There’s Something About Variance

Imagine practicing hitting a target using darts, bow and arrow, pistol, cannon, missile launcher, or whatever. You aim for the center of the target. If your shots land where you aimed, you are considered to be accurate. If all your shots land near each other, you are considered to be precise. The two properties are not linked. You can be accurate but not precise, precise but not accurate, neither accurate nor precise, or both accurate and precise.

Accuracy and precision also apply to statistics calculated from data. If you’re trying to determine some characteristic of a population (i.e., a population parameter), you want your statistical estimates of the characteristic to be both accurate and precise.

The same also applies to the data themselves. When you start measuring data for an analysis, you’ll notice that even under similar conditions, you can get dissimilar results. That lack of precision
is called variability. Variability is everywhere; it’s a normal part of life. In fact, it is the spice in the soup. Without variability, all wines would taste the same. Every race would end in a tie. Even statistics might lose its charm. Your doctor wouldn’t tell you that you have about a year to live, he’d say don’t make any plans for January 11 after 6:13 pm EST. So a bit of variability isn’t such a bad thing. The important question, though, is what kind of variability?

The Inevitability of Variability

Before going further, let me clarify something. Statisticians discuss variability using a variety of terms, including errors, uncertainty, deviations, distortions, residuals, noise, inexactness, dispersion, scatter, spread, perturbations, fuzziness, and differences. To nonprofessionals, many of these terms hold pejorative connotations. But variability isn’t bad … it’s just misunderstood.

Suppose you’re sitting in your living room one cold winter night contemplating the high cost of heating oil. The thermostat reads 68 degreesF, but you’re still shivering. Maybe the thermostat is broken. Maybe the heater is malfunctioning or you need more insulation. You need a warmer place to sit while you read An Inconvenient Truth, so you grab a thermometer from the medicine cabinet and start measuring temperatures around the room. It’s 115 degrees at the radiator, 68 degrees at your chair, 59 degrees at the window, and 69 degrees at the stairs. You keep measuring. It’s 73 degrees at the fish tank, 67 degrees at the couch and bookcase, 82 degrees at the TV, and 60 degrees at the door. That’s a lot of variation!

Think of those temperature readings as the summation of five components:

  • Characteristic of Population—the portion of a data value that is the same between a sample and the population. This part of a data value forms the patterns in the population that you want to uncover. If you think of the living room space as the population you’re measuring, the characteristic temperature would be the 68 degrees at your chair where you want to read.
  • Natural Variability—the inherent differences between a sample and the population. This part of a data value is the uncertainty or variability in population patterns. In a completely deterministic world, there would be no natural variability. You would read the same value at every point where you took a measurement. But in the real world, if you made the same measurement again and again, you probably would get different values. If all other types of variation were controlled, these differences would be the natural or inherent variability.
  • Sampling Variability—differences between a sample and the population attributable to how uncharacteristic (non-representative) the sample is of the population. Minimizing sampling error requires that you understand the population you are trying to evaluate. The sampling variability in the living room would be attributable to where you took the temperature readings. For example, the radiator and TV are heat sources. The door and window are heat sinks. Furthermore, if all the readings were taken at eye level, the areas near the ceiling and floor would not have been adequately represented. The floor may be a few degrees cooler because the more dense cold air sinks displacing the warmer air upward, which is why the air at the ceiling is warmer.
  • Measurement Variability—differences between a sample and the population attributable to how data were measured or otherwise generated. Minimizing measurement error requires that you understand measurement scales and the actual process and instrument you use to generate data. Using an oral thermometer for the living room measurements may have been expedient but not entirely appropriate. The temperatures you wanted to measure are at the low end of the thermometer’s range and may be less accurate than around 98 degrees. Also, the thermometer is slow to reach equilibrium and can’t be read with more than one decimal place of precision. Use a digital infrared laser-point thermometer next time. More accurate. More precise. More fun.
  • Environmental Variability—differences between a sample and the population attributable to extraneous factors. Minimizing environmental variance is difficult because there are so many causes and because the causes are often impossible to anticipate or control. For example, the heating system may go on and off unexpectedly. Your own body heat adds to the room temperature and walking around the living room taking measurements mixes the ceiling and floor air which adds variability to the temperatures.

When you analyze data, you usually want to evaluate characteristics of some population and the natural variability associated with the population. Ideally, you don’t want to be mislead by any extraneous variability that might be introduced by the way you select your samples (or patients, items, or other entities), measure (generate or collect) the data, or experience uncontrolled transient events or conditions. That’s why it’s so important to understand the ways of variability.

Variability versus Bias

Remember target practice? If there is little variation in your aim, the deviations from the center of the target would be random in distance and direction. Your aim would be accurate and precise. But what if the sight on your weapon were misaligned? Your shots would not be centered on the center of the target. Instead there would be a systematic deviation caused by the misaligned sight. Your shots would all be inaccurate, by roughly the same distance and direction from the center. That systematic deviation is called bias. You may not even have known there was a problem with the sight before shooting, although you would probably suspect something after all the misses.

Bias usually carries the connotation of being a bad thing. It usually is. It may be why 19th Century British Prime Minister Benjamin Disraeli mistakenly associated statistics with lies and damn lies. But if the systematic deviation is a good thing because it fixes another bias, it’s called a correction. For example, you could add a correction, an intentional bias in the direction opposite the bias introduced by the weapon sight, to compensate for the inaccuracy. So bias can be good (in a way) or bad, intentional or not, but it’s always systematic. On the other hand, a bias applied to only selected data is a form of exploitation, and is nearly always intentional and a very bad thing.

So the relationships to remember are:

Variance ↔ Imprecision

Bias ↔ Inaccuracy

Most statistical techniques are unbiased themselves, as long as you meet their assumptions. If something goes wrong, you can’t blame the statistics. You may have to look in the mirror, though. During the course of any statistical analysis, there are many decisions that have to be made, primarily involving data. Whatever the decisions are, such as deleting or keeping an outlier, there will be some impact on precision and perhaps even accuracy. In an ideal world, the sum of the decisions wouldn’t add appreciably to the variability. Often, though, data analysts want to be conservative, so they make decisions they believe are counter to their expectations. But when they don’t get the results they expected, they go back and try to tweak the analysis. At that point they have lost all chance of doing an objective analysis and are little better than analysts with vested interests who apply their biases from the start. Avoiding such analysis bias requires no more than to make decisions based solely on statistical principles. This sounds simple but it isn’t always so.

Sometimes bias isn’t the fault of the data analyst, as in the case of reporting bias. In professional circles the most common form of reporting bias is probably not reporting non-significant results. Some investigators will repeat a study again and again, continually fine-tuning the study design until they reach their nirvana of statistical significance. Seriously, is there any real difference between probabilities of significance of 0.051 versus 0.049? But you can’t fault the investigators alone. Some professional journals won’t publish negative results, and professionals who don’t publish perish. Can you imagine the pressure on an investigator looking for a significant result for some new business venture, like a pharmaceutical? He might take subtle actions to help his cause then not report everything he did. That’s a form of reporting bias.

Perhaps the most common form of reporting bias in nonprofessional circles is cherry picking, the practice of reporting just those findings that are favorable to the reporter’s position. Cherry picking is very common in studies of controversial topics such as climate change, marijuana, and alternative medicine. Virtually all political discussions use information that was cherry picked.

Given that someone else’s reporting bias is after-the-analysis, why is it important to your analysis? The answer is that it’s how you can be misled in planning your statistical study. Never trust a secondary source if you can avoid it. Never trust a source of statistics or a statistical analysis that doesn’t report variance and sample size along with the results. And always remember: statistics don’t lie; people do.

Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data Analysis at Wheatmark, amazon.combarnesandnoble.com, or other online booksellers.

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About statswithcats

Charlie Kufs has been crunching numbers for over thirty years. He currently works as a statistician.
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14 Responses to There’s Something About Variance

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