# Measures of Central Tendency

## ACOS Objective 12

Determine measures of central tendency (mean, median, and mode) and the range using a given set of data or graphs, including histograms, frequency tables, and stem-and-leaf plots.
• Creating histograms

ARMT Possible Points = 7 (MC, GR, OE)
• Word problems/real-life situations may be used.
• The word "mean" will be used for the arithmetic average.

Sample problems from Item Specs

Content previously taught

## NCTM Principles and Standards for School Mathematics

• "...students can use and further develop their emerging understanding of proportionality in various aspects of their study of data and statistics." (249)
• "Middle-grades students should formulate questions and design experiments or surveys to collect relevant data so that they can compare characteristics within a population or between populations." (249)
• "Students [should] plan experiments in which they collect data..." (249)
• "...mathematics teachers may find it useful to collaborate with science teachers [to] collect the data for an experiment in science class and analyze it in mathematics class." (249)
• "...students should learn to find relevant data in other resources, such as Web sites or print publications." (249)
• "When using data from other sources, students need to determine which data are appropriate for their needs, understand how the data were gathered, and consider limitations that could affect interpretation." (249)
• "Middle-grades students should learn to use absolute- and relative-frequency bar graphs and histograms to represent the data they collect and to decide which form of representation is appropriate for different purposes." (249)
• "Students need to understand that the mean "evens out" or "balances" a set of data and that the median identifies the "middle" of a data set. They should compare the utility of the mean and the median as measures of center for different data sets." (251)
• "...the teacher has an important role in providing experiences that help students construct a solid understanding of the mean and its relation to other measures of center. " (251)
• "Students also need to think about measures of center in relation to the spread of distribution. In general, the crucial question is, How do changes in data values affect the mean and median of a set of data?" (251)
• "[S]tudents [need to] understand the different sensitivities of the mean and the median; for example the mean is very sensitive to the addition or deletion of one or two extreme data points, whereas the median is far less sensitive to such changes." (251)
• "Students should consider how well different graphs represent important characteristics of data sets." (251)
• "In collecting and representing data, students should be driven by a desire to answer questions on the basis of the data." (251)
• "Teachers should encourage students to plot many data sets and look for relationships in the plots; computer graphing software and graphing calculators can be very helpful in this work." (253)

## Textbook Resources

Teaching Mathematics with Manipulatives, p. 37
Story Graph: Make up a story to fit the given graph.

Wequest: Lions, Tigers, and Bears

Are you ready to join a team of animal experts? As part of your application to be the zoo's new coordinator, you must complete several challenging tasks. You'll make decisions about what animals to purchase for the zoo based on financial information provided to you. You'll gather specific data about the animals you choose, including their weight and expected lifespan. Finally, you'll present your findings to the hiring committee. So pack up your gear and don't forget your algebra tool kit. This adventure is going to be wild!

Zoo Challenge 1:
First, read a letter from the zoo Animals Inc. The letter is found in the Process section. Then, create a spreadsheet according to the request of the zoo.
Zoo Challenge 2:
Next, research the average weight, lifespan, and gestation and/or incubation period of each animal in your newly created zoo.
Zoo Challenge 3:
Last, create a presentation that explains why you chose each particular animal for the new zoo. You will give this presentation to your classmates, the board members of Animals Inc.

Math content: mean, median, mode, box-and-whisker plot, bar graph, one other graph range, scale,
upper quartile, lower quartile, upper extreme, lower extreme, interquartile range, x-axis, and y-axis.

Technology skills: spreadsheet, presentation (PowerPoint, poster, brochure, video, article, or portfolio)

Interdisciplinary: science

## Resources from Books

Math in the Real World
• Where We Live (33-36) - percents, bar graph
• Education Pays (46-48) - interpreting charts, making predictions, mean
• About My Class (125-126) - survey, bar graph

Math Projects
• Analyzing Graphs (3)
• Collecting Data (7)
• Comparing Prices and Ingredients of Cereals (9)
• Conducting and Measurement Olympics (10)
• Producing a Math Survey (42)
• Ratios of Age to Height (43)

Teamwork Test Prep
• Sporty Statistics (56) mean, median, mode, internet research, writing, box-and-whisker plot
• What's Your Top 10? (60) pictographs, internet research
• Measure Up! (72)

Using Assessment to Improve Middle-Grades Mathematics Teaching and Learning
• Bar Averages - great for teaching open-ended questions. Includes rubric and sample scored answers.
Mathematics Teaching in the Middle School, August 2006
• Families Ask Take-Home Page (49): Isn't statistics something that people learn in college? What does my child need to know about statistics in middle school?

## Online Resources

Bar Grapher
Graph data sets in bar graphs. The color, thickness and scale of the graph are adjustable. You can input your own data, or you can use or alter pre-made data sets.

Circle Grapher
Use this tool to graph data sets in a circle graph. You can input your own data or alter a pre‑made data set.

Histogram Tool
This tool can be used to create a histogram for analyzing the distribution of a data set using data that you enter or using pre‑loaded data that you select.

State Data Map
Information can be represented in many ways, and this applet allows the user to represent data about the states using colors. The state with the highest data value is darkest; other states are shaded proportionally. Investigate any of the data provided—or enter data of your own!

Math Stock Market
By studying the stock market, students will have an opportunity to become familiar with the newspaper, the concept of the stock market, fractions, mean, median, mode, range, and line graphs.

Introduction to Statistics: Mean, Median, and Mode

MMMR Rap
Lyrics to a rap explaining mean, median, mode, and range.

Grab Bag
Lesson plan using Unifix cubes.

Box Plots and Histograms

TI-84 lesson: Create and explore a box plot and histogram for a data set. Then compare the two data displays and use the comparison to draw conclusions about the data.

## Cool Problems

### Slanted Statistics

What is wrong with the reasoning in the following statement?
"Last Monday, my team of researchers randomly called 1000 people at home between the hours of 11 a.m. and 3 p.m. to determine the percentage of the population that has full-time jobs. Of the people we surveyed, only 60% have full-time jobs. There is clearly a crisis of unemployment in this country."
Answer: The Daily Spark, p. 33.