What is a Normal Distribution?

Posted by  Adam Hilliard

One of the most common questions I receive from clients when implementing a new assessment, is “How do you know how many people will pass before anyone takes it?” This is certainly a reasonable question, considering that within seconds of looking at a scoring profile for any of our assessments, I can give a client a reasonable expectation of their pass/fail rate.  The answer is, simply, because our assessments are “normally distributed.”

The term normal distribution is defined as “a function that represents the distribution of many random variables as a symmetrical bell-shaped graph” . See the figure below to get a visual image of a normal frequency distribution. Without getting too technical, because of the Central Limit Theorem (CLT), we are surrounded by bell curves. Just about anything you can measure is normally distributed. You might ask yourself why you would care if something is normally distributed, but when something is normally distributed you can apply a set of probability rules to it that allows you to interpret the information. It’s pretty powerful.

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Let’s look at height as an example. Let’s say you’re standing in line at a store behind a woman who is 6’ 2”. You know this because you asked her. You realized soon thereafter that you had never met a woman that tall before. You started thinking, how many women are that tall? Because height is normally distributed, you can easily answer this question if you know some information about the height of all American women.

The first thing you need is a very large sample (thousands) of American women to have their height measured and recorded. Then, calculate the average height (µ) and the standard deviation (σ) for that group. Luckily, this has been done and made public. The average American woman’s height is 5’4” or µ =64 inches and the standard deviation is σ=2.5 inches. 

How does this information help you? Looking at the figure again - the line that bisects the curve is the mean or average. In a normal distribution, 50% of the people fall above the mean and 50% fall below. That is a rule that applies to all normal distributions. The standard deviation helps you better understand a score within a normal distribution by providing an idea of how many people are likely to be that score. The rules of a normal distribution are as follows: 68.2% of people will score within one standard deviation (1SD) of the mean, 95% within 2SD and 99.6% within 3SD. So, here is what you now know about height - there is a 68.2% chance that any American woman you pick off the street will be between 61.5” and 66.5”, a 95% chance of being within 2SDs (59” to 69”); and a 99.6% chance of being within 3 SDs (56.5” and 71.5). So, the woman you met, who was 74” tall, is 4 standard deviations away from the mean – that’s less than half of one percent chance of seeing an American woman that tall. You should go buy a lottery ticket.

Height is an example of an external characteristic that is normally distributed, but internal characteristics (e.g., personality traits) are also theorized to be normally distributed. We specialize in measuring psychological constructs through pre-employment assessments. The CLT and normal distributions are fundamental to how we approach our scoring. Our assessments provide final competency scores on a 10-pt scale that approximate a normal distribution. This is why I can estimate a pass rate with such speed. So, anyone who looks at the scores across any of our assessments can easily interpret scores in the middle (5-6) as average or common and scores on the end (1-2 and 9-10) as rare. Scores below 5 are considered below average and scores above 6 are above average. Low scores, such as a 1, should be treated as a red flag, especially if the competency is key to success in the job, because a vast majority of candidates did not score that low.

To ensure that our clients can rely on this ease of interpretation, our R&D team is consistently reviewing our candidate data to ensure that normal distributions are there. We collect data on all of our measurements and load them into a “normative database.”

So, the next time you see super tall (or short) woman when you’re out shopping, consider yourself one of the lucky few who had the chance to observe such a rare thing!

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Adam Hilliard

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