数学英语 23 What are the Range and Standard Deviation?

by Jason Marshall

We’ve now talked about three methods for calculating average values: the mean, median, and mode (affectionately known as the three Ms). So, are we done? Is that everything we could ever want to know about interpreting average values? No, there’s a bit more to it. Today, we’re going to talk about two values that complement¹ averages and help us interpret what they mean: range and standard deviation².
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Do Average Values Tell Us Everything We Need to Know?
Why don’t average values alone tell us everything we need to know about a set of data? Let me answer this question with an example. Imagine you’ve been given the job of comparing the performances of students in high school math classes at two competing schools. Much to the disappointment of the administration at both schools (they’re always looking for good news to boast about), you find that the average math scores are identical for both schools.
Naïvely, it’s tempting⁴ to conclude from this that the students at the two schools are performing similarly. But you, being a clever and mathematically informed individual, decide to look a little further into the scores of the individual students. You find that the five math students at the first school (yes, it’s a very small school) received scores of 40, 55, 70, 85, and 100; while the five math students at the second school received scores of 68, 69, 70, 71, and 72. Both of these schools, therefore, have mean and median math scores of 70 (see the articles on how to calculate mean values and how to calculate median values for more information on these two averages), but the individual scores are very different. It’s clear that any fair and complete comparison of the students at these two schools will require more than a simple calculation of average values.
What is the Range?
Let’s start digging deeper by looking at the most obvious difference between the scores of the students from the two schools. Namely, the scores from the first school range between 40 and 100, whereas those from the second school range between 68 and 72. My use of the word “range” here is no coincidence—it’s a word we use in English, but it also has a mathematical definition: the range is simply the difference between the highest value and the lowest. So, for the first school, the range is 100 – 40 = 60; whereas for the second school, the range is 72 – 68 = 4. The range is the crudest way to estimate the spread of the data, also known as the dispersion. In other words, it tells us how much variation there is in the scores. Student scores at the first school are therefore extremely varied—two students failed (with scores of 40 and 55), and one student got a perfect score of 100. In contrast to this, the five students at the second school—with scores ranging between 68 and 72—all performed very similarly.
Okay, so are we done now? Does the combination of the average value and range tell us everything we need to know to fairly compare the two sets of data? For our particular example here, it probably does. But this example is a little ideal and oversimplified since most classes have more than five students in them. Imagine instead that the class at the first school had 30 students in it instead of 5, and that 28 of these 30 students had scores ranging between 66 and 85. This leaves us with two other students—one scored a rather miserable⁵ 15 and the other scored a perfect 100. The mean value of the scores that ranged between 15 and 100 could still be 70, but the range would now be 85! That in spite of the fact that the range of scores of 28 of the 30 students (that’s about 93% of them) is only 19. So, in this more realistic scenario⁶, with a range of 85, the range does not provide a good measure of the typical spread in scores.
What is the Standard Deviation and How is it Calculated?
And that’s exactly where the standard deviation comes in. In a nutshell, the standard deviation provides an approximate measure of the mean distance between each data point and the mean of all the data points. I know this can be a mouthful to digest at first, so let’s take a minute to walk through the calculation slowly. Let me first say this isn't going to be the exact method for calculating the true statistical⁷ standard deviation. The real calculation is a little too complex to cover in the time we have right now, but if you'd like to see a demonstration⁸ of this real method, check out this week's Math Dude "Video Extra!" epi-sode on YouTube. In the meantime, here's an approximate calculation that gives the idea of what the standard deviation tells you (we'll call it the quasi-standard deviation). For this example, let’s look at the scores of the five students from the second school: 68, 69, 70, 71, and 72. Here’s the quick and dirty tip for calculating the approximate quasi-standard deviation of these five scores. First, the mean value of the five scores is 70, and the quasi-standard deviation is the mean distance between each data point and this mean value of 70. Now, the distance from both 68 and 72 to 70 is 2, the distance from both 69 and 71 to 70 is 1, and the distance between 70 and 70 is, of course, 0 (ignore positive and negative values here—think of it as being interested in the distance, not the direction). Now, all we have to do is find the mean of these five distances: 2 + 1 + 0 + 1 + 2 = 6, so the approximate quasi-standard deviation of the data set—in this case test scores—is therefore 6 / 5 = 1.2
What Does the Approximate Quasi-Standard Deviation Mean?
But what does the approximate quasi-standard deviation mean? Well, for comparison with the value we just calculated, let’s quickly calculate the approximate quasi-standard deviation of the scores in the first class—the one with the widely varying scores of 40, 55, 70, 85, and 100. The mean is again 70, and the distance from each data point to this mean is 30, 15, 0, 15, and 30. The approximate quasi-standard deviation is then just the mean of these five numbers, which is 18. So now, what do these two approximate quasi-standard deviations⁹ tell us? Well, as with the range, the approximate quasi-standard deviation measures how much variation there is in the data. But whereas the range tells us the total amount of variation, the approximate quasi-standard deviation tells us the average variation; and, therefore, the value of the approximate quasi-standard deviation won’t be skewed nearly as much by a few extreme data points. In the end, the approximate quasi-standard deviations for the two schools confirm in a well-defined way that the scores from the first school vary a lot more on average than those from the second.
Wrap Up
Okay, that’s all the math we have time for today. Needless to say, there’s a lot more to talk about. Next time we’ll put a dent³ in that list by looking at some ways that the standard deviation is used in science and politics—applications that are sure to have an impact on your life.
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Please email your math questions and comments to。。。。。。You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.
Finally, if you like what you’ve read and have a few minutes to spare, I’d greatly appreciate your review on iTunes. And while you’re there, please subscribe¹⁰ to the podcast to ensure you’ll never miss a new episode.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
 

1 complement   

n.补足物，船上的定员；补语；vt.补充，补足

参考例句：

• The two suggestions complement each other.这两条建议相互补充。
• They oppose each other also complement each other.它们相辅相成。

2 deviation   

n.背离，偏离；偏差，偏向；离题

参考例句：

• Deviation from this rule are very rare.很少有违反这条规则的。
• Any deviation from the party's faith is seen as betrayal.任何对党的信仰的偏离被视作背叛。

3 dent   

n.凹痕，凹坑；初步进展

参考例句：

• I don't know how it came about but I've got a dent in the rear of my car.我不知道是怎么回事，但我的汽车后部有了一个凹痕。
• That dent is not big enough to be worth hammering out.那个凹陷不大，用不着把它锤平。

4 tempting   

a.诱人的, 吸引人的

参考例句：

• It is tempting to idealize the past. 人都爱把过去的日子说得那么美好。
• It was a tempting offer. 这是个诱人的提议。

5 miserable   

adj.悲惨的，痛苦的；可怜的，糟糕的

参考例句：

• It was miserable of you to make fun of him.你取笑他，这是可耻的。
• Her past life was miserable.她过去的生活很苦。

6 scenario   

n.剧本，脚本；概要

参考例句：

• But the birth scenario is not completely accurate.然而分娩脚本并非完全准确的。
• This is a totally different scenario.这是完全不同的剧本。

7 statistical   

adj.统计的，统计学的

参考例句：

• He showed the price fluctuations in a statistical table.他用统计表显示价格的波动。
• They're making detailed statistical analysis.他们正在做具体的统计分析。

8 demonstration   

n.表明，示范，论证，示威

参考例句：

• His new book is a demonstration of his patriotism.他写的新书是他的爱国精神的证明。
• He gave a demonstration of the new technique then and there.他当场表演了这种新的操作方法。

9 deviations   

背离，偏离( deviation的名词复数 )； 离经叛道的行为

参考例句：

• Local deviations depend strongly on the local geometry of the solid matrix. 局部偏离严格地依赖于固体矩阵的局部几何形状。
• They were a series of tactical day-to-day deviations from White House policy. 它们是一系列策略上一天天摆脱白宫政策的偏向。

10 subscribe   

vi.(to)订阅，订购；同意；vt.捐助，赞助

参考例句：

• I heartily subscribe to that sentiment.我十分赞同那个观点。
• The magazine is trying to get more readers to subscribe.该杂志正大力发展新订户。