Standard Deviation vs Coefficient of Variation

by | Jan 26, 2018 | CPCU 500

Two of the more difficult concepts to grasp in CPCU® 500 are standard deviation & coefficient of variation. The textbook says they are both measures of variation, but what does that mean exactly & why does that matter? This tip will provide a better explanation to help you understand these two tricky concepts better.

Standard deviation is a measure of how much variation there is within a data set. This is important because in many situations, people don’t want to see a lot of variation – people prefer consistent & stable performance because it’s easier to plan around & less risky. For example, let’s say you are deciding between two companies to invest in that both have the same number of average monthly sales. Store A is fairly consistent, selling roughly the same amount of product each month. Store B, however, is all over the place: sometimes they do okay, some months they might even lose money, but once a year they get really lucky because a parade runs in front of their store so they get tons of sales that day which brings their average monthly sales figure way up. Obviously, Store A is far more attractive because you can feel more confident that they’ll be able to pay their expenses, bring in profit regularly, and their business pattern is less risky (imagine what would happen to Store B if the parade got cancelled!). This is why looking at variation, not just the average, is important.

To calculate standard deviation, you’d compare each data point to the average to see how far apart they are, then average all the differences. Don’t worry – you won’t have to calculate standard deviation on your CPCU 500 test, but it is important to understand how the calculation works. Going back to the example above, since Store A’s sales are pretty consistent, the figures for each month aren’t going to be too far off from the annual average, so their standard deviation is going to be small. On the other hand, Store B is going to have big differences from month to month, so their standard deviation ends up being quite high, even though in the end, their average monthly sales is similar to Store A’s (so misleading!). Remember that with standard deviation, smaller is better because smaller standard deviation = less variation = more consistency & stability.

The main limitation with standard deviation is that it can only be used to compare two things that similar – “apples to apples.” It should not be used to compare “apples to oranges,” so that is where coefficient of variation comes in. What you need to know about coefficient of variation is this: you calculate coefficient of variation by taking the standard deviation & dividing it by the mean or average to get a percentage, which allows you to compare variation between two things that aren’t similar enough to just look at their standard deviation.

To give an example: let’s say you are going to award an athletic award at a school. One of the judging criteria is consistency in athletic performance, but your two top candidates are a basketball player & a swimmer. The basketball player averages 30 points per game, with a standard deviation of 15 points per game, and the swimmer has an average swim time of 1 minute per race with a standard deviation of 5 seconds per lap. Comparing the standard deviations of “15 points” to “5 seconds” doesn’t really tell you anything because they are completely different. If we use the coefficient of variation, however, we can see that the basketball player variation is 50% (15 points per game divided by average of 30 points per game) whereas the swimmer’s variation is only 8.3% (5 seconds per lap divided by average swim time of 60 seconds per race). Thus, having a percentage makes things easier to compare.

Fortunately, when it comes to the CPCU 500 exam, the focus is more on understanding the theory rather than making these statistical calculations. The main things you’ll need to know for the test are:

  • Standard deviation
    • used to compare variation among similar things (“apples to apples”)
    • calculated by taking the average of all the differences between each data point compared to the overall average
  • Coefficient of variation
    • used to compare variation among dissimilar things (“apples to oranges”)
    • calculated by taking standard deviation and diving by the mean or average

Learn these and other hard concepts in easy-to-understand language!

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