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

How Adding Data Changes Things

Understand how additional data added to data sets may affect these computations of measures of central tendency.

Understand that the measures of central tendency can change when new data is added to a data set. Though all measures of central tendency can change, they won’t always. For example, if the mode of a set of data is 12 and another 12 is added to the data set, the mode will not change. The median is another measure that may not change. For example, if 15 was the median of a set of data with a total a 7 numbers in the data set and a 15 was added as the 8th number in the data set, the median would remain at 15. Also, the range of a data set will remain always the same, as long as the data added to the set is not larger or smaller than the data set. The mean of a data set is usually the most affected by the addition of new data.

The table below shows the miles Tisha has run for one week. Calculate the mean, median, mode and range of the data in the table. Then calculate the changes in each measure if she ran 10 miles on Saturday.

Monday

Tuesday

Wednesday

Thursday

Friday

4 miles

5 miles

5 miles

5 miles

7 miles

(mean 5.2, median 5, mode 5, range 3; mean 6, median 5, mode 5, range 6)

(2)

Alex’s science test scores are: 80, 70, 75, 82, and 68. What is his average score? How will his grade change if he receives a 95 on his next test?

(75, 78.3)

(3)

Tyson scored 150, 200, 166 and 211 points on his video game. Will the mode change if he scores 155 points next time he plays? Explain.

(No, there is no mode.)

(4)

Sasha earned $4, $5, $4, $5, $10 for washing dogs around the neighborhood. Will her median income change if she earns $5 washing the next dog? Explain.

(No, it will still be $5.)

(5)

Ian is growing a plant for a science project. He’s been keeping track of its growth in the table below. Which measure of central tendency will change the most if the plant measures 11.5 cm on the 5^{th} week?

It is important to understand the difference between the different measures of central tendency. The meaning of each and the steps needed to calculate them must be memorized. Here are some mnemonic devices to help with the memorization.

Mean: Most Exact Average Number or “It’s mean to add up all the numbers and then divide them again.”

Median: A median in the street is the center divider, it goes down the middle of the road. Likewise, a median in math is in the middle. Also, median, shares sounds with medium. A medium soda is the middle size. The median is the middle number. Visual learners may want to draw this. Median Medium

Mode: The word mode has the same beginning sound as most. The mode is the number you read the most in a data set. Mode Most

Range: Since the range is the difference between the least and the greatest, this visual may help to aid in memorization:

least-RANGE -greatest (the range is between the least and greatest, - stands for difference)

For more specific steps on how to solve each, visit the skill titled: Compute the range, mean, median, and mode of data sets.

(2)

Strong organization is needed when comparing data from an original set to additional data. Since, some measures of central tendency may change, while some may stay the same, a strong system of organizing your data will improve time management and accuracy. One system of organization is a two-column chart, as shown below.

Data Set: 10, 3, 5, 7, 5

Additional Data: 12

Original Data Set

Original Data Set with

Additional Data Added

Mean

5 (total numbers in data set)

10+3+5+7+5 = 30

30/5 = 6

Original Mean is 6.

Mean

Use the information form the original set to save time.

30 (sum from original data set)

12 (new data to be added)

6 (total numbers in data set)

30 + 12 = 52

52/6 = 8.6666666

New Mean is about 8.7

Median

3, 5, 5, 7, 10

Original Median is 5.

Median

3, 5, 5, 7, 10, 12

The median will certainly change because there are now two middle numbers that are not both 5.

Middle numbers 5 + 7 = 12

12/2 = 6

New Median is 6.

Mode

Original mode is 5.

Mode

The mode will remain the same because there are no other 12s in the original data set.

Mode stays 5.

Range

10 (largest number)

-3 (smallest number)

7 is the original range.

Range

This will change, because 12 is now the largest number.

12 – 3 = 9

New range is 9.

(3)

Use your steps and calculations from an original data set to find the new numbers. For example, if you’ve already placed all numbers in numerical order to find the median, you can use this to find the median with added data. You won’t need to rewrite all the numbers in numerical order if your work is neat. Simply, fill in the new data where it will fit numerically that then find the median. Also, when finding the mean, don’t recalculate the sum of all the numbers along with the new data. Just add the new data to the sum you got when finding the original mean. However, don’t forget that your divisor will change when you add data!

(4)

Many children forget to change their divisor when finding the mean of a data set with additional data added. This will result in a wrong answer every time. It is important for children to work slow and count out all the numbers in a given set of data. So, if my original data set is: 12, 13, 14, 3, 5, 7, there are 6 numbers in the set, so my divisor would be 6. However, if I’m asked to add the number 12 and 16 to that data set, my new divisor is 8, because I’ve added two more to the original 6.

(5)

The range should be the easiest in which to determine change. Simply take a quick glance at the additional data being added to determine if it is larger or smaller than the largest or smallest numbers in the original data set. If it is not, your range will remain the same.

Use the data in the table to find the mean, median, mode and range of Dennis’ points earned in a basketball game in the first three quarters. Then, recalculate these measures of central tendency, assuming he had scored 12 additional points in the last quarter.

Quarter 1

Quarter 2

Quarter 3

Quarter 4

12 points

4 points

10 points

(mean 8.7, median 10, no mode; mean 9.5, median 11, mode 12)

(2)

Sami received $100, $50, $80, $10, and $50 as graduation gifts. What was average gift amount given to her? How would this change if her grandpa sent her a check for $200?

($58; $81.67. It would increase by almost $24.)

(3)

Jake collected data to find number of computers in each of the eight houses on his street. His data set included: 1, 2, 3, 2, 1, 2, 1, 1. What was the mean, median, mode and range of his data set? How would these measures change if the 9^{th} house he visited had 1 computer and the 10^{th} had 4?

(1.625, 1.5, 1; 1.8, 1.5, 1; Only the median would be increased.)

(4)

Derek conducted a survey to find the number of pets for 10 of his friends. His data set was: 2, 3, 1, 4, 0, 5, 1, 2, 3, 2. What is the mode for this data set? Would it change if he asked 5 more friends and got the answers: 1, 1, 4, 2, 1? Explain.

(2; Yes, the mode would become 1.)

(5)

Derek conducted a survey to find the number of pets for 10 of his friends. His data set was: 2, 3, 1, 4, 0, 5, 1, 2, 3, 2. What is the range of this data set? Would it change if he asked three more friends who said: 5, 2, and 1? Explain.

(2.3; Yes, it would become 2.38.)

(6)

If the number 35 were added to the data set below, which would be true?

28, 24, 30, 28

The mean would decrease.

The mean would increase.

The mode would increase.

The median would increase.

(B)

(7)

Danni tracked her money earned for the month of June. Use the table below to find the mean of her monies earned. Then use the table for the monies earned in July to determine if the mean changed from June to July.

June 1

June 2

June 3

June 4

$40.50

$35.35

$12.75

$30.00

July 1

July 2

July 3

July 4

$0.00

$40.50

$40.50

$40.50

(June: 29.65; July: 40.50; She made more money on average in July.)

(8)

If the number 62 were added to the data set below, which measure of central tendency would change the most?

56, 55, 47, 50

(The median would increase the most, by going up 2.5. The mean increase by 2.)

(9)

The ages of students on Johnny’s soccer team are listed below. What measure(s) of central tendency will change the most if the coach’s son, who is 16, joins the team? Which measure(s) will change the least?

12, 12, 11, 10, 12, 13, 10, 9, 14, 12, 12, 11, 11

(The mean, and range will change. The range will change the most. The median and mode will change the least, because they won’t change at all.)

(10)

The ages of students in Angelica’s dance class are listed below. What measure of central tendency best describes the typical age of students in the class? Would this measure still work if 5 new students had the ages of 19, 18, 12, 17 and 13?

11, 11, 13, 11, 12, 12, 11, 13, 11, 12

(mode, no)

(11)

The heights of four family members are: 48 inches, 22 inches, 62 inches and 70 inches. How will the median of this data set change if a 5^{th} family member measures in at 45 inches?

(The median will decrease.)

(12)

A store manager has been keeping track of her sales of the newest video game. The first week he sold 45, the second week 58, the third 23, fourth 17, fifth 20, sixth 17, seventh, 22, and eighth 11. What is the average and median number of games sold in the past eight weeks? How will this change if she sells 4 in the ninth week and 0 in the tenth week?

(The first 8 weeks: 26.625 average and 21 is the median. Both numbers will decrease to 21.7 average and 18.5 median.)

(13)

A store manager has been keeping track of her sales of the newest video game. The first week he sold 45, the second week 58, the third 23, fourth 17, fifth 20, sixth 17, seventh, 22, and eighth 11. Which statement(s) is/are true as the weeks increase?

A) The mode will increase.

B) The mean will increase.

C) The mean will decrease.

D) The median will increase.

(c)

(14)

Find the mean, median, mode and range of the Tigers’ homerun hits.

Tigers’ Homerun Hits: 41, 25, 44, 22, 32, 33, 60.

Which measure(s) will not change if the following data was added? 33, 32, 40.

(mean 36.7, median 33, no mode, range 38. The range and median will not change.)

(15)

Find the mean, median, mode and range of the Tiger’s homerun hits.

Tiger Homerun Hits: 41, 25, 44, 22, 32, 33, 60. At the end of the season, the homerun hits: 40, 41, 32 and 30 were added. Which measure(s) of central tendency would best describe the data set with the new data added in? Explain.

(mean 36.7, median 33, no mode, range 38; The mean would best describe the data set, because there are a wide range of numbers to be averaged.)