Mathematics I—Semester B

Sample Lesson - Introduction

Lesson 8.08: Choosing and Using the Appropriate Centers and Spreads of Data

The meeting is 15 minutes away and we have 7 to get there. Buckle up.

(Source)

These days, pretty much everybody is in a hurry. We've overbooked ourselves to the point where a red light is just an opportunity to check our email and our hair. We just haven't got time for long, drawn out discussions of data trends. We need the basic gist of the thing, and we need it yesterday. That's where summaries of the center and spread come in handy. They're a 10-second sound bite of everything that we think is going on. If the data are a 100-page dissertation, the center and spread are the summary in Tweet form.

If we want to sum up our data in one number, we have a couple options to choose from. You've probably been introduced to these guys before. But there's more to this than just picking one at random and going with it. Each measure has a strong suit, just like how a track coach picks certain team members for a specific event every time. It doesn't make sense to have your marathoner run a sprint. In today's reading, we'll give you the lowdown on each measure’s skill set so that you can make intelligent choices about when to call them each to the plate. (Our sports metaphors got jumbled. But you know what we mean.)

Sample Lesson - Reading

Reading 8.8.08: Making the Right Choice

To find the center of a dataset, don't ask that jerk Mr. Owl. Instead, we can use the mean, median, or mode. (If you want a review of how to find them, check here.)

Mean: Mean is our default average, but it's not always the best choice. It's strongly influenced by outliers and skewing. That means that it's really only accurate for data that is nice and well behaved. And by well behaved we mean it's symmetric around the mean and it only has one hump in the data. The mean can handle large datasets easily, but if things start to get tricky, we hand things off to median.

Median: This measure takes longer to find than mean, but it isn't as influenced by skew or outlier weirdos, so it's a good choice if the data has some wonkiness to them.

Mode: The mode is the neglected third wheel for analyzing a dataset. However, for some qualitative datasets, it doesn't make much sense to take the mean or median. If you want to know what the most popular ice cream around is, for instance, you should take your data à la mode.

We also have choices when it comes to describing the spread of data. Lucky us. 

Range: Range is a nice, simple way to sum up how much variation there is in our data, but it can be influenced by outliers, just like the mean. Outliers will make the data look much more varied if we only know the range. In situations like that, interquartile range is the better choice.

Interquartile range: This is the middle 50% of the data, calculated using median. It's the stuff in the box on a box and whiskers plot. We chop off all the outliers and are left with just the good, juicy data center. Yum. It's great for when outliers are totally harshing our buzz.

We've talked a bit before about the range and interquartile range, but there's a new measure on the block, the standard deviation. You can think of it like the counterpart to the mean: super useful when we can use it, but it doesn't feel so good when outliers or skew show up.

Standard deviation: If you're wondering why we haven't mentioned this measure before if it's so useful, that's because the formula for it is a bit tough to chew through. Don't worry, we'll walk you through it.

The standard deviation, s, when we have n data points, is:

Let's take this piece-by-piece. Start with the numerator, where we have x_i and x. x_i stands for "the ith data point," while x is a fancy way of writing the mean of the data.

(x_i – x) means that we subtract the mean from every data point in our dataset. If we measured the height of 100 cocker spaniels (we were bored at the dog park, okay), then we'll have 100 values of (x_i – x).

Then we take our 100 values, square them, and add them all up (that's what that Σ means). If you don't have a calculator or a spreadsheet program, we suggest not even bothering.

After everything is added together, we divide our honking number by (n – 1). It's not quite the number of data points we have, just a liiittle short. Then we're just a quick square root away from getting the standard deviation. Congrats!

What do we get for all of our troubles? A super-accurate measure of the spread of a dataset. Hooray! Unless there are outliers or skew, in which case it gets becomes really, really inaccurate. Yarooh! (That's the opposite of hooray. Outliers and skew make us the opposite of happy).

Recap

When we want to sum up our data in just one phrase, we want to talk about the data's center and spread. Center can be measured with mean, median, or mode, and spread is measured with the range, interquartile range, or standard deviation.

Depending on how the data are shaped, some measures will be more correct than others. If our data is symmetrical, single humped, and free of outliers, the mean and standard deviation are preferred. But if our data is skewed or peppered with far-flung points, the median and interquartile range are safer choices. Mode is kind of just along for the ride, in case we need to know our most common response. He's a little bit of a third wheel.

Sample Lesson - Activity