## Three Approaches to Average Sleep

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As a new parent, sleep is one of the most important commodities in my life. Sympathetic friends often ask how much sleep I’m getting. If I want to respond with the amount of sleep I’m getting “on average” there are several different approaches I can take. Consider the following data:

The D row tracks the days being studied. The H row tracks the longest stretch on uninterrupted sleep on that day (in hours).

If we take the traditional meaning of average and take the mean, we add all the totals together and divide by the number of days. When we do that we find that I had an average of 4.44 hours of sleep per night. But that doesn’t tell the only possible story. Perhaps you can see from the data that my wife and I alternate nights on baby duty. On odd numbered nights I tended to get less sleep and on even numbered nights I tended to get more sleep. Although 4.44 is the mean, there is quite a bit of spread around that number.

A second approach to average would be to take a median. Perhaps that would give a better picture. When we take a median we simply take the middle term of the set, or when there is an even number of terms the average of the two middle terms. For this full set, that yields a median of 4.75 hours- a slightly rosier picture of my sleep situation. However, in a set like this the selection can influence the results. If we take the median of days 1-7 we get 4.5 hours and if we take the median of days 2-8 we get 5 hours. Trust me, those 30 minutes can make a lot of difference.

A third approach to finding average can be to find a mode. A mode is simply the most common number found in a set. In this case, I got 5 hours of sleep twice over this 8-day period making the mode 5. However, if we ignore all half hours and simply round everything down, we would have 2 modes: 2 and 5.

Here you see three different ways that we can discuss a data set in order to find an average or a middle. Hopefully this discussion has helped you to see how they work, how they can give different results, and what those results mean!

# Post-Holiday Discount Candy

I have a sweet tooth. I have a budget. These two facts are often at odds with each other when I’m out shopping. I was out at the drugstore the other day when I noticed the discounted Valentine’s Day candy in the heart-shaped boxes.

Now, perhaps you think it’s silly for a man to by himself candy in a heart-shaped box nearly two weeks after Valentine’s Day, but I am nothing if not a pragmatist. If I was getting the right value, I was ready to buy.

Value in the pre-holiday market is almost impossible to find. That’s because of the large mark-up on the candy due to decorative wrapping. But, as a non-sentimental pragmatist I get no value out of the wrapping. Let’s say that there is a 50% mark-up on all candy in decorative wrapping. (The actual number is probably higher, but for the sake of this example and for the sake of not sending me into a rant about inefficiency and marketing, let’s say it’s 50%). For instance, if the fair price of the candy is $2, it will be priced at$3. The question is this: if the fair value of the candy is marked up by 50%, how much does the candy need to be discounted in order for me to get a fair price?

Many of you probably defaulted to a quick answer of 50%. Let’s take a moment and see why that’s not correct.

Let’s call the fair value of the candy f. If we increase that by 50%, we get the original price at the store: 1.5f. Now, if the store decides to put candy on sale for 50% off of the sales price, you end up paying 1.5f * 0.5 = 0.75f. So, a sale of 50% actually allows you to pay a price below the fair price… a good deal! If we want to figure out the discount we need from the store in order to get a fair price, we use the following equation:

1.5f * (1-y) = f

1.5f – 1.5fy = f

1.5fy = 0.5f

1.5y = 0.5

y = 1/3

So, if the candy is 1/3 off, or approximately 33% off, you’re getting a fair price. This answer that may not have been intuitive comes from the fact that in a percent change problem you always need to identify your base price. The first time we increase the price by 50%, we’re using the fair price as our base. When we decrease the price by 50%, we’re using the store price as our base. That’s a big difference and something to keep in mind the next time you face a percent change problem… or go to buy some post-holiday candy!

# Point-saving Fact of the Day

Let’s take a moment to talk today about something that comes up on every single test that I see. Here is a simple, yet extremely useful fact: two is the smallest and only even prime number. That may not seem like some mind-blowing revelation, and yet it’s a concept that’s tested in many different ways. Let’s explore a few of them so that you’ll be ready when you see it on your test.

First take the following example: X and Y are unique prime numbers whose product is Z. It’s a fairly innocuous looking sentence, and yet our simple fact of the day can help unpack it quite a bit. This, like many problems that you’ll encounter requires a basic understanding of number properties. If, for instance we found out that Z was an even number, we would then know that either X or Y must be two. We would know this because and even product requires that one or both of Z’s factors be even. In this particular case we know that the only way for X or Y to be both prime and even is for them to be equal to two. We further know that since X and Y are unique prime numbers, only one of them could be equal to two. So, this first example shows how a relatively small piece of information about Z, combined with this basic fact that two is the smallest and only even prime number can yield quite a bit of help in a short time.

Another example where this fact comes into play is the rather straightforward situation where you find that P is a prime number less than 38. It’s the sort of situation that you might encounter three or four or even more times on your test. When considering prime numbers are first instinct is to look for odd numbers, and as you count off the primes or lists them on the page you need to be very careful and very intentional about remembering to include two. I’m often amazed, and usually horrified, when I watch my students to practice problems of this type. Everything will be going along smoothly, every step is executed to perfection, and then we get to the very last thing that needs to be done before the correct answer and it all goes wrong. “So our primes are 37, 31, 29… 7, 5, and 3.” All that work goes down the drain at the very last minute, and the correct answer is off by one. You can get so caught up in this pattern of all primes being odd that you miss the number two.

Don’t let it happen to you! You can avoid this simple mistake by simply forcing yourself to focus on today’s fact: two is the smallest and only even prime number. Keep that in mind and save yourself some points on test day.

# Mental Math Magic

I have amazing magical powers. Well, not really, but that’s what many of my students seem to think when we start working together. Why do they think I have these amazing magical powers? It’s simply due to the fact that I can do relatively large math operations in my head. I promise you, it’s not magic and I can teach you the basics of my technique today and also why these techniques are useful.

What is 324*96? Don’t panic. We can figure this out. If you’re asking yourself why we should bother, that may be because your test allows you to use a calculator. If that your situation, obviously mental math is going to be less useful to you, but that doesn’t mean you should tune out. By taking the first steps in this process you’ll be able to understand the approximate value that you seek and more easily discover any data entry errors you have with the calculator.

So, back to our problem. When we look at 324*96 we see a difficult-looking problem, but to start let me ask a different question: about how much is 324*96? The answer there is a little less than 32,400 which is 324*100. That’s easy. We’re multiplying by 100, which just means adding a couple zeroes to the end of the number. But we’re looking for something a little bit better than about, so let’s move to the next stage. To get to an exact answer we can re-state our problem like this:

$324*96=\left ( 324*100 \right )-\left ( 324*4 \right )$

So, after starting with our estimation, if we can figure out the value of 324*4 we’ll be left with a straightforward subtraction problem. To do that, we’ll break down the problem again. Let’s start with 24*4. I know that 25*4 is 100, so 24*4 must be 4 less or 96. That just leaves 300*4, which is 1,200. Very quickly that gets us to 1,296.

Now we’re down to a subtraction problem: 32,400 – 1,296. Again, let’s simplify these numbers. Instead of subtracting 1,296, let’s subtract 1,300 and then add 4. 32,400 – 1,300 = 31,100. Add our 4 back in and we get our answer 31,104.

I know there are a lot of steps there, and that’s by no means an easy example, but hopefully you can see how these methods are useful. First, they give us an idea of about how big the answer is going to be. Second, they break down one big complicated problem into several smaller more manageable problems. Practice these techniques and you’ll be way on your way to your own mental math magic.