COMPARISON OF MEASURES OF CENTER AND VARIABILITY
INTRODUCTION
The objective for this lesson on Comparison of Measures of Center and Variability is, the student will use measures of center and variability for numerical data and informally compare and contrast two numerical data distributions.
The skills students should have in order to help them in this lesson include, measures of center and variability, dot plots, mean, median, IQR, which is Interquartile Range, MAD, which is Mean Absolute Deviation, and box plots.
We will have three essential questions that will be guiding our lesson. Number one, what measures of center and variability can I use to compare two populations? Number two, how can I compare two numerical data distributions using the measure of center? And number three, how can I compare two numerical data distributions using the measure of variability?
Begin by completing the warm-up for this lesson on box plots and dot plots to prepare for the comparison of measures of center and variability.
SOLVE INTRODUCTION
The SOLVE problem for this lesson is, Coach Miller is getting ready for an important basketball game. He is comparing the heights of the players on his team and the opposing team, so he can create the best plays. The following tables show the heights of the five starting players on each team. What are the mean and median height of each team?
In Step S, we Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, what are the mean and median height of each team?
Now that we have identified the question, we need to put this question in our own words in the form of a statement. This problem is asking me to find the mean and median height of each team.
In the O Step, we Organize the Facts. First we identify the facts. We go back to our original problem and read it again, and place a vertical at the end of each fact. Coach Miller is getting ready for an important basketball game./ Fact. He is comparing the heights of the players on the team/ and the opposing team, so he can create the best plays./ The following tables show the heights of the five starting players on each team./ What are the mean and median height of each team?
After we identify the facts, we eliminate the unnecessary facts. In this word problem each of the facts given we can delete or draw a strike through, because all the information we need is contained in the two tables.
Then we list the necessary facts. Our necessary fact from this problem is the height of all the players.
In the L Step, we Line Up a Plan. Write in words what your plan of action will be. Determine the mean of the heights for each team by adding together the heights and dividing by the number of players. Determine the median of each team by listing the values in order from least to greatest and finding the middle value.
Choose an operation or operations. Addition and Division.
V, Verify Your Plan with Action. We first estimate your answer.
The estimates that we have given are Team one will have a mean of seventy, and a median of seventy.
Team Two will have a mean of seventy-two; and a median of seventy-three.
Carry out your plan. For team one we determine the mean by adding all of the heights and dividing by the number of players. The median we find by writing the values in order from least to greatest and identifying the middle value.
We follow the same process for team two.
In Step E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, because I was looking for the mean and the median of each team.
Is your answer reasonable? Compare your answer to the estimate. Yes, because both sets of answers are close to my estimates.
Is your answer accurate? Check your work. Yes.
Write your answer in a complete sentence. For Team one the mean is sixty-nine point four, and the median is sixty nine. For Team two the mean is seventy-three, and the median is seventy three.
COMPARING DATA DISTRIBUTIONS
We will be using information from the SOLVE problem to compare the data from Team one and Team two using a dot plot, the mean, and mean absolute deviation.
How many pieces of data are being used for each team? Twelve
Let’s plot the data for Team oneon the dot plot form Team one. Then we’ll plot the data for Team two on the dot plot for Team two.
Is there overlap in the two data sets? Yes
Explain how you know. Both sets of data have values from sixty eighty through seventy-four.
Because of the overlap and the two data sets being closely clustered around the same values, we are going to use the mean and the mean absolute deviation to compare the two data sets.
Let’s list the data values from Team one in the first column in order from least to greatest. How do we find the mean height? Add the values in the first column and divide by twelve, which is the number of data values.
What is the mean? Seventy point seven five.
What is meant by deviation from the mean? How far a data value is from the mean.
What is the deviation from the mean for the value of sixty-eight? Negative two point seven five
Why is it a negative value? Because the data value is less than the mean.
Complete the second column with values of deviation from the mean.
What is the label for the third column? Absolute Deviation
What is the absolute deviation? The absolute value of the deviation form the mean.
Complete the data in column three.
What is the mean absolute deviation? The mean of the absolute deviation values.
How do we determine the mean absolute deviation? Add all the values of the absolute deviations and divide by the number of data values.
What is the mean absolute deviation for Team one? One point nine six. Record the value in the table.
Now complete the data table for Team two.
Take a look at the graphic organizer at the top of the next page. Record the mean for Team one and for Team two that you found on the previous page. What do the two values mean? Team two has a greater mean height than Team one.
How do we determine the difference in the means? Subtract the values of the two means. We subtract the two means and find the difference.
What is the difference in the means? One point two five
Now, record the mean absolute deviation for Team one and Team two from the previous page.
What do the two values of mean absolute deviation show? There is a moderate variation in both sets of data.
The next row in the graphic organizer shows the relationship between the difference of the means and the Mean Absolute Deviation of each data set.
How do we describe the relationship between the difference in the means and the Mean Absolute Deviation? Divide the difference in the means by the Mean Absolute Deviation.
Explain how to determine the relationship for Team one. One point two five divided by one point nine six is equal to zero point six four.
Explain how to determine the relationship for Team two. One point two five divided by one point eight three is equal to zero point six eight.
What conclusion can you draw about the relationship between the difference in the means and the mean absolute deviation variability? The difference between the mean heights of the team one point two five is approximately zero point seven times the variability of the data sets. Record in the graphic organizer.
Let’s take a look back at the dot plots on the previous page. Both sets of data have values from sixty-eight to seventy-four.
The means for the two teams are only one point two five apart.
The difference between the mean heights of the teams is approximately zero point seven times the mean absolute deviation.
The separation between the distributions of the data of the heights on the two dot plots is not very noticeable.
Based on our conclusions in Question six, let’s make some predictions about a new situation.
Coach Miller wants to compare the height of Team one with the height of twelve players on a professional team. Here is the data for the two teams listed in order form least to greatest.
What do you notice as you compare the data values in the two lists? The values for Team one range fro sixty eight to seventy four, and the Pro Team ranges from seventy four to eighty one.
What do you think will happen when we plot this data on a dot plot? Why? The overlap will be less, because the values are farther apart.
CHOOSING THE MEASURE OF CENTER WITH DATA SETS
Sometimes when we compare data sets, there is a value that is far from the other values. This value can affect the measure of center that we use to describe the data set.
Here we have a Data Set listed: First we list the data set in order from least to greatest. Work together to determine the mean and the median of the data set.
What is the mean of the data set? Ninety point nine
What is the median of the data set? Ninety-four
What value is far away from the other values? Sixty-four
What is the mean without the value of sixty-four? Ninety-three point eight nine
What is the median without the value of sixty-four? Ninety-five
How did the mean change when the value of sixty-four was removed? The mean increased by almost three points.
How did the median change when the value of sixty-four was removed? The median increased by only one point.
Which measure of center was affected more by the value of sixty-four? The mean
Which measure of center would be more accurate to use when there is an extreme value included in the data set? The median
COMPARING DATA SETS WITH BOX PLOTS AND MEASURES OF CENTER AND VARIABILITY
Place a dot for each of the five values median, minimum, maximum, Quartile one and Quartile three on the first box plot.
How do we represent the IQR? By drawing a rectangle that goes from Quartile one to Quartile three. Draw the rectangle!
How do we represent the median? With a vertical line inside the rectangle directly above the value for the median.
Draw the median.
What is the last step for our box plot? Draw a horizontal line from the rectangle to the minimum value and a horizontal line from the rectangle to the maximum value. Draw the two lines.
Use the data from Class Two in the data table to create a box plot to represent Class Two.
Then, complete the table at the bottom of the page using the information from the box plot and the data table to compare the two data sets.
How many pieces of data are in the total data set for Class One? There are twelve.
How many pieces of data are in the total data set for Class Two? There are twelve
Fifty percent of Class One read more than how many books? Thirteen or more
Fifty percent of Class Two read more than how many books? Fifteen or more
How many books did the upper twenty five percent of Class One read? Fourteen or more
How many books did the upper twenty five percent of Class Two read? Sixteen or more
How many books did the lower twenty five percent of Class One read? Ten or less
How many books did the lower twenty five percent of Class Two read? Eleven or less
How many books were read by the middle fifty percent of Class One? Between ten point five and fourteen.
How many books were read by the middle fifty percent of Class Two? Between eleven point five and sixteen.
COMPARISON OF SITUATIONS WITH MEASURE OF CENTER AND VARIABLILTY
When we are comparing numerical data, we can look at the data set and situation to determine what is the most appropriate measure of center, measure of variability, and data display to compare the two sets.
Read Example one
What is the average price of a home in an area? We have Data Set one and Data Set two.
What are the two data sets we are comparing? Average home prices in an area.
What do you notice about the four values in Data Set One? Three of them are grouped closely together, and one is much lower.
What do you notice about the four values in Data Set Two? Three of them are grouped closely together, and one is much lower.
What will be a consideration for comparing those two data sets? Both data sets have a low value that would affect the outcome.
Which measure of center would be more accurate? The median
Which data display would be most appropriate? The box plot
Explain why we would want to use a box plot. It will allow us to easily identify the median and other values by percentages.
Number two. Want to determine the time of student’s five k run over a one-month period of time.
What is the situation in Example Two? We want to determine the time of a student’s five k run over a one-month period of time.
What is different about this example than Example One? Only the situation is given, not the actual data values.
What would we need to determine in Example Two? The average time over the month.
Which measure of center would be more accurate? Mean. Explain why. We need to look at each piece of data and see how it affects the measure of center.
Which data display would be most appropriate? The dot plot
Explain why we would want to use a dot plot. On a dot plot, each data value is displayed.
SOLVE CLOSURE – MEASURE OF CENTER AND VARIABILITY WITH TWO DATA SETS
What data are you comparing in the situation on the next page? Test Scores for math tests between two different groups.
What is the firsts step in order to compare the data sets? Write the data in order from least to greatest.
For Set One we need to complete the table for the Range, Median, Mean, list all the values and determine the Absolute Deviation from the Mean.
Let’s complete the same information for Table Two.
What is the difference in the means? The difference in the means is twenty-nine point three.
What conclusions can you make about the mean test scores for both sets? The scores in Set One are much lower than the scores in Set Two.
What conclusions can you make about the mean absolute deviation? The scores in Set Two are much higher than the scores in Set One.
How do the two sets of data compare? The scores overlap only with the score of thirty.
How do the mean scores compare? The mean scores of Set One are almost half of Set Two.
Explain the relationship between the mean absolute deviation for the two sets of scores. The mean absolute deviation for the scores in Set Two is almost three times as large as Set One, so the separation between the two sets of data is noticeable.
What data are we comparing in this example? Laps run by two groups of students.
What is the first step in order to compare the data sets? Write the data in order from least to greatest.
Complete the tables and create box plots for each group.
How many pieces of data are in Group One? Twelve
How many pieces of data are in Group Two? Twelve
If we look at the data values and box plots, fifty percent of Group One ran more than how many laps? Five or more
How did you determine the answer? Look at the values that are above the median which marks the middle value.
If we look at the data values and box plots, fifty percent of Group Two ran more than how many laps? Four or more
How did you determine the answer? Look at the values that are above the median which marks the middle value.
How many laps did the upper twenty five percent of Group One run? Six or seven
How did we determine that answer? By looking at the data value of Quartile three and identifying what values were greater than that.