Range, Mean, Median, Mode, Outliers and Central Tendency

Choose Between Mean, Median and Mode

Know why a specific measure of central tendency (mean, median) provides the most useful information in a given context.

Understand that specific measures can be used to manipulate the view of a data set to work for a given context. The presentation of the measure used for mean or median can represent data in a fair light or manipulate data to persuade the viewer of the data to think a certain way. Know when to use the mean or the median to give the most useful information for a given situation.

Brix College would like to prove that students with a college degree make more money on average than people with no degree. According to their data, the salaries are: $50,000, $50,000, $55,000, $45, 000, $49,000 and $200,000. Would it be best for the Brix to tell students the mean or median of the data if they want to show a high average of pay. (mean)

(2)

Tom is graduating from Brix College and would like a realistic estimate of the amount of money most new graduates earn. According to the data at Brix, the starting salaries of the last five graduates were: $50,000, $50,000, $55,000, $45, 000, $49,000 and $200,000. Would it be best for Tom calculate the mean or the median to find the most realistic view of starting salaries? (median)

(3)

Brianna is arguing with her brother, T.J., about who is on the better soccer team. She’s using the data from their last five games. The table below shows their scores. Brianna says her team is better because they have the highest average of points. Is this correct? What measure of central tendency could T.J. use to prove her wrong?

Briana

4

5

4

2

12

T.J.

5

7

6

5

7

(Brianna is incorrect, because the outlier skewed the mean. T.J. could use the median to prove her wrong.)

(4)

Would the mean or median give the most accurate representation of the typical number of computers in a household on Montair Street. Could the mode be used to give an accurate representation? Explain.

(The mean, because there are no outliers. It would not be best to use the mode, because a true average can be found when no outliers are involved.)

(5)

Mr. Brown recorded the hours his students spend on homework. The times in hours were: 1, 1.25, 1, 1.5, 1, 3.5, 1.25, 1.5, 1, 1, 1, 1.5, 1, 1.25, 1.75, 1.25, 1. If Mr. Brown wants a clear view of how long the average student takes to complete homework, should he calculate the mean or median? Could the mode be used? Explain.

(Median, because the outlier of 3.5 will skew the mean. The mode could be used in the case, since it is a dominant mode and all other data is in close proximity to it.)

Read the problem carefully. Think about who is presenting the data and whom they are presenting it to. Consider all motives of the person presenting the data. Ask yourself if he/she is trying to convince someone of something. Let’s look at an example. A real estate agent is trying to convince a buyer to make a purchase in a particular neighborhood homes recently sold for: 300,000; 350,000; 400,000; 350,000 and 1,000,000. The agent could find the mean of all these sales and lead the buyer to believe that the average property value is higher than it actually is by including the one million dollar sale. Even though the agent is not lying on calculating the mean, it is not a fair representation. A better representation for the typical sale value of homes in this area would be to use the median.

(2)

Remember to watch out for outliers when looking at data and a problem situation. An outlier is a number that is much smaller or much larger than the other data. An outlier can affect the accuracy and fairness of the computation of the measures of central tendency. This is particularly true for the mean. An outlier that is much larger that the rest of the numbers will increase the mean, while a much smaller outlier will do the opposite. For this reason, the median is most often used to find a more typical measure when an outlier is involved. However, from time to time the mode can be used if it is repeated significantly in the data set. An example of how the mode can be used can be seen in the data set that follows: 4, 4, 3, 5, 4, 4, 6, 4, 4, 4, and 16.

(3)

If you need to determine which measure of central tendency could best be used to represent a set of data fairly, think about the problem and all that ways the data could be represented (mean, median, or sometimes mode). Eliminate the measures that won’t give a broad overall view of the data and a fair representation of the data. Now try computing the data with that measure. You can then eliminate outlier from the data set and compute the mean. Lastly, compare the two results. They should be fairly close to one another.

(4)

The kinesthetic learner will benefit from acting out problem situations. You can use the sample problems. Begin by discussing who is in the problem. So, for the problem with Mr. Brown, you child would act out being a teacher that wants to know the typical hours spent on homework by an average student. Remind your child that most of the students spend in the one hour range, this makes the student that spends 3.5 hours doing homework above the average. Make sure your child knows his/her motive when role playing each problem situation.

(5)

The visual learner can use color-coded underlines to identify important parts of the problem that will be need to be considered. Some suggestions for color-coding are; one color for each of the following: who is presenting the data measure, why – what is the motive in presenting the data measure, which numbers are in the data set, and the outlier (if present). The information that is color-coded by underlines or highlights can then be rewritten in a neat visual. Below is a sample problem and one possible way to organize the data visually.

Mr. Brown recorded the hours his students spend on homework. The times in hours were: 1, 1.25, 1, 1.5, 1,3.5,1.25, 1.5, 1, 1, 1, 1.5, 1, 1.25, 1.75, 1.25, 1. If Mr. Brown wants a clear view of how long the average studenttakes to complete homework, should he calculate the mean or median? Could the mode be used? Explain.

wants a clear view of how long the average studenttakes to complete homework

The table can now be used to think in steps about each part of the problem and draw a conclusion about which unit of measure would give the best results to meet Mr. Brown’s goal.

The hourly rates of pay for 6 store employees are listed below. Which measure of central tendency best describes the typical rate of pay?

$7.25, $7.00, $7.50, $6.50, $8.25, $8.00, $6.75, and $7.25

(mean)

(2)

The hourly rates of pay for 6 store employees are listed below. Which measure of central tendency best describes the typical rate of pay?

$7.25, $7.00, $7.50, $6.50, $15.25, $8.00, $6.75, and $7.25

(median)

(3)

If the number five were added to the data set below, which would decrease the most the mean or median?

55, 42, 60, 49

(mean)

(4)

If the number 99 were added to the data set below, which would increase the most the mean or the median?

22, 42, 33, 18, 20, 19, 39, 29

(mean)

(5)

The prices of concert tickets for sale are listed below. What would be the best measure of central tendency to find the average cost of a concert ticket.

$20.00, $30.00, $40.00

(mean)

(6)

The prices of theatre tickets for different plays are listed below. Which measure of central tendency could best be used to show the typical price of a theatre ticket?

$300, $25, $50, $25, $35, $55

(median)

(7)

The homeruns for a baseball team for the season are 5, 11, 19, 10, 16, 7, 9, 11, 19, 11. What measure of central tendency could be used to determine the average homeruns made?

(mean)

(8)

Dyllan’s soccer goals scored for the first 6 games of the season were: 2, 4, 2, 5, 3, 12. What measure of central tendency could best be used to determine Dyllan’s typical goals scored on a given day?

(median)

(9)

Sylvia’s teacher gave her a list of her Science scores, as shown below. Would Sylvia best benefit from her teacher finding the mean of her scores or the median? Explain.

75, 80, 82, 76, 70, 20

(Median, if the teacher averaged in her score of 20 her grade would drop to 67% or a D. However, if the teacher finds the median Sylvia will have 75.5%, a C.)

(10)

John’s teacher gave him a list of his Language Arts scores, as shown below. Would John most benefit from his teacher using the mean or median to determine his final grade? Explain.

75, 80, 85, 78, 80, 100, 78

(John would receive a higher percent if his teacher found the mean, however his grade would remain at a B with either score. This is because the outlier is not hugely different than the other numbers in the data set.)

(11)

L.A. University would like to prove that students with a college degree make more money on average than people with no degree. According to their data, the salaries of recently graduated students are: $50,000, $50,000, $55,000, $45, 000, $49,000, $60,000, $55,000, $40,000, $52,000 and $500,000. Would it be best for the L.A. University to tell students the mean or median of the data if they want to show a high average of pay. Explain.

(L.A. University should present the data as the mean, because the outlier of $500,000 will help increase the average earnings.)

(12)

Juan is graduating from L.A. University and would like a realistic estimate of the monthly income he could bring home as a new graduate. According to the data at L.A. University, the starting salaries of the last five graduates were: $45,000, $40,000, $555,000, $45, 000, $49,000 and $30,000. Would it be best for Tom calculate the mean or the median to find the most realistic view of starting salaries? Explain.

(Juan should calculate the median income, because the outlier of the graduate that is earning $555,000 is nowhere near that of the other graduates and would increase the mean by too much. This would give Juan an exaggerated view of how much money he could expect to earn.)

(13)

Would the mean or median give the most accurate representation of the typical number of pets in each household on Willard Street? Could the mode be used to give an accurate representation? Explain.

(The mean should be used because there are no outliers to skew the data results.)

(14)

Mrs. Grimes is calculating Jessica’s math scores, as listed below. She noticed that Jessica had one bad test at the beginning of the unit and improved as the unit progressed. Should she use mean or median to determine a score that will best represent how Jessica is doing in math overall?

10, 60, 75, 75, 80, 75

(Mrs. Grimes should use the median to determine Jessica’s score, so that her first test score doesn’t ruin all her hard work at the end of the unit. Calculating the mean would give Jessica a grade of a D, but finding the median would give her a grade of a C. This is fair since Jessica’s score just keep increasing).

(15)

Nico insists that The Tigers are a better baseball team than The Bears because they have a higher points average per game. He used the table to calculate the mean for each team’s average scores. Is Nico’s statement valid if we only base consider average team points per game?

Tigers

Bears

10

9

11

12

15

5

11

10

(Nico can use the mean to prove his point. The mean is the best representation for both teams in this situation because neither team has a clear outlier.)