Data: Data Analysis, Probability and Statistics, and Graphing
Adults make decisions based on data in their daily lives and in the workplace. Reading charts and graphs, interpreting data, and making decisions based on the information are key skills to being a successful worker and an informed citizen. Being an informed citizen includes understanding statistics and probability as well. Adults cannot make reasonable decisions unless they understand from where the statistics come.
Charts and graphs are essential in the workplace. Data from charts and graph are used to make decisions. Graph are useful tools in that they organize data so the information becomes clearer. This organized information can then be used to craw conclusions, to make decisions, or to influence others. Data is organized in a variety of fashions, from charts and graphs, to computer-generated spreadsheets.
Data collection, analysis, and graphing are essential in the workplace. Many industries, manufacturing in particular, now use statistical control processes to monitor their processes in order to ensure quality products. Often the front line employee is required to collect the data used for charting the manufacturing process; therefore, employees at all levels should be knowledgeable about and comfortable with using a variety of charts. As more and more quality teams, consisting of a variety of employees, are charged with the task of ensuring quality products, employees will need to have an understanding of probability and sampling.
There is also the need to have the ability to read and interpret statistical process control charts. Employers want everyone to understand quality. Any chart or graph that shows production uses statistics. Other forms of charting are also used in the workplace to make decisions as well as gauge accuracy.
Statistical knowledge is important in problem solving and decision making. Adults, often without realizing it, make decisions, based on statistical information. It may be via the television, radio, or it may be through print materials. Statistical information is used to communicate information and sometimes influence others. Understanding the flood of statistical information allows adults to make more informed decisions.
Graphs, tables, and statistics make data easier to understand. Adults create graphs for clarity and understanding, for themselves and for others.
Sometimes seeing the data in chart form makes the decision making process easier since the information is clearer. Even when charts and graph are not initiated by adults, they do tend to make the information easier to digest. Charts and graphs are also used for record keeping such as spreadsheets and data bases.
There is a concern for the lack of understanding and ability to read and interpret statistical information, including charts and graphs. There is also worry about use and misuse of statistical information. Transferability is hard for many adults. To know a concept is one thing, but to be able to look at a table and understand and interpret it, is hard to do.
Adults use charts, graphs, and statistical information in their roles as workers, parents, and citizens. As workers, adults use data monitor the quality of the products being made. They also make decisions based on the data. As citizens, adults need to understand the data that they are continually being bombarded with.
IMPLICATIONS FOR LEARNING AND TEACHING
We must introduce more work-related charts and graph and other statistical information to better prepare adults learners for the word of work. Adult learners need much more than simple activities where they are asked to find literal bits of information in more than simple activities where they are asked to find literal bits of information in charts and graph. They need opportunities to collect their own data, then create their own charts and graph. In designing their own charts, adult learners begin to understand how data can be represented. Employees at all levels are begin required to read and interpret charts and graphs, so adult learners need to be prepared.
We must provide hands-on experiences collecting, organizing, and interpreting data. It is not enough that adult education classes give learners practice in simply reading and finding literal information based on charts and graph. Providing adults learners with the actual experience of gathering data, deciding on how to represent the data, and interpreting the results will give them a deeper understanding of statistical information. Adult learners need opportunities to interpret charts and graphs and discuss their finding and implications with others.
TITLE - Percentages, projections, and multi-colored candies
MATERIALS LIST - small bags of multi-colored candies like M & M's for each student, 1 large bag of M & M's for the class, Tables I and 2 - provided Optional - calculator
TARGET AUDIENCE - ABE/pre-GED/GED
MODE OF INSTRUCTION - Class or large group
2. Open bag and count individual colors in each bag. Figure out the fractions of different colored candies -- the number of candies of each color divided by the total number of candies and then change it to the percent of each color. Plain M & M's come in six different colors. (Use the calculator, if desired, to figure the percent.)
Put your percents into Table 1.
3. Average the percents in each column to find the calculated average percent for each color. (Add all the percent numbers in each column and divide by the total number of percents; this will be the same as the total number of participating students).
Put your averaged percents into Table 1.
4. Open up the large bag of candies and count them without respect to color. Then calculate how many candies you think that there will be of each color, using the average percent you had calculated in step 3 of this activity. Multiply each percent for the different colors times the total number of candies you have just counted in the large bag.
Fill in Table 2 with your proposed number of candies of each color.
5. Separate the candies into groups by color, and count the candies in each group.
Log in the actual number of each color of candy in Table 2.
6. Calculate the actual percentage of each color candy in the large bag now by figuring out the fractions and then changing it to a percent.
Enter these actual percentages in Table 2.
7. Compare your percentage from steps # 2, # 3, and # 6 with the company's official percentage of each color (put this into Table 2). (There is a toll free number on a bag of M&M's that you may call for information.) As of May, 1996, the percents for plain M&M's are - red-20%, blue-10%, yellow-20%, green-10%, orange-10%, and brown-30%.
8. Discuss the results. Was the larger bag closer in color percentages than each individual small bag was to the company's official numbers? How did the average of the percents of the smaller bags found in the bottom row in Table 1 compare to the company's percents? Do you think that these averages would come closer to the company's percents with the more students involved in the project? (probably) How did your proposed number of multi-colored candies in the large bag compare with the actual number of candies, by color, counted in the large bag. See Table 2.
9. Eat the candies and enjoy!
WHY I LIKE THIS ACTIVITY - I liked this activity as it is fun, and the student gets experience in actual data collection and in making projections.
SUGGESTIONS FOR EXTENSION - Have the students think of how they could display their data. A bar graph of the data could show the percents of each color with a different bar for the averaged percents of the small bags, the large bags,, and the company's percents. The horizontal axis could show the color names and the vertical axis could give the percent.
TIME - Approximately two hours.
HIGH CRIME REGIONS IN CITIES
Newspapers are wonderful in the classroom. Begin collecting local crime reports from daily newspapers. Discuss what areas of the city that they expect to find a high incidence of crime. Discuss how crime is related to such things as population density, price of average house, distance from neighbors, etc. Make predictions and see what happens!
You will need a large detailed map of your city. Practice finding streets on the map. Discuss the kinds of crime students expect to read about and color-code the crime to a color of map pin. Have groups take turns collecting articles and placing pins on the map
Data should be collected for at least a month to allow for patterns to form. Make smaller outline maps and have students use different techniques to illustrate the pattern of crime. Discuss who is interested in such statistics. Invite a police officer to close this activity by bringing in statistics for a longer period of time to see how their short sampling compares.
CONNECTIONS: Daily reading is encourage through this activity. Drawing conclusions and illustrating the data will make students more comfortable attempting to make sense out of social studies and science data. Writing can follow in a number of ways. The data could be used to write a comparison of two areas of the city. An opinion essay could be written on whether pizza delivery services should be allowed to exclude certain neighborhoods from their route.STATISTICS Provided by: Susan A. Mellott
CONNECTIONS BETWEEN EDUCATION AND POVERTY FOR WOMEN
- 40 % of female single parents have an eighth grade level education or less.
- 35% of displaced homemakers have less than an eighth grade education.
- In 1990, over 67.3 % of women who worked without a high school diploma earned less than $12,400.00 per year.
- One in Eight women workers has less than a high school diploma.
People with less than a high school education will only be able to fill 14 % of the jobs that will be available in the future.
INCOME AND WAGE GAPS CONCERNING WOMEN
- In 1 9 8 5- women narrowed this gap by increasing their wages to sixty-four cents for every dollar a man earned.
- Our country offers women less income than other industrialized nations:
-in 1980: Sweden- women earned eighty one cents for every dollar earned by a man. Britain- women earned sixty-six cents for every dollar earned by a man.
In 1 9 8 4- the average yearly income for a woman was $14,479.00, as compared to $23,218 which is the average yearly income for an average man.
GIVE EXAMPLES OF SOME PROBABILITY QUESTIONS
Games are usually interesting. For example, if a friend challenges you to draw an ace on your first pick for a free lunch, how good are your chances of winning? Since there are 4 aces in 52 cards, your chances of winning are 4/52 or 1/13. So the probability of not picking an ace is 48/52 OR 12/13! The odds are against you, but not too impossible.
Now let's say the friend has challenged you to toll dice - two of them. He will buy you lunch if you roll a six on each dice. So what is the probability of getting a free lunch? Your chances of rolling a six on one dice is 1/6. But how about both at the same time? Now we have to multiply the probability of getting a six on each die. Now your chances are 1/6 x 1/6 or 1/36. Your chances of rolling 2 sixes are much worse that drawing an ace out of a deck of cards.
Another example would be a "Pick Three" in the Lottery. If your chances are 1/10 for each number, then your chances of picking all three correct would be 1/10 x 1/10 x 1/10 or 1/1000! Save your money!
In Figmentland, USA, marketing specialists decided that they would promote Smarties Cereal by giving away coupons in specially marked boxes. These coupons could be redeemed for passing scores on the GED Test, Writing Skills, Essay, Social Studies, Science, Literature, and Mathematics coupons were put in Smarties Cereal. Equal numbers of each coupon were placed in boxes and distributed evenly to area stores.
Since your chances of getting any of the coupons are the same, how many boxes of Smarties Cereal would you expect to have to buy in order to collect all six coupons?
Is it possible to get all six coupons in only six boxes? Is it possible that you might buy 100 boxes and still not get all six? Let's set up a simulation or "act out" the problem.
SPINNERS WITH 6 EQUAL AREAS
HANDMADE COUPONS ON 3 X 5 CARDS
OVERHEAD DICE (TRANSPARENT) if doing project with whole class together. Each group should run three trials with their materials and keep track of the number of times it took to get all 6 coupons.
Tallying the results of the simulation can be done in several different ways. I have used "100" GRID" PAPER. Using that, they star a number when it is a coupon that they have not yet gotten. Then on spins that are duplicates, either just mark out the number or record the duplicate subject name. By recording the name of each spin, at the end other numbers can be tabulated such as how many times was each subject area spun. Also the last number starred will be the total number of attempts that it took to get coupons for all sections of the GED Test.
A.B.L.E. STUDENTS EXAMINE TAXES
Rationale and Overview:
"In this world nothing Is certain but death and taxes." -- Benjamin Franklin
Adult students deal with tax costs and benefits regularly. Gaining greater understanding of the tax system, where tax money comes from and how it's used, can help them understand and appreciate the application of classroom mathematics to real life situations.
A study of taxes can help students integrate understandings from the areas of social studies and mathematics. Students are interested In how their own money Is spent. This unit could logically follow a study of family budgets, emphasizing the need for the government unit to also budget its income and expenses. As students develop an interest In how tax money is spent, they may also develop an Interest in participating In the system to help influence how the government allocates financial priorities.
When my 14-year-old daughter collected her first paycheck, she quickly noticed that several deductions had lessened the amount she'd expected. she wanted to know why that money was taken away from her, who said they could do that, who decided how much money to take, what did those people do with her money, and what did she get out of the-deal. After a lot of discussion, her final comment was, "No wonder you guys vote." We and our students have wondered the same things, and this unit gives everyone an opportunity to develop and express opinions, quantify them, and compete their ideas to reality.
Discussing what students already know about taxes, using math skills to analyze their opinions, and comparing their concepts to real life examples gives them many opportunities to build new understandings and grow.
Connections to past learning:
Give students some time to consider what they already know about taxes. Encourage them to share their experiences with taxes -both as payers and consumers. This is a good opportunity to correlate previous learning about taxation and government and information about the history of the United States that relates to this issue. Students use critical thinking, communication skills and reasoning during the discussion. If the need arises, they may want to use Information gathering and research skills.
As the unit develops, students practice previously learned math skills in estimating, calculation of whole numbers and percents, averaging, and chart and graph development.
Connections to the real world;
Taxes are a certainty. Income taxes have been collected in the U.S. since 1913, and the Social Security Act of 1935 created the Internal Revenue Service which supervises the collection of federal income taxes. Looking at a sample paycheck will show many deductions for various taxes. Students will be able to name other kinds of taxes from their experience.
Even If a student doesn't currently pay income taxes, other taxes are a reality. Each student can consider how government activities are funded. Then each can consider himself as a consumer of tax money in benefits. The A.B.L.E. classroom itself may enter the discussion.
Students can be encouraged to analyze their own paychecks or tax forms. And they can look for ways their lives benefit from government spending.
Each student has the opportunity to think about how tax money should be allocated, then to compare those opinions to the others in the group and to the real applications of government. Students can be encouraged to understand how taxes (collection and expenditure) may influence Individual lives.
Connections to adult life skills:
Adult students participating In this unit will practice language skills. During discussions they'll offer opinions and adjust those opinions based on other input. They'll also use critical thinking and analysis skills as they compare their ideas to actual tax sources and expenditures. Learning about the tax system will help them develop their ability to read, evaluate and understand information from periodicals and broadcasts. Perhaps they will want to express the opinions they develop to their government representatives.
During the unit each student will have many opportunities to apply math skills In reading and writing numbers, comparing and estimating amounts, calculating totals and averages, converting whole numbers to fractions and percents and reading various charts and graphs. Some students may want to learn how to fill out their own IRS forms.
Connections to work:
Using a paycheck to compute the amount or percent of tax contribution can emphasize the importance of learning about taxes. Some students may want information on adjusting their deductions.
As students become aware of how tax money is spent, they may become quite interested in how government spending of tax money both creates and influences their own Jobs and the Jobs of others. Some students may want to learn more about the Jobs of the people who collect, disburse and allocate their tax money.
Create graphs from the class or student charts (on computer?).
Fill out sample tax forms.
Compare graphs showing spending of tax money in previous years to current graphs. Compute the changes and/or discuss what has influenced the differences.
Consider how to reduce the deficit and pay off the federal debt by reducing taxes or reducing spending in certain categories.
Find tax-related articles in periodicals and compare them to the worksheet information.
Contact the budget office for the government unit and ask for budget projections for the coming year.
TITLE: STATISTICS, PROBABILITY AND CONNECTIONS USING THE MORTALITY TABLE
Materials: Mortality tables from different periods Computer software for creating tables or spreadsheets Calculators Graph paper Color markers or pencils
Target Audience: ABLE/JOBS Adult Students
Mode of Instruction: Small group or whole class
Rationale: Adult education should put less emphasis on teaching isolated mathematical skills and increase emphasis on teaching the math of life skills and the world of work. Investigation of statistics and probability should actively engage learners in exploring events and making predictions about situations relevant to their daily lives. Adults know that decisions made on the basis of various statistics affect them daily. Collection, organization, calculation, and interpretation of data are fundamental to our personal lives and the lives of most adults in the workplace. Adults use and analyze statistics and, formally or informally, predict outcomes daily.
Therefore, it is important that adult learners understand how statistical representations and calculations are used. There is nothing more indigenous or relevant to human life than mortality. Using mortality tables from different time periods is an effective way to investigate changes and predict future change. It also is a means of getting adults who smoke (many of our students do) or have other dangerous lifestyles to consider their own mortality
Statistics, Probability, Data Collection, and Insurance
One of the largest businesses in the United States which relies heavily on statistics and probability is insurance. It is very important to an insurance company to know how to measure the risks against which people are buying the insurance. In order to set the premiums, a fire insurance company must have some way of knowing how many fires will occur. An automobile insurance company must be able to predict the number of accidents involving injury, loss of life, and property damage. A life insurance company must know what the expected number of deaths will be in a given group of policyholders.
1) Class or Small Group Discussion - List of Ideas or Plans- Discussion of Feasibility of Ideas, Collection of Data
How then will companies such as these make their estimates? How will they, how would you estimate the probability:
-that a new all brick house in your community will burn?
-that a 70-year old man will be hospitalized this year.?
-that a 16-year old person will die before reaching age 17?
You know immediately that in order to arrive at estimates such as these, some data must be collected. The automobile Insurance company will have to gather data on the 18-year old male drivers in order to tell what the experience is likely to be - how many accidents this group of drivers has, on the average. Or the fire insurance company must compile statistics on fires among all-brick houses in communities having adequate fire departments. And life insurance companies must have available some statistics that will show how many 16-year olds die, on the average. Data such as these must be based on large numbers of events. Operating in cases such as these is the Law of Large Numbers, which, stated simply, means that with large groups we can predict fairly accurately what is likely to happen. With a large number of experiments, the ratio of the number
Life insurance companies use sets of statistics called mortality tables to predict how many people of the same age will die in a particular year. The companies assume that what will happen in the future will be similar to what happened in the recent past. A mortality table is based on the lives and deaths of policyholders of several large life Insurance companies. It has a margin of safety added, and although it might not be used for premium calculations, it is used for making other calculations necessary to life insurance company operations. It is a table of probabilities, also.
2) Creating Tables and Graphs - Organizing Data if available, have students use computer programs to create tables and graphs of the selected information below. 9 computers are not available, have students draw a table of the Information below and create a graph, using graph or lined paper and color pencils or markers, based on the table.
A 1958 United States Commissioners Standard Ordinary Table of Mortality (Table I ) listed the following selected probabilities:
-At age 1, the, probability that death would occur within 1 year was 1.76/1000
-At age 15, the probability that death would occur within 1 year was 1.46/1000
-At age 30, the probability that death would occur within 1 year was 2.13/1000
-At age 45, the probability that death would occur within 1 year was 5.35/1000
-At age 60 the probability that death would occur within 1 year was 20.34/1000
-At age 75 the probability that death would occur within 1 year was 73.37/1000
-At age 90 the probability that death would occur within 1 yr. was 228.14/1000
-At age 99 the probability that death would occur within 1 yr. was 1000/1000
It is customary in the table to speak of the annual rate of death per 1000 persons rather than to call this the probability of death at a given age.
3) Interpreting Data - Questions for class or small group discussion, group reports, or writing assignment.
-Why would the probability almost quadruple between ages 45 and 60?
-Why would the probability almost quadruple between ages 75 and 90?