Statistics: Two-variable Data

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Unit 3_2 Variable Data.notebook 1 March 19, 2013 Dec 16:28 PM Statistics: Two-variable Data Key Ideas: 1. Interpreting Scatter Plots Independent vs. Dependent variables Correlation Line of Best fit Outliers Interpolation/Extrapolation 2. Statistical Literacy & Data Analysis Quartiles & percentiles Population vs. Sample Analyzing bias in surveys Dec 17:43 PM Part 1: TwoVariable Data

Transcript of Statistics: Two-variable Data

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Statistics:Two-variable DataKey Ideas:

1. Interpreting Scatter Plots• Independent vs. Dependent variables• Correlation• Line of Best fit• Outliers• Interpolation/Extrapolation

2. Statistical Literacy & Data Analysis• Quartiles & percentiles• Population vs. Sample• Analyzing bias in surveys

Dec 1­7:43 PM

Part 1: Two­Variable Data

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What you'll learn ...

To distinguish types of data and to analyse and represent two­variable data from primary and secondary sources.

And why ...

Analysing data to look for relationships is part of many college courses and professions. Fishery and Forestry Managers, Sports Trainers, Medical Researchers and Lab Technicians all work with two­variable data.

Oct 12­7:36 PM

Activate Prior Knowledge

• Interpreting Data GraphsData can be presented in a variety of way. Graphical representations can include bar graphs, histograms, scatter plots, and circle graphs.

Example:

Solution:

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• Working with Slope an Line Graphs

Example:

Solution:

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Of the following, only one graph displays two­variable data.Which one do you think it is? Why?

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In statistics, a variable is an attribute that can be measured.

One­variable data sets give measures of one attribute. This is what you studied in grade 11 . You can recognize one­variable situations when you see:

• Tally charts • Histograms

• Frequency tables • Pictographs

• Bar graphs

• Circle graphs

One­variable data can be analyzed using mean, median, or mode.

Dec 1­8:22 PM

• Two­column (or row) tables of values

Two­variable data sets give measures of two attributes for each item in a sample. You can recognize two­variable situations when you see:

• Ordered pairs i.e. (age, height)

• Scatter plots

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Example 1:

State whether each situation involves one­variable or two­variable data. Justify your answers.

a) Noah researches annual hours of sunshine in Canadian cities.

Noah’s research could be analysed using mean, median, or mode. So, the situation involves one­variable data.

b) A study compared the length of time children spend playing video games and the time they spend reading.

The study involves two pieces of information for each child, time spent on video games and time spent reading. These data could be represented in a scatter plot. So, the study involves two­variable data.

Identifying Situations Involving One­ and Two­Variable Data

Dec 1­8:44 PM

Example 2: Deciding Which Type of Graph to Draw

For a class project, Dylan surveyed students about their part­time jobs.

a) What type of graph would be best to show how many hours each student worked on the weekend? Justify your choice. Does the graph display one­variable or two­variable data?

A bar graph would display all the information together so that the reader can easily compare the number of hours for each student. One piece of data would be displayed for each student. So, the graph would display one­variable data.

b) What type of graph would best show a possible relationship between weekday and weekend hours? Justify your choice.

A scatter plot could show a possible relationship between weekday hours and weekend hours. Since each point on the scatter plot would display two pieces of information about a student, the graph would reflect two­variable data.

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ActivityWork with a partner to collect data from at least 10 students.

1. For each student, measure and record the distance:

• From the elbow to the outstretched tip of the middle finger • From the knee to the ankle 2. Graph your data set, and describe the results. 3. Do you think the two body measurements are related? Why or why not?

Dec 1­9:00 PM

Compare graphs with other pairs. Do your graphs appear to showthe same relationship between the measurements?

Identify at least one reason the relationship you identified may notbe true in general. What could you do to find out?

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Scatter plots represent two­variable data as points. Scatter plots may reveal a relationship between the two variables.

In two­variable situations, one variable may be dependent on another; its value changes according to the value of the independent variable.

For example, the value of a car depends on its age.

3.2 Using Scatter Plots to Identify Relationships

Can you think of any more examples?

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Typically, we plot the independent variable on the horizontal axis, and the dependent variable on the vertical axis.

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Example: Interpreting a Scatter Plot

Jay researched estimates for a jobpainting his house.

The scatter plot shows Jay’s results.

a) Which is the dependent variable?

Employees are paid for the time they work, so the cost depends on the time required for the job. The dependent variable is cost.

b) Which two companies will take the longest? Which of these is cheaper?

Company D and Company E take the longest time for the job. Of Companies D and E, Company E is cheaper.

c) Which two companies charge the same amount?

Company B and Company D charge the same amount.

d) Why might you pick company E? Company B?

You might pick Company E if you are not in a hurry. They take a long time, but are the cheapest. Company B is the second fastest and charges a low price. So, you might pick Company B if you want the job done fairly quickly, but economically.

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A correlation is a relationship between two variables.Graphing two­variable data on a scatter plot may showa correlation between the variables.

Types of correlations

A positive correlation describes a situation in which both variables increase together.

On a scatter plot, the points go up to the right.

A negative correlation describes a situation in which one variable decreases as the other variable increases.

On a scatter plot, the points go down to the right.

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WARNING!

Observing a relationship between two variables does not mean that one variable causes a change in the other. Other factors could be involved, or the correlation could be a coincidence.

Demonstrating cause and effect conclusively is a challenging task that requires careful analysis and specialized statistical tools.

A correlation does not necessarily indicate a cause­and­effect relationship.

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State whether the claim in each situation is reasonable.

a) A scientific study showed a negative correlation between aerobicexercise and blood pressure. It claimed that the increase in aerobicactivity was the cause of the decrease in blood pressure.

It is reasonable to think there may be a cause­and­effect relationship. There are many factors that affect blood pressure, however. Since this is a scientific study, we might reasonably expect that the researchers made efforts to neutralize the other factors, for example by studying subjects in a very close age and fitness range.

b) Mila discovered a positive correlation between gasoline price and average monthly temperature. She concluded that temperature determines the price of gasoline.

It is not reasonable to say there is a cause­and­effect relationship between temperature and gasoline prices. A more likely explanation for the correlation is that higher temperatures occur in the summer, when more people are travelling out of town for weekends and vacations. This increased demand could cause the price increase.

c) Since the 1950s, the concentration of carbon dioxide in the atmosphere has been increasing. Crime rates in many countries have also increased over this time period. Does more carbon dioxide in the atmosphere cause people to commit crimes?

It is not reasonable to say there is a cause­and­effect relationship. It is much more likely that carbon dioxide levels and crime rates are each determined by many other factors, such as increasing populations.

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Graphing the Line of Best Fit

1. Graph the data. Use axes that show house sizes up to 2500 square feet andprices of up to $400 000.

2. Describe any trend you see in the graph.With a coloured pencil, draw the line that you think comes closest to matching the trend.

3. Use the line to estimate how much a 2000 square foot house might cost in Sabah’s area.

Work with a partner. Sabah researches the cost of houses in a new area. She is looking for:

• A two­storey, detached house• Asking prices of $300 000 or less• 2000 square feet of living space

Size (sq. ft.) Price ($)1700 271 9001850 289 9001600 277 9001650 289 9001700 279 0001800 294 9001550 269 900

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Sabah notices a new For Sale sign on a house in the area.At 2300 square feet and $384 500, it does not match her criteria.However, she decides to add the data to her collection anyway. Add the point (2300, 384 500) to your scatter plot. With a different colour, draw the new line of best fit. Use this line to estimate the price for a 2000 square foot house inSabah’s neighbourhood. Compare this estimate to your first estimate.

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• How does the arrangement of plotted data affect the way you draw a line of best fit? How does it affect your confidence in using the line to make predictions?• A real estate agent tells Sabah that 2000 square foot houses often come up for sale in this area, usually at prices from $295 000 to $305 000. How close were your estimates to these prices? Which line of best fit helped you make a closer prediction?

Reflect

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Line of Best Fit

3.3 Line of Best Fit

A line of best fit is a line drawn through data points to best represent alinear relationship between two variables. Other names for a line of bestfit are regression line or trend line.

NOTE: Drawing the line of best fit involves more than just finding a pathwaythrough the middle of the data. The line of best fit is the line that isclosest to each point. The more varied the position of points, the moredifficult it is to draw the line of best fit.

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OutliersIn a scatter plot, a point that lies far away from the main cluster ofpoints is an outlier. An outlier may be caused by inaccuratemeasurements, or it may be an unusual, but still valid, result.

Outliers and the Line of Best FitThe line of best fit should reflect all valid points from a data set. Thisincludes outliers.

If most of the plotted points are clustered along a linear path, evena single outlier can affect the path of the line of best fit.

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Example 1: Exploring the Affect of Outliers and the Line of Best Fit

Which line is the line of best fit? Justify your choice.

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Solution

The line in Graph C seems most likely to be the line of best fit.

• The line in Graph A passes through the middle of the main cluster ofdata. It suggests the two outliers are not present or are invalid.

• The line in Graph B follows a path exactly halfway between the maincluster of data and the outliers. It suggests that the outliers and themain cluster of data are equally important.

• The line in Graph C passes just below the middle of the main clusterof data. Its path is affected by the outliers, but it is affected more bythe main cluster of data. It is the most reasonable choice.

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Interpolating and Extrapolating from a Line of Best Fit

A line of best fit can be used to estimate or predict values.

• Estimating values that lie among the known values on the graph isinterpolation.

• Predicting values that lie beyond the known values is extrapolation.

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Example 1: Using a Line of Best Fit to Make Predictions

These are the pre­exam term marks and exam marks for some students in a Grade 12 math course.

Term mark (%) 84 76 70 95 92 61 25 55 51 73 62

Exam mark (%) 80 72 68 96 9 58 29 60 53 77 607

a) Graph the data and draw the line of best fit.b) Determine the equation of the line of best fit.c) Use the data to predict the exam mark of a student with a pre­examterm mark of 98%.d) Use the data to predict the exam mark of a student with a pre­examterm mark of 10%.

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a) Graph the data and draw the line of best fit

Term mark (%) 84 76 70 95 92 61 25 55 51 73 62

Exam mark (%) 80 72 68 96 9 58 29 60 53 77 607

b) Determine the equation of the line of best fit.

c) Use the data to predict the exam mark of a student with a pre­exam term mark of 98%.

d) Use the data to predict the exam mark of a student with a pre­exam term mark of 10%.

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How confident can we be of predictions made from scatter plots?

Data spread and reliabilityA model with data spread over a small interval is less reliable than a model based on data spread over a larger interval.The farther we get from the main cluster of data, the less confidence we should have on predictions made from that model.

Sample size and reliabilityThe more data we use, the more reliable the prediction should be.

Non­linear dataNot all relationships between variables are linear. Over a small interval, a linear model may provide a reasonable fit for non­linear data, but it will notbe reliable at the extremes.

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Example 2: Deciding if a Correlation Is Linear

Reanna created a scatter plot using data about the world record timesfor women’s 500­m speed skating. If the world record changed morethan once in a year, she used the best time for that year.

Describe the relationship between the two variables. Does it appear to be linear? If not, describe how Reanna could better model the data.

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SolutionAs the years increase, the time decreases; there is a strong negative correlation between the variables.

Although the relationship appears to be linear in some places, theoverall relationship is non­linear. The world record times decreasedmost rapidly in the four­year period from 1970 to 1976. A model that decreases steeply at first and then more slowly would better represent the data.

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Assessing a linear modelA linear model may be unreliable in these situations:

• The model is based on too few pieces of data.• The model is based on data that are clustered together.• There does not appear to be any correlation.• There are outliers.• The data do not appear to be linear.

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Part 2: Statistical Literacy and the Media

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The mean, median, and mode are measures of central tendencyfor a data set. They represent a typical value for the set.

The range is a measure of spread. It tells you the difference between the greatest and least numbers in the set.

Measures of Central Tendency and RangeReview

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Eight students received these marks on a test: 85, 76, 91, 65, 68, 72, 78, 43

a) Calculate the mean, median, and mode mark.

b) Which measure of central tendency best describes these data? Explain.

Example:

c) Determine the range of the data set.

The mark 43 is much less than the other marks. It reduces the mean, but has no effect on the median. There is no mode. So, the median best describes the data.

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Percent Increase and Decrease

Percent increase and decrease are often used to describe how quantities have changed over time. They describe the change in a quantity as a percent of its original value.

For each calculation, write the difference as a fraction of theoriginal. Then multiply by 100 to determine the percent increase ordecrease.

To calculate:

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Example: In August, a sports store priced a shoe at $125.99.The shoes were so popular that the store increased the price to $134.99 inSeptember. Shoe purchases declined. So, the store reduced the price to $119.99 in October.

a) What was the percent increase in price from August to September?

b) What was the percent decrease in price from September to October?

c) What was the percent decrease in price from August to October?

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InvestigateOrganizing and Describing Data

Work with a partner.

The 16 students in Jesse’s math class measured their heights to the nearest centimetre.

Determine the measures of central tendency and the range for thisset of data. What percent of the class is shorter than each measure of centraltendency? Rylan is taller than 65% of the class. How many students are shorterthan he is? What is Rylan’s height?

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Reflect

Why might people want to know how they compare in height toclassmates or to other people their age?

How might information about typical heights and weights atdifferent ages be of use to a clothing manufacturer?

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Television, radio, newspapers, and Web sites often report statistical data.To understand these reports, you need to be familiar with the statisticallanguage they use.

Statistical Literacy

PercentilesA percentile tells approximately what percent of the data are less than aparticular data value. Percentiles are a good way to rank data when youhave a lot of data or want to keep data private.

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QuartilesA quartile is any of three numbers that separate a sorted data set intofour equal parts.

The second quartile is the median. It cuts the data set in half.So, it is the same as the 50th percentile.

The first, or lowest, quartile is the median of the data values less thanthe second quartile. It separates the lowest 25% of the data set.So, it is the same as the 25th percentile.

The third, or upper, quartile is the median of the data values greaterthan the second quartile. It separates the highest 25% of the data set.So, it is the same as the 75th percentile.

Dec 8­7:42 PM

Example:Working with PercentilesHere are the hourly pay rates, in dollars, for 17 high­school studentswith part­time jobs.

a) What are the quartiles for this data set?

Hint: Start by finding the median. This is the 2nd Quartile.

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b) Damien’s pay is in the 85th percentile for this group.What does thepercentile mean? What is Damien’s hourly pay rate?

So, Damien is the 15th student in the ordered list. He earns $11.50 per hour.

The 85th percentile means approximately 85% of the students in thegroup earn less money per hour than Damien.

17 × 0.85 = 14.45

Round down to the nearest whole number to determine the numberof students who earn less money per hour than Damien: 14

Dec 8­7:54 PM

Data ReliabilityWhen you read statistical data, you need to think about the reliability of the source. Data from a government agency are usually more reliable than data from someone who is trying to sell a product or promote a point of view.

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Example: Comparing Data Sources

In each case, a research topic and two sources of information are described. Decide which data source is more likely to provide reliable data.

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Solution:a) The animal rights group is promoting a particular point of view andmay not be objective. The Food Guide was developed in consultationwith thousands of dietitians, scientists, physicians, and public healthpersonnel from across Canada. It will represent a more balanced view.

b) Both the wildlife organization and the forestry company arepromoting particular points of view. It would be best to look foradditional sources.

c) While the Ministry of Health Web site promotes the use ofimmunizations, it also describes possible complications. A groupopposing immunizations is more likely to present only one sideof the issue.

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PollsPolling companies conduct interviews with randomly selectedCanadians to determine their opinions about a variety of topics.These surveys are called polls.

The results of polls are often reported in the media, particularly duringelections. Poll results usually state a margin of error that describes how reliable the data are. As a media consumer, you need to know how to interpret these results.

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Example: Interpreting Poll Results

The results of a poll conducted by EKOS in 2005 are shown.

“Canada should increase its humanitarian aid to poor countrieseven if it means less spending in other important areas”

a) What question were people asked?

People were asked whether they agreed with the statement: “Canadashould increase its humanitarian aid to poor countries even if itmeans less spending in other important areas.”

b) How did the favourable responses compare in January and August?

In January, 31% of the people polled agreed with the statement. In August, the percent agreeing had increased to 43% of those polled.

c) A line below the graph stated “The results are valid within a marginof error of plus or minus 2.5 percentage points, 19 times out of 20.”What does this mean?

If you had taken another poll, there is a 19 out of 20, or 95% chance that the results would be within 2.5 percent of these results. That is, in August, somewhere between 40.5% and 45.5% of the people polled would agree with the statement.

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