Mean or Median?
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[Originally posted in the Discussions forum]
Module 0 Week 2 Day 5 Your Turn Explanation Part 1
When he described how 1 plus 2 plus 3... plus 9 was the same as “finding the average and How many of them there are”, did he mean median instead of average?
Hi there,
Professor Loh meant the average! Here's why: When you find the average of a bunch of numbers, you add them all up and then divide by how many numbers there are. So for example, (1+2+3+4+5+6+7+8+9) / 9 = 5, where 5 is the average. But if you simply move the 9 over, you get that 1+2+3+4+5+6+7+8+9 = 9*5. So, what Professor Loh is saying is that the sum of the numbers is the number of numbers times the average of the numbers.
Coincidentally, if my numbers form an arithmetic sequence (So that the difference between every two numbers next to each other is the same), then the median is the same as the mean (or average). So, if it's easier you can think of it as the median too.
I hope that helped. Happy learning!
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Hi,
I understand that the mean is the sum of the numbers divided by how many numbers there are in total. Is there a more intuitive explanation of what average actually means?
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@ingeniousosprey Yes! Means (averages) are everywhere, and we don't always notice that they're there.
Suppose you're watching the news, which is discussing the ages of children who get sick from a certain disease in various countries. If it were to simply tell you the age of every kid who gets sick, it might look something like this:
Hmm... What can this information tell us, and how useful is it? The graphs tell us that in Country A, a 10-year-old got sick, a 4-year-old got sick, an 8-year-old got sick, a 1-year-old got sick, a 10-year-old got sick (not the same one as before), and a 3-year-old got sick. This might be useful if you're a teacher keeping track of every single one of your students. However, what if you want to compare the three countries to look for a general trend or pattern? Or, imagine if instead of six kids per country, you had 1 million kids per country? How could you even fit all of that information onto one page? That's where means and averages come in!
If we could just replace each of these "towers" with different towers that are easier to visualize, but still give the same data overall, that would be great. Let's try looking at Country A. The ages of all the kids in Country A, added up together, is
$$ 10 + 4 + 8 + 1 + 10 + 3 = 36 $$
There were \(6\) kids in Country A, so we could pretend that each kid was actually \(\frac{36}{6} = 6\) years old. Let's draw the towers again, replacing them with an army of identical towers, each \(6\) tall:
Why do the towers look sort of angry? Because they're mean towers!
Sometimes I like to conjure up the image here of a ferocious, mean-looking army of soldiers, all looking identical.
But... wait a second. There's a problem with that picture. Do you see it? There are only 5 towers. Oops! There are supposed to be 6 towers, because we can directly replace each of the previous kids with a 6-year-old kid, like this:
Great, that's better. So, remember, the number of mean towers should always be equal to the number of data points that we had before. This is the motivation for the equation
$$ \text{ mean } = \frac{\text{total amount}}{\text{number of things}} $$
Now, using the mean, we can see that the
$$ \text{average of ages of sick kids in Country B } = \frac{7 + 1 + 6 + 1 + 8 + 1}{6} = \frac{24}{6} = 4 $$
And we could replace the sick kids in Country B with a set of mean towers, each with height 4, like this:
Finally, for Country C
$$ \text{ average of ages of sick kids in Country C } = \frac{10 + 6 + 8 + 4 + 12 + 8}{6} = \frac{48}{6} = 8 $$
And we could replace each of the sick kids in Country C with an army of mean towers, each of height 8:
Now, if you look in the aggregate, it's much easier to see that in Country C, on average, the sick kids are older, and, on average, the sick kids in Country B are younger:
I hope this helps! This was a great question, and one of my favorites to answer. I love drawing the mean faces on all of the identical mean towers.
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Thank you very much. The answer was very useful. There might be a typo in the fourth paragraph and you need to replace There were "6 kids in Country C". I think you might have meant country A (the country that has an average of 6).
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@ingeniousosprey Thanks for letting me know about the typo; I've fixed it!
