Finding the Averages of the Ordinary
Toothpaste is something we use everyday (or at least, I hope you do)! But did you know that there are many different kinds of toothpastes that are available to us? We will learn how to use data collected by 177 people to find the averages (or most common) and plot this data on histograms and different charts.Standard
CCSS.Math.content.6.SP.B.5.A: Reporting the number of observations
CCSS.math.content.6.SP.B.4: Display numerical data in plots on a number line, including dot plots, histogram and box plots.
Where did we first learn about averages?
The Greek historian, Herodotus, explained in his analysis of the of Egyptian King, that three generations are roughly 100 years! While we now know that this is accurate, between 485-420 BC, this was not common knowledge. We know that a generation can be about 100 years, give or taken a few years, but this is a great representation of averages.
Herodotus explained, "They declare that three hundred and forty-one generations separate the first king of Egypt from the last mentioned (Hephaestus) - and that there was a king and high priest corresponding to each generation. Now reckon three generations as a hundred years, three hundred generations make ten thousand years, and the remaining forty-one generations make 1,340 years more; thus one gets a total of 11,340 years...".
What can you gather from the above statement? We look at the first king of Egypt and the last mentioned (those are the qualifier, the first and the last and everyone in between). Then we look at the number of 340 generations separating the first king and the last.
Math Antics
Let's watch this video on Mean, Mode and Median to get a better understanding!
Introduction of Data
I polled 178 people in which is their favorite toothpaste. Here is the collection of data
| Favorite Toothpaste | Votes |
| Crest | 76 |
| Colgate | 63 |
| Rembrandt | 1 |
| Aim | 2 |
| Tom's | 7 |
| Arm & Hammer | 1 |
| Sensodyne | 24 |
| Jason | 1 |
| Schmidt's | 1 |
| Ultrabrite | 1 |
| Aquafresh | 1 |
| Total | 178 |
Finding the Mean
Now, let's find the mean. We already have the total number of votes, which is 178. How did we find this? We found this by adding all of the votes together. Next, we must divide the total by the number of items, which is 11. Our equation is as follows:
178 / 11 = 16.18
This means our mean is 16.18, or the average number we have is 16.18.
Finding the Mode
The mode is the number that appears the most often in a number set. Can you guess what number appears the most often? First, we must put our numbers in numerical order.
1, 1, 1, 1, 1, 1, 2, 7, 24, 63, 76
Let's see which number appears the most in our list:
Number: Amount in which it appears:
1 6
2 1
7 1
24 1
63 1
76 1
This list demonstrates that the number 1 is our mode.
Finding the Median
The median is the number in the middle. This of a 4 lane highway. What do we call the the grass or concrete in the middle of those 4 lanes? We call it a median.
It is the same thing for a list of numbers. Whatever the middle number in a number set is the median. Let's look back at our list of favorite toothpastes.
1, 1, 1, 1, 1, 1, 2, 7, 24, 63, 76
Because we already know there are 11 numbers in this particular number set, we would find the number after the fifth number, which is sixth number.
1, 1, 1, 1, 1, 1, 2, 7, 24, 63, 76
1 is our Median.
It is important to remember that mean, mode and median are NOT the same thing. Let's review our numbers.
Mean: 16.18
Mode: 1
Median: 1
Graphing
When you look at data in the future, you will see graphs and will be able to construct the same mean, mode and median from bar graphs and histograms and even pie charts. We will dive further into pie charts next week. Using the same data we collected before, we come up with these graphs:
This is a bar graph. It has all the same numbers as before, but it looks a little bit cooler!
Next we have a pie chart. This is pretty neat looking as well, just a little more crowded due to the number of toothpastes we used.
And lastly, we have a line graph. I would like you to create your own, but here is an example.
Tonight, I want you to look at the latest Detroit Tiger's Batting Average and find the mean, mode and median. You will be able to find the team's total average (mean), the median (remember, the middle number) and the most common number is the number set. You will be able to do this with any batting average for any baseball team from now on.
| Player Name | Batting Average |
| Miguel Cabrera | 0.323 |
| Nicholas Castellanos | 0.319 |
| John Hicks | 0.294 |
| Leonys Martin | 0.29 |
| Jeimer Candelario | 0.277 |
| James McCann | 0.273 |
| Victor Martinez | 0.26 |
| Grayson Greiner | 0.25 |
| JaCoby Jones | 0.245 |
| Jose Iglesias | 0.22 |
| Dixon Machado | 0.203 |
| Niko Goodrum | 0.189 |
| Victor Reyes | 0.143 |
| Mikie Mahtook | 0.129 |
Detroit Tigers 2018 Batting/Hitting Scores. (n.d.). Retrieved May 07, 2018, from http://www.espn.com/mlb/team/stats/batting/_/name/det/detroit-tigers
Egyptian King Picture. (n.d.). Retrieved May 07, 2018, from https://klimtlover.files.wordpress.com/2012/09/death-mask.jpg
Grade 6 » Statistics & Probability. (n.d.). Retrieved May 07, 2018, from http://www.corestandards.org/Math/Content/6/SP/
Math Antics - Mean, Median and Mode. (2017, March 03). Retrieved May 07, 2018, from https://youtu.be/B1HEzNTGeZ4
The Early History of Average Values and Implications for Education. (n.d.). Retrieved May 07, 2018, from http://ww2.amstat.org/publications/jse/v11n1/bakker.html
Understanding a question is half an answer - ancient Greek philosopher.. (n.d.). Retrieved May 07, 2018, from https://www.123rf.com/photo_41982954_understanding-a-question-is-half-an-answer-ancient-greek-philosopher-socrates-quote-printed-on-grung.html
Egyptian King Picture. (n.d.). Retrieved May 07, 2018, from https://klimtlover.files.wordpress.com/2012/09/death-mask.jpg
Grade 6 » Statistics & Probability. (n.d.). Retrieved May 07, 2018, from http://www.corestandards.org/Math/Content/6/SP/
Math Antics - Mean, Median and Mode. (2017, March 03). Retrieved May 07, 2018, from https://youtu.be/B1HEzNTGeZ4
The Early History of Average Values and Implications for Education. (n.d.). Retrieved May 07, 2018, from http://ww2.amstat.org/publications/jse/v11n1/bakker.html
Understanding a question is half an answer - ancient Greek philosopher.. (n.d.). Retrieved May 07, 2018, from https://www.123rf.com/photo_41982954_understanding-a-question-is-half-an-answer-ancient-greek-philosopher-socrates-quote-printed-on-grung.html
Detroit Tigers 2018 Batting/Hitting Scores. (n.d.). Retrieved May 07, 2018, from http://www.espn.com/mlb/team/stats/batting/_/name/det/detroit-tigers
Egyptian King Picture. (n.d.). Retrieved May 07, 2018, from https://klimtlover.files.wordpress.com/2012/09/death-mask.jpg
Grade 6 » Statistics & Probability. (n.d.). Retrieved May 07, 2018, from http://www.corestandards.org/Math/Content/6/SP/
Math Antics - Mean, Median and Mode. (2017, March 03). Retrieved May 07, 2018, from https://youtu.be/B1HEzNTGeZ4
The Early History of Average Values and Implications for Education. (n.d.). Retrieved May 07, 2018, from http://ww2.amstat.org/publications/jse/v11n1/bakker.html
Understanding a question is half an answer - ancient Greek philosopher.. (n.d.). Retrieved May 07, 2018, from https://www.123rf.com/photo_41982954_understanding-a-question-is-half-an-answer-ancient-greek-philosopher-socrates-quote-printed-on-grung.html
