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!
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!
The best stories are often the improbable ones. At the Opening Ceremonics of the 2014 Olympic Games in Sochi, Russia, Vladimir Putin had the best seat in the house. There is nothing improbable about this. Putin is the President of Russia, and arguably the most powerful man in the world. But seated next to Putin was Irina Skvordsova. Her place in that seat was very improbable.
Skvordsova was a Russian bobsledder who had dreams of attending the Opening Ceremonies as an athlete, marching into the stadium with her teammates under the Russian flag to the applause of her countrymen. A training accident in November 2009 changed everything. Skvordsova’s bobsled overturned on a training run in Germany—a reasonably common occurrence in a sport where speeds are pushed right to the edge. Then things got bad. The German track official responsible for starting the teams was unaware of the crash down the track and gave the go ahead to the next sled to start down the course.
The loaded bobsled—which travels between 50-90 miles per hour in competition—barreled into Skvordsova as she lay on the track. She was rushed to a hospital with injuries that were called “incompatible with life”. She was put in an induced coma as doctors performed surgery after surgery in an attempt to save her life.
Somehow Skvordsova survived. She will never be able to bobsled again, but she was able to regain enough strength to continue a somewhat normal life. Normal, that is, until she was selected to take the seat of honor next to Putin at the Opening Ceremonies as she was repeatedly captured on TV broadcasts that beamed around the world.
The day of the crash life seemed an optimistic goal for Skvordsova. Now, she has a platform that seemed unthinkable.
The best stories are often the improbable ones. Maybe you’re starting from a really low starting score. Maybe you have major obstacles that you need to overcome in order to achieve your goals. Hard work and perseverance won’t make your success likely, but it will make success possible.
And just think of the story you’ll have.
For much of the country, the weather outside is awful. I’m fortunate enough to live in a place with very mild winters, but I did spend a few years in a more snowy clime so I understand how many of you are feeling. Perhaps you’ve even had a few snow days recently. It must be a nice feeling to know that rather than needing to slog to school, you have the opportunity to climb back under the warm covers in sleep a little longer. Or maybe instead you can catch up on that TV series or that videogame. Because it’s a free day, right?
There’s a concept in economics called opportunity cost. It says that when making decisions about what to buy you have to consider what you could have bought instead had you not made that purchase. We can think about our time in much the same way. A snow day might seem like a free day, but it’s also a day with very low opportunity cost. Anything that she would have done outside is basically off the table, so your options are already limited. What a great time to break out the study books!
You don’t have to spend all day with your nose in a book, by getting your studying done now means they’ll be less to do when the sun is out, or when your friends want to go out, or when you just rather go to bed.
Sure, a snow day is a free day, but don’t waste all your freedom! Using an hour or two productively now can free up hours down the road that you’d much rather use other ways.
So, if you do end up with more snow days this winter think about grabbing your tablet to watch some Barron’s test prep videos on the way back to that warm bed.