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

How Far Away Points Change Things

Understand how the inclusion or exclusion of outliers affects measures of central tendency.

Know that in a data set, an outlier is a number that is numerically distant from the rest of the numbers in a data set. This is, it is far below or far above the middle percent of the data terms. For example, the number 99 is an outlier in the data set: 5, 4, 13, 11, 15, and 99. Understand that using an outlier can cause statistical data to be misleading. The mean will always be affected most by an outlier. For example, if you were asked to find the typical age of people on a playground and used the data set above to calculate the mean, the outlier would completely skew your view of the age. Using the measure of the median age and including the outlier would give you a more accurate view of the typical age. So, the outlier could be excluded to find the mean in this example, or it could be included in using the median. Logic will need to be called upon to decide whether to include or exclude an outlier for a given measure of central tendency.

Determine if the statement below is true or false.

The average salary of an employee at J.M. Mills is $1767 per month.

Salaries: $100, $150, $200, $100, $50, $10,000

(false)

(2)

What measure could best be used to find the typical salary of employees at J.M. Mills, without excluding the outlier, if the salaries are $100, $150, $200, $100, $50, $10,000.

(median)

(3)

What measure could be used to determine the average salary at J.M. Mills if the outlier were excluded if the salaries are $100, $150, $200, $100, $50, $10,000?

(mean)

(4)

Rick’s science scores were: 88, 98, 75, 80 and 0. How will the outlier affect his grade if his teacher averages his scores? What measure of central tendency could be used to find his typical science grade with the outlier included? (The outlier will bring his grade down. The median could be used to find his typical grade.)

(5)

Carolina was charting the weather in June. Which temperature will most affect the mean when she calculates the average temperature?

To better understand what an outlier is, be sure your child understands that a outlier will always be the smallest or largest number in a data set. However, in order to be considered an outlier, it must be much larger or much smaller that the rest of the numbers in the set. The outlier can be described as a lonely number, because it is left out of the group. The outlier is left out. It does not have any numbers that are close it nearby. Have your child practice identifying outliers in lists on numbers. It is helpful to some to think of the numbers as ages. So if we have a list of numbers, such as: 4, 6, 12, 14, 102, 15, 18. We would notice that 102 year old would stand out in a crowd of kids.

(2)

It is important for children to practice finding the mean, median, mode and range of data with the outlier and again without it. This will make very clear how the inclusion or exclusion of the outlier affects each measure of central tendency.

(3)

For a more kinesthetic approach to finding outliers, you can use index cards. Write several numbers on the cards. For example: 4, 5, 4, 6, 7, 8, 19, 18, 42, 15, 10, 16, 20, 15… Have your begin by drawing 3 cards from the stack. Ask him or her if there are any outliers in the three. If you’re using the cards above in order, the answer would be no. Next have him/her draw 3 more and place them next to the original 3. Have your child look for an outlier amongst the 6. There is still no outlier. Now have your child draw a 7^{th} card. When he/she places the 19 next to the rest of the pile, he/she should recognize it as the outlier. The cards can be placed in a separate pile and the process can be repeated for the rest of the cards. It would be a good idea to incorporate cards with decimals and positive and negative numbers as you child becomes more comfortable with identifying an outlier.

(4)

It is a good idea to get your child thinking logically about problems before beginning any calculation. You can have him/her look at a data set with an outlier to make predictions as to whether the mean would increase or decrease with an outlier added in to the data. Making predictions will increase your child’s success when finding the actual difference between a data set with an outlier and one without.

(5)

Don’t forget that if you are asked to find the average of a set of data, you will need to find the mean. If you are asked to find the “typical” salary, etc… you can either find use the mean or median. You will need to think about which will give the most fair answer before you decide.

(6)

Review how to find mean, median and mode.

You can find the mean of a number, using two operations (+ then ), in three simple steps.

Add up all the numbers in the data set to calculate the sum.

Count up how many numbers are in the data set. This is your number of addends.

Divide the sum from step 1 by the number of addends from step 2.

Here’s an example of how to use the steps above for the following data set:

2, 4, 5, 6, 2, 4, 5

Add all: 2 + 4 + 5 + 6 + 2 + 4 + 5 = 28, the sum of the addends is 28.

Count the addends, only the numbers next to a + sign. There are 7 addends above.

Divide the sum, 27 by the addends, 7. 28 7 = 4

The mean is 4.

The median can be found in two steps if the total numbers in your data set are odd and three steps if even. Here’s an example of each.

Odd Data Set: 12, 15, 7, 4, 8

There are a total of 5 numbers in this data set. Since 5 is an odd number, we can find the median in two steps:

Rearrange the numbers so that they are in numerical order from least to greatest. 4, 7, 8, 12, 15

Circle the middle number. This is the median. In the set above, 8 is in the middle of two numbers on each side. So, 8 is the median.

Even Data Set: 11, 15, 9, 15, 5, 10

There are a total of 6 numbers in this data set. Since 6 is an even number, we can find the median in three steps:

Rearrange the numbers so that they are in numerical order from least to greatest. 5, 9, 10, 11, 15, 15

Circle the middle numbers. There will always be two middle numbers if the data set has an even number of data values. In this problem, 10 and 11 are in the middle.

Add the two middle numbers, 10 + 11 = 21. Then divide by 2 to find the average of these numbers, 21 2 = 10.5. The median of this data set is 10.5 or 10 ½. Note, you will always divide by two, because you are finding the average of the two middle numbers.

The mode(s) of a data set can be calculated by finding the number or numbers that are repeated most often in a data set. The mode for the data set: 2, 3, 2, 4, 2, 4, 3, 3, 2, is 2.

Find the mean, median, mode and range of the data set with and without the outlier.

230, 100, 250, 212, 216

(14)

Find the mean, median, mode and range of the data set with and without the outlier. Round the mean to the nearest tenth.

1.25, 1.5, 1.25, 1.30, 4.50, 1.80

(with: 1.9, 1.4, 1.25; without: 1.42, 1.3, 1.25)

(15)

Find the mean, median, mode and range of the data set with and without the outlier.

-12, 5, 7, 9, 5, 8, 5, 7

(with: 4.25, 6, 5; without: 7.7, 7, 5)

(16)

Saul has been keeping track of the points he’s scored in each basketball game this season in the table below. Use the data to find his average and median score for the games so far. How will these change if he scores 52 points in the next game? Round the average to the nearest one.

Game 1

Game 2

Game 3

Game 4

Game 5

Game 6

30 points

28 points

26 points

32 points

26 points

30 points

(so far: 29, 29; with 52 points: 32, 30)

(17)

Jesse’s Math scores for the second trimester are: 50, 55, 50, 65, 60, 60. What is her average score for math? What will happen to Jesse’s average if she gets a score of 100 on the next test? Write scores as the nearest percent.

(57%, 63%)

(18)

Claudia is playing a game with her sister. The table shows her scorecard for each round. What is Claudia’s score range? Median? Mode? Mean?

How will these change if she scores 150 points in the last round?

Round 1

65

Round 2

70

Round 3

70

Round 4

85

Round 5

90

(without 150: median 70, mode 70, mean 76; with 150: median 77.5, mean 70, mode 88)

(19)

Brendan has recorded the height of his puppy, Chino for the first five months he owned him. He wrote down: 1.2 ft, 1.5 ft, 1.5 ft, 1.7 ft, 2 ft. He then recorded his height again at two years as 3.5 feet. Explain how the last measurement would affect each of the measures of central tendency.

(The last height would make the mean increase.)

(20)

Monica has a summer dog-walking job. She’s recorded her earnings as: $4.50, $5.25, $5.25, $6.50, $6.25, $8.00, $7.50 and $25.00. How did her last week’s earnings affect the mean and median of the monies she earned over the summer.