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Fundamentals of Statistics 2: The Normal Distribution :: The Normal Distribution and Zscores
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 bellshaped 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 zscore.
Zscore
The zscore is just a fancy name for
standard deviations.
So a zscore of 2 is like saying 2 standard deviations above and below the the mean. A zscore of 1.5 is 1.5 standard deviations above and below the mean. A zscore of 0 is no standard deviations above or below the mean (it's equal to the mean).
You can also just have zscores on one side of the mean: 1 standard deviation below the mean is a zscore of 1 and a zscore of 2.2 can be 2.2 standard deviations above the mean. A zscore of 3 is 3 standard deviations below the mean. One and 2sided zscores will be addressed in subsequent lessons when they will be referred to as 1 and 2sided or 1 and 2tailed tests. For now,
when you see z, think sas 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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July 18, 2015  Greg wrote:
Why would I want to know the zscore. Understanding the empirical rule makes a lot of sense, but having a hard time understanding why anyone (other than a teacher/professor) would think in terms of standard deviations and want to know the value within a specific sd/sd range is a bit puzzling. It's another one of those "so what" questions.