Standard #: MA.912.DP.1.1

• While the benchmark states that students select an appropriate data display, instruction also includes cases where students must create the display. 
• This benchmark is closely linked to MA.912.DP.1.2, where students interpret displayed data using key components of the display. 
• Instruction includes student discussions (MTR.1.1) regarding the strengths and weaknesses of each data display, and includes the use of appropriate units and labels (MTR.4.1). 
  • Numerical univariate is data that consists of one numerical variable, and an important feature of the data is its numerical size or order. Examples include height, weight, age, salary, speed, number of pets, hours of study, etc. Displays include histograms, stem-and-leaf plots, box plots and line plots. 
    • Histograms 
      • Good for large sets of data. 
      • Shows the shape of the distribution to determine symmetry. 
      • Data is collected in suitably-sized numerical bins with equal ranges. 
      • Because of the bins, only approximate values of individual data points are displayed. 
    • Stem-and-Leaf Plots 
      • Good for small data sets. 
      • Shows the shape of a data set and each individual data value. 
      • Lists exact data values in a compact form. 
    • Box Plots 
      • Beneficial when large amounts of data are involved or compared. Used for descriptive data analysis. 
      • Shows multiple measures of variation and/or spread of data. 
      • Shows one measure of central tendency (median). 
      • Individual data points are not shown. 
      • Presents a 5-number summary of the data. 
      • Can indicate if a data set is skewed or not, but not the overall shape. 
      • Can be used to determine if potential outliers exist. 
    • Line Plots (Dot Plots) 
      • Used for small to moderate sized data sets in which the numerical values are discrete (often integers, or multiples of ½). 
      • Shows the shape of the distribution and the individual data points. 
      • Useful for highlighting clusters, gaps, and outliers. 
  • Numerical bivariate is data that involves two different numerical variables that have a possible relationship to each other. Displays include scatter plots and line graphs. 
    • Scatter Plots 
      • Good for large data sets, and for data sets in which it is not clear which variable, if any, should be considered the independent variable. 
    • Line Graphs 
      • Good for showing trends or cyclical patterns in small or medium-sized data sets in which there is an independent variable and a dependent variable. Often the values of the independent variable are chosen in advance by the person gathering the data. Examples of independent variables may be points in time or treatment amounts and examples of dependent variables might be total sales or average growth. 
  • Categorical univariate is non-numerical data of only one variable that can be categorized/grouped. Displays include bar charts, line plots, circle graphs, frequency tables and relative frequency tables. 
    • Bar Charts (Bar Graphs) 
      • Good for showing comparisons between categories or between different populations. A bar chart may show frequencies (counts) or relative frequencies (percentages) in each category. 
    • Circle Graphs 
      • Good for illustrating the percentage breakdown of items and visually representing a comparison. Not effective when there are too many categories. Shows how categories represent parts of a whole. A circle graph may show frequencies (counts) or relative frequencies (percentages) in each category. 
    • Frequency Tables and Relative Frequency Tables 
      • This is often the easiest way to display bivariate categorical data. The categories for one variable are listed in the header row of the table and the categories for the other variable are listed in the header column. The frequencies (counts) or relative frequencies (percentages) are listed in the cells for each of the indicated joint categories. Total counts or percentages for the rows may be listed in the final column of the table and total counts or percentages for the columns may be listed in the final row. 
    • Segmented Bar Charts 
      • Comparison of more than one categorical data sets. 
      • Good for showing the composition of the individual parts to the whole and making comparisons. 
• Non-numerical data may consist of numbers if the categories are not primarily determined by the numerical size or order of the numbers. 
  • For example, the data may answer the question “What is your favorite real number?” and the categories could be “Integers,” “Rational numbers that are not integers” and “Irrational numbers.” 
• Using the same real-world data (MTR.7.1), encourage students to create a variety of data displays appropriate for the data given (MTR.2.1). This makes the discussion of the similarities and differences of the displays more robust and allows students to visualize and justify their responses (MTR.3.1). 
  • This strategy might work best if you present the class with a set of data, group students and ask each group to create a different display using the same data. 
  • Each group can then present the strengths and weaknesses of their display as compared to the others (MTR.5.1). 
  • This should be repeated for each separate data category, see examples above. 
• This benchmark references bar charts; however, other benchmarks and the glossary (Appendix C) reference bar graph, these terms are used interchangeably without difference.

Common Misconceptions or Errors

• Students may not know how to label displays appropriately or how to choose appropriate units and scaling. 
  • For example, they may not know how to create or scale the number line for a line plot, they may confuse frequency and actual data values, or they may not understand that intervals for histograms should be done in equal increments. 
• Students may not understand the meaning of quartiles in the box plot. 
• Students may not know how to calculate the median with an even number of data values. 
• Students may not accurately place data values in increasing order when there are many data points. 
• Students may confuse bar charts (for categorical data) and histograms (for numerical data). 
• Students may be confused when categorical data consists of numbers that have been categorized in ways that do not primarily reflect the numerical size or order of the numbers. In such cases, it will be helpful to have the student think about whether any of the measures of center (mean, median) or variability (quartiles, range) are meaningful for the data set. If they are, then the data can be considered numerical, because these measures are concerned with the numerical size and order of the data points. If not, then it can be considered categorical.

Strategies to Support Tiered Instruction

• Teacher co-creates anchor charts that include appropriate units of measure. 
  • For example, time measurement units include seconds, minutes, hours, days, weeks, etc. 
• Teacher provides numerical univariate, numerical bivariate, categorical univariate and categorical bivariate examples. Each example should include scaling to ensure that students have experience scaling for graphs and tables that are in each category. 
  • For example, employee ages for the company AdvertiseHere can be displayed using a box plot as shown. 

• Teacher reviews the difference between histograms and bar graphs, creating an anchor chart with properties of a histogram for students to refer to. 
• Teacher reinforces how scales are represented with specific endpoints. The endpoints they chose to use, or as defined in a problem, tell them if the point is included in the bin or not. Include notation of endpoints on anchor chart to display in the classroom. 
• Teacher co-constructs vocabulary guide/anchor chart with students who need additional support understanding the vocabulary for measures of center and variation. 
  
  • Examples of guides and charts are shown below. 
    
    
    
    

• Teacher models ordering data sets in ascending order before finding a median, quartile or range. 
• Teacher provides a chart to display calculating the median with an even and odd data set.

       

• Instruction includes discussions about whether any of the measures of center (mean, median) or variability (quartiles, range) are meaningful for the data set. If they are, then the data can be considered numerical, because these measures are concerned with the numerical size and order of the data points. If not, then it can be considered categorical.

Instructional Tasks

• The number of cars sold in a week at a large car dealership over a 20-week period is given below.