysis?
• How does the type of data determine the type of numerical analysis?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Statistical Reasoning

• Conclusions/Predictions
• Data
• Statistical Representations

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Univariate data involves one variable or one type of data and represents the simplest form of data to be analyzed.

• What is meant by univariate data?
• What are two types of univariate data?
• What are some examples of categorical, univariate data?
• What are some examples of numerical, univariate data?

Bivariate data involves two numerical variables and the primary purpose of bivariate data is to determine the relationship between the variables.

• What is meant by bivariate data?
• How is bivariate data different than univariate data?
• What are some examples of bivariate data?
• How can a scatterplot be used to determine the relationship between bivariate data?
• How can the strength of the relationship between bivariate data be determined?

Data can be organized and represented using different graphical representations to emphasize and interpret various aspects of the data.

• What graphical representations can be used to represent data?
• When representing data graphically, how is the best representation determined?
• What are the characteristics of the following types of graphs: line graphs, bar graphs, circle graphs, histograms, scatter plots, dot plots, stem-and-leaf plots, and box and whisker plots?
• What processes are used to construct the following graphs: line graphs, bar graphs, circle graphs, histograms, scatter plots, dot plots, stem-and-leaf plots, and box and whisker plots?
• What aspects of a data distribution can be emphasized with the following types of graphs: line graphs, bar graphs, circle graphs, histograms, scatter plots, dot plots, stem-and-leaf plots, and box and whisker plots?
• What type of conclusions can be drawn from the different representations and how are the conclusions used to make predictions?
• How can representations of data influence conclusions and/or predictions?
• How can graphical representations reveal strengths and weaknesses of a set of data?

 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Statistical Reasoning

• Conclusions/Predictions
• Data
• Summary Statistics

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

The three measures of central tendency of numerical data (mean, median, and mode) reveal different information about relationships between and among the data within the data set.

• What is the mean, how is it determined, and what does it represent in the data set?
• When is the mean most helpful in drawing conclusions about a set of data?
• What is the median, how is it determined, and what does it represent in the data set?
• When is the median most helpful in drawing conclusions about a set of data?
• What is the mode, how is it determined, and what does it represent in the data set?
• Does every data set have a mode? Could there be more than one mode?
• How can the measures of central tendency be used to summarize data?

The measures of variability (range, interquartile range, and standard deviation) reveal how the data within the set of data is distributed.

• What is the range of the data set, how is it determined, and what does it represent in the data set?
• What conclusions might be drawn from the range?
• What is the interquartile range, how is it determined and what does it represent in the data set?
• What conclusions might be drawn from the Interquartile range?
• What is the standard deviation (poulation standard deviation and sample standard deviation), how is it determined, and what does it represent in the data set?
• What conclusions might be drawn from the standard deviation?
• How can the data with variability be summarized?
• What types of situations yield data with variability?
• How can data without variability be summarized?
• What types of situations yield data without variability?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may struggle with the difference between univariate and bivariate data, especially when observing certain graphs.

Underdeveloped Concepts:

• Some students may think that categorical and numerical data can always be displayed by the same representations rather than realizing that the appropriate representation for a set of data depends on the type of question being asked about the data (e.g., Bar graphs and frequency tables can represent categories of data that do not have numeric ordering, such as favorite colors or type of transportation to get to school, as well as representing numeric data such as shoe size or number of family members.)
• Some students may determine that fractions and decimals cannot be represented on a stem-and-leaf plot rather than using the whole number as the stem and fraction or decimal amount as the leaf.
• Students may choose one of the two center numbers in an even number of values as the median instead of finding the average of the two middle values.
• Although some students may be proficient at displaying data using different representations, they may lack the experience to solve problems by analyzing the data.
• Some students may have difficulty constructing graphs due to the scales, keys, and other characteristics of the graph.
• Some students may not remember how to determine or know the difference between mean, median, mode, and range.

#### Unit Vocabulary

• Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other to show the frequency (number of times) that each category occurs
• Box and whiskers plot – a graphical representation that displays the centers and range of the data distribution on a number line
• Circle graph – a circular graph with partitions (sections) that represent a part of the total. Each section is proportional to the magnitude of the quantity. The total percentage is equal to 100. The angle measure of the section(s) is between 0° and 360°
• Dot plot – a graphical representation to organize small sets of data that uses dots (or Xs) to show the frequency (number of times) that each number occurs. One dot (or X) for each element in the data set is placed above the corresponding category or numerical value on the horizontal axis. When categorical data is used, it is orderly and not arbitrary.
• Histogram – a graphical representation of adjacent bars with different heights or lengths used to represent the frequency of data in certain ranges of continuous and equal intervals
• Interquartile range (IQR) – The difference between the first quartile and the third quartile of a set of numbers. The second and third quartiles make up the middle half of the data in the set.
• Line graph – a graphical representation used to display the relationship between discrete data pairs with a line connecting data points
• Mean of a set of numbers – average of a set of data found by finding the sum of a set of data and dividing the sum by the number of pieces of data in the set
• Median of a set of numbers – the middle number of a set of data that has been arranged in order from greatest to least or least to greatest. The median is a number that is larger than half the numbers and smaller than the other half of the numbers.
• Mode of a set of numbers – most frequent value in a set of data. There may be more than one mode or no mode, if all numbers appear the same number of times.
• Normal distribution – even dispersement of data that form a bell shaped curve
• Quartiles – the groups of data within a set; 1st quartile is the first 25th percentile and 3rd quartile is the 75th percentile of the data
• Range of a set of numbers – the difference in the value of the largest number and the smallest number in a set
• Scatterplot – a graphical representation used to display the relationship between discrete data pairs
• Standard deviation – average amount either above or below the mean that the data deviate from the mean
• Stem-and-leaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating one place value from another place value of a data set. The larger of the two place values is called the stem and the smaller of the two place values is called the leaf

Related Vocabulary:

 Bivariate data Categorical data Deviation Discrete data Distribution Inference Keys of graphs Numerical data Population standard deviation Prediction Sample standard deviation Scales of graphs Trend Univariate data
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Mathematical Models with Applications Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
M.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
M.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.9 Mathematical modeling in social sciences. The student applies mathematical processes and mathematical models to analyze data as it applies to social sciences. The student is expected to:
M.9A Interpret information from various graphs, including line graphs, bar graphs, circle graphs, histograms, scatterplots, dot plots, stem-and-leaf plots, and box and whisker plots, to draw conclusions from the data and determine the strengths and weaknesses of conclusions.

Interpret

INFORMATION FROM VARIOUS GRAPHS, INCLUDING LINE GRAPHS, BAR GRAPHS, CIRCLE GRAPHS, HISTOGRAMS, SCATTERPLOTS, DOT PLOTS, STEM-AND-LEAF PLOTS, AND BOX AND WHISKER PLOTS (SOCIAL SCIENCES)

Including, but not limited to:

• Interpreting  information from various graphs
• Univariate or bivariate data
• Categorical or numerical data
• Scales
• Keys
• Types of graphs
• Line graph – a graphical representation used to display the relationship between discrete data pairs with a line connecting data points
• Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other to show the frequency (number of times) that each category occurs
• Circle graph – a circular graph with partitions (sections) that represent a part of the total. Each section is proportional to the magnitude of the quantity. The total percentage is equal to 100. The angle measure of the section(s) is between 0° and 360°
• Histogram – a graphical representation of adjacent bars with different heights or lengths used to represent the frequency of data in certain ranges of continuous and equal intervals
• Scatterplot – a graphical representation used to display the relationship between discrete data pairs
• Dot plot – a graphical representation to organize small sets of data that uses dots (or Xs) to show the frequency (number of times) that each number occurs. One dot (or X) for each element in the data set is placed above the corresponding category or numerical value on the horizontal axis. When categorical data is used, it is orderly and not arbitrary.
• Stem-and-leaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating one place value from another place value of a data set. The larger of the two place values is called the stem and the smaller of the two place values is called the leaf.
• Box and whiskers plot – a graphical representation that displays the centers and range of the data distribution on a number line

To Draw

CONCLUSIONS FROM THE DATA

Including, but not limited to:

• Categorical or numerical data
• Univariate or bivariate data
• Scale and keys of graphs
• Conclusions about the relationships between and among the different data points.
• Determination of patterns and trends
• Reasonableness of predictions beyond the data shown on the graph.

Determine

THE STRENGTHS AND WEAKNESSES OF CONCLUSIONS

Including, but not limited to:

• Clarity of display of data
• Reasonable conclusions for data collected
• Knowledge of the subject of the data

Note(s):

• Grade 2 introduced bar graphs.
• Grades 3 – 7 learned and used dot plots to organize and analyze data.
• Grade 6 introduced box plots, histograms, and stem-and-leaf plots.
• Grade 7 introduced circle graphs.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• C1 – Use estimation to check for errors and reasonableness of solutions
• II. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships
• IV. Measurement Reasoning
• A1 – Select or use the appropriate type of unit for the attribute being measure.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.9B Analyze numerical data using measures of central tendency (mean, median, and mode) and variability (range, interquartile range or IQR, and standard deviation) in order to make inferences with normal distributions.

Analyze

NUMERICAL DATA USING MEASURES OF CENTRAL TENDENCY (MEAN, MEDIAN, AND MODE) AND VARIABILITY (RANGE, INTERQUARTILE RANGE OR IQR, AND STANDARD DEVIATION)

Including, but not limited to:

• Population – total collection of all elements of a set of data
• Sample of a population – subset of a population
• Mean of a set of numbers – average of a set of data found by finding the sum of a set of data and dividing the sum by the number of pieces of data in the set
• Mean of a population ()
• Mean of a sample ()
• Median of a set of numbers – the middle number of a set of data that has been arranged in order from greatest to least or least to greatest. The median is a number that is larger than half the numbers and smaller than the other half of the numbers.
• Mode of a set of numbers – most frequent value in a set of data. There may be more than one mode or no mode, if all numbers appear the same number of times.
• Range of a set of numbers – the difference in the value of the largest number and the smallest number in a set
• Quartiles – the groups of data within a set; 1st quartile is the first 25th percentile and 3rd quartile is the 75th percentile of the data
• Interquartile range (IQR) – the difference between the first quartile and the third quartile of a set of numbers. The second and third quartiles make up the middle half of the data in the set.
• Standard deviation – average amount either above or below the mean that the data deviate from the mean
• Population standard deviation formula: , where  = population standard deviation,  = sum or “add them all up”, x = individual data values,  = mean, and N = number of terms
• Sample standard deviation formula:  ,  where Sx = sample standard deviation,  = sum or “add them all up”, x = individual data values,  = mean, and N = number of terms
• Calculator or computer technology
• Identification of the measures of central tendency (mean, median, and mode)
• Identification of the measures of variability (range, interquartile range or IQR, and standard deviation)

To Make

INFERENCES WITH NORMAL DISTRIBUTIONS

Including, but not limited to:

• Normal distribution – even disribution of data that form a bell shaped curve
• The larger the number of samples taken, the more normal data becomes.
• For normal data, approximately 68% of the data fall within one standard deviation on either side of the mean, 95% of the data fall within two standard deviations of the mean, and 99.7% fall within three standard deviations of the mean.

Note(s):

• Grade 2 introduced bar graphs.
• Grades 3 – 7 learned and used dot plots to organize and analyze data.
• Grade 6 introduced box plots, histograms, and stem-and-leaf plots.
• Grade 7 introduced circle graphs.
• Middle School addressed mean, median, and mode.
• Middle School addressed variance, including range and interquartile ranges.
• Mathematical Models with Applications introduces standard deviation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• C1 – Use estimation to check for errors and reasonableness of solutions
• II. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships
• IV. Measurement Reasoning
• A1 – Select or use the appropriate type of unit for the attribute being measure.
• VI. Statistical Reasoning
• B1 – Determine types of data.
• B2 – Select and apply appropriate visual representation of data.
• B3 – Compute and describe summary statistics of data
• B4 – Describe patterns and departure from patterns in a set of data.
• C1 – Make predictions and draw inferences using summary statistics.
• C2 – Analyze data sets using graphs and summary statistics.
• C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections