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Fundamentals of Statistics 2: The Normal Distribution :: The Normal Distribution and Z-scores
The normal distribution is a statistical model. If we can show that our data is approximately like this model then we can make all sorts of inferences about other things in the population that we're unsure of.

We saw that the empirical rule states that approximately 68%, 95% and 99% of values fall with 1,2 and 3 standard deviations of the mean. The normal distribution shares this property as well, in fact that's where it comes from since as was stated earlier, all bell-shaped symmetrical distributions share this property. It's also important because it also does a good job of approximating discrete distributions when there are a large enough number of outcomes.




Photo By: Greg O'connell
For example, North American adult men on average are 70 inches with a standard deviation of 4 inches, then we can infer that approximately 68% of all North American men will be between  66 and 74 inches (between 5'6 and 6'2). So if you're an adult male from North America, then chances are pretty good that you're in between these two heights.  In fact, I'm 95% sure your height is between 5'2 and 6'6 and I know nothing about you!

That last example showed us how knowing something about the distribution allows us to make inferences about any single person in the population. But what if we want to know what percent fall within say 1.22 standard deviations? Enter the z-score.




Z-score

The z-score is just a fancy name for standard deviations. So a z-score of 2 is like saying 2 standard deviations above and below the the mean. A z-score of 1.5 is 1.5 standard deviations above and below the mean. A z-score of 0 is no standard deviations above or below the mean (it's equal to the mean).

You can also just have z-scores on one side of the mean: 1 standard deviation below the mean is a z-score of -1 and a z-score of 2.2 can be 2.2 standard deviations above the mean. A z-score of -3 is 3 standard deviations below the mean. One and 2-sided z-scores will be addressed in subsequent lessons when they will be referred to as 1 and 2-sided or 1 and 2-tailed tests. For now, when you see z, think s--as in standard deviations.  And remember, the standard deviation is just the average distance values are from the mean.

More formally, what was just described can be written as :
  • μ + z σ : mean plus z standard deviations
  • μ z σ: mean plus or minus z standard deviations
  • μ - z σ : mean minus z standard deviations



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