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The Standard Deviation
What is normal? In sociology, the term “normal”, “norms” and “normative” are used in many contexts but what is a normative behavior? What is a normal human attribute? There are a number of ways to determine normalcy including the use of the standard deviation.
Would you consider an adult male who weighs more than 200 pounds abnormal? What about a male who weighs 300 pounds? What is normal, what is abnormal? What is acceptable, what is deviant? The standard deviation can be helpful in providing normative information in a number of situations, such as the posed obesity question.
Conceptually, the standard deviation is a measure of how “messy” a data set is. A set of data that has a lot of different values relative to the average value (the mean value) has a relatively large standard deviation. Conversely, a data set with very little variance in the data set values with respect to the mean has a small standard deviation. To calculate the standard deviation for a set of values, you can use the STDEV.S function in Excel. For mathematical information, see Standard deviation.
But how is the standard deviation connected to normalcy? Consider figure 1, highlighting the standard deviation σ:
Figure 1 depicts a normal curve for a large set of data. The average (mean) value of the data occurs at 0 on the horizontal axis (the peak of the curve). As can be seen, 34.1% + 34.1% = 68.2% ≈ 68% of the data occurs within ±1 standard deviations of the mean. Similarly, 13.6% +68.2% + 13.6% = 95.4% ≈ 95% of the data occurs within ±2 standard deviations of the mean. A rule of thumb is that anything outside of 2 standard deviations is abnormal (i.e., an outlier). Let’s apply this rule of thumb to our obesity question above:
Figure 2 Generated using Wolfram Mathematica 10.1.0.0
Using our rule of thumb, we see that a 200-pound adult male is within normal. However, a 300-pound male is not considered normal – from the table, we see that the average male weight is 156 pounds and that +2 standard deviations from this average is 228 pounds. That is, only 2.5% of the male population weighs more than 228 pounds, implying that a 300-pound male is abnormally heavy. Does this seem overly analytical, even dehumanizing? That is precisely what it’s important to understand this common method of determining “normalcy”.
Consider the DSM II, which considered homosexuality as a form of paraphilia (a condition characterized by abnormal sexual desires). From a purely analytical perspective, it’s easy to see how the DSM II may have considered homosexuality as abnormal – 4.5% of adult Americans identify as LGBT (Wikipedia), below the 5% threshold of “normalcy” in the normal curve of figure 1. This highlights how the standard deviation and normal curve can be misapplied and why having a basic understanding of it is important.