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Statistics Resources

This guide contains all of the ASC's statistics resources. If you do not see a topic, suggest it through the suggestion box on the Statistics home page.

Empirical Rule

The standard normal distribution is actually a probability distribution. By identifying points on the curve, we can determine the probability associated with lying at that point. How these probabilities were determined would take a lot of explanation. Suffice it to say that that work has already been done, so understanding how these values were derived is not the important part here. :)

When we first start talking about probability for the normal distribution, we're often introduced first to The Empirical Rule. This states the following:

  • approximately 68% of the distribution lies within 1 standard deviation of the mean (that is between 1 st. dev. below and 1 st. dev. above)
  • approximately 95% of the distribution lies within 2 standard deviations of the mean
  • approximately 98.7% of the distribution lies within 3 standard deviations of the mean

normal distribution showing percentages of data that lie between standard deviations

So, by determining how many standard deviations a value is from the mean, we can determine the probability of obtaining that value on that distribution.

 

Example:

IQs are normally distributed with a mean of 110 and a standard deviation of 15. How many standard deviations is an IQ of 95 from the mean?

We completed the computation for this on the previous tab, but we can also use critical thinking here: 

  • if a standard deviation is 15 points, how many times do I have to take 15 away from 110 to reach 95?
    • 110 - 15 = 95
    • so I take 15 away 1 time, which means 95 is one standard deviation below the mean of 110

What's the probability of picking a person that has an IQ of 95 or less?

  • now that we know that 95 is one standard deviation below the mean, we can use the Empirical Rule to estimate the probability
    • note this will be cumulative probability because it's 95 or less
  • start with the basic characteristic that 50% of the distribution lies below the mean
  • from the Empirical Rule (shown above), we know that 34% of the distribution lies between the mean and 1 standard deviation below it.
  • to find the probability of having an IQ that 95 or less, we can take 50% - 34% = 26%

Typically probability is reported as a decimal. To convert a percentage to a decimal, we can divide the percentage by 100:

  • 26/100 = .26

So the probability of picking someone with an IQ of 95 or less is .26. We could also say that the probability of lying more than 1 standard deviation below the mean is .26.