Naked Statistics: Stripping the Dread from the Data

by Charles Wheelan

Therein lies the insight: Even though you will continue moving forever—with each move taking you half the remaining distance to the wall—the total distance you travel can never be more than 2 feet, which is your starting distance from the wall. For mathematical purposes, the total distance

And when I do, it will probably make sense. In my experience, the intuition makes the math and other technical details more understandable—but not necessarily the other way around.

end, I hope to persuade you of the observation first made by Swedish mathematician and writer Andrejs Dunkels: It’s easy to lie with statistics, but it’s hard to tell the truth without them.

I am a Chicago Bears fan. During the 2011 playoffs,

something like the Gini index, which is a standard tool in economics for measuring income inequality. I’ll

Is the Gini index the perfect measure of inequality? Absolutely not—just as the passer rating is not a perfect measure of quarterback performance. But it certainly gives us some valuable information on a socially significant phenomenon in a convenient format.

Descriptive statistics exist to simplify, which always implies some loss of nuance or detail.

Probability is one weapon in an arsenal that requires good judgment.

Regression analysis is the tool that enables researchers to isolate a relationship between two variables, such as smoking and cancer, while holding constant (or “controlling for”) the effects of other important variables, such as diet, exercise, weight, and so on.

(1) quantify the association observed between eating bran muffins and contracting colon cancer (e.g., a hypothetical finding that people who eat bran muffins have a 9 percent lower incidence of colon cancer, controlling for other factors that may affect the incidence of the disease); and (2) quantify the likelihood that the association between bran muffins and a lower rate of colon cancer observed in this study is merely a coincidence—a quirk in the data for this sample of people—rather than a meaningful insight about the relationship between diet and health.

Regression Analysis

statistical tools to answer important social

“statistically significant”

affluent have the strongest incentive to change society. These individuals may also be particularly rankled by suppression of freedom, another factor associated with terrorism. In Krueger’s study, countries with high levels of political repression have more terrorist activity (holding other factors constant).

The point is to learn things that inform our lives.

Alan Krueger’s study of terrorists did not follow thousands of youth over multiple decades to observe which of them evolved into terrorists. It’s just not possible.

Researchers did a large-scale study on whether or not prayer reduces postsurgical complications, which was one of the questions raised earlier in this chapter. That study cost $2.4 million. (For the results, you’ll have to wait until Chapter 13.)

We conduct statistical analysis using the best data and methodologies and resources available.

How to Lie with Statistics, which was first published in 1954

“middle” of a set of data, or what statisticians might describe as its “central tendency.”

For distributions without serious outliers, the median and the mean will be similar.

Because the distribution is nearly symmetrical, the mean and median are relatively close to one another. The

What becomes clear is that your firm does not have a uniform quality problem; you have a “lemon” problem;

The benefit of these kinds of descriptive statistics is that they describe where a particular observation lies compared with everyone else. If I tell you that your child scored in the 3rd percentile on a reading comprehension test, you should know immediately that the family should be logging more time at the library.

Here is a good point to introduce some useful terminology. An “absolute” score, number, or figure has some intrinsic meaning. If I shoot 83 for eighteen holes of golf, that is an absolute figure. I may do that on a day that is 58 degrees,

the standard deviation, which is a measure of how dispersed the data are from their mean. In other words, how spread out are

The standard deviation is the descriptive statistic that allows us to assign a single number to this dispersion around the mean. The

“The standard deviation for the HCb2 count is 18,” the technician informs you curtly. What the heck

There is natural variation in the HCb2 count, as there is with most biological phenomena (e.g., height). While the mean count for the fake chemical might be 122, plenty of healthy people have counts that are higher or lower. The danger arises only when the HCb2 count gets excessively high or low.

Data that are distributed normally are symmetrical around their mean in a bell shape that will look familiar to you.

According to the Wall Street Journal, Americans even tend to park in a normal distribution at shopping malls; most cars park directly opposite the mall entrance—the “peak” of the normal curve—with “tails” of cars going off to the right and left of the entrance.

The beauty of the normal distribution—its Michael Jordan power, finesse, and elegance—comes from the fact that we know by definition exactly what proportion of the observations in a normal distribution lie within one standard deviation of the mean (68.2 percent), within two standard deviations of the mean (95.4 percent), within three standard deviations (99.7 percent), and so on.

The mean is the middle line which is often represented by the Greek letter µ.

In both the sodium and the income examples, we’re missing context.

Assume that a department store is selling a dress for $100. The assistant manager marks down all merchandise by 25 percent. But then that assistant manager is fired for hanging out in a bar with Bill Gates,* and the new assistant manager raises all prices by 25 percent. What is the final price of the dress? If you said (or thought) $100, then you had better not skip any paragraphs.

The final price of the dress is actually $93.75. This is not merely a fun parlor trick that will win you applause and adulation at cocktail parties.

The formula for calculating a percentage difference (or change) is the following: (new figure – original figure)/original figure. The numerator (the part on the top of the fraction) gives us the size of the change in absolute terms; the denominator (the bottom of the fraction) is...

The point is that a percentage change always gives the value of some figure relative to something else.

Illinois personal income tax, which was raised from 3 percent to 5 percent. There are two ways to express this tax change, both of which are technically accurate. The Democrats, who engineered this tax increase, pointed out (correctly) that the state income tax rate was increased by 2 percentage points (from 3 percent to 5 percent). The Republicans pointed out (also correctly) that the state income tax had been raised by 67 percent. [This is a handy test of the formula from a few paragraphs back: (5 – 3)/3 = 2/3, which rounds up to 67 percent.]

The advantage of any index is that it consolidates lots of complex information into a single number.

Alas, the disadvantage of any index is that it consolidates lots of complex information into a single number.

(He comes down particularly hard on the college rankings.)

Any index is highly sensitive to the descriptive statistics that are cobbled together to build it, and to the weight given to each of those components.

The HDI provides a handy and reasonably accurate snapshot of living standards around the globe.

answer. To assess the economic health of America’s “middle class,” we should examine changes in the median wage (adjusted for inflation) over the last several decades. They also recommended examining changes to wages at the 25th and 75th percentiles (which can reasonably be interpreted as the upper and lower bounds for the middle class).

The variance, which is often represented by the symbol σ2, is calculated by determining how far the observations within a distribution lie from the mean.

Variance is rarely used as a descriptive statistic on its own. Instead, the variance is most useful as a step toward calculating the standard deviation of a distribution, which is a more intuitive tool as a descriptive statistic.

We ought to begin with the crucial distinction between “precision” and “accuracy.”

Accuracy is a measure of whether a figure is broadly consistent with the truth—hence the danger of confusing precision with accuracy. If an answer is accurate, then more precision is usually better. But no amount of precision can make up for inaccuracy.

Joseph McCarthy, the Red-baiting senator from Wisconsin, reached the apogee of his reckless charges in 1950 when he alleged not only that the U.S. State Department was infiltrated with communists, but that he had a list of their names. During a speech in Wheeling, West Virginia, McCarthy waved in the air a piece of paper and declared, “I have here in my hand a list of 205—a list of names that were made known to the Secretary of State as being members of the Communist Party and who nevertheless are still working and shaping policy in the State Department.”1 It turns out that the paper had no names on it at all, but the specificity of the charge gave it credibility, despite the fact that it was a bald-faced lie.