# Standard Deviation and Standard Error

The standard deviation is how the various items in a population vary from the mean. In graph terms, the standard deviation is how tall/thin or short/fat the graph is. In comparison, the standard error is looking at a sample set from the larger population. In essence the standard error is a standard deviation, but of the the way the sample means lay out. The formula isstandard error = (standard deviation) / (square root of observations in each sample)

When you're using a Z formula to look at a single X against a whole population, you're determining what proportion of values have that number. So if you are drawing a sample student in a class and measuring his height, and you use the Z formula, you're finding out what proportion of students have that height or less. (That also tells you the inverse, how many students have that height or more). The formula is

Z = sample - mean / standard deviation

When you're using a Z formula to look at a standard error and Xbars, what you are finding out is how likely your current set of results are with the sample set you happened to draw, given all the various results options you might have gotten. The formula here is

Z = (Xbar - mean) / (standard deviation / square root of # of observations)

So let's say you drew a sample set of 10 students out of a population of 1,000 college students and measured their heights. You got an average height of 4' with your 10 students. You want to know if this is a really unlikely situation, or if it really is a reasonable sample result to have gotten. The Z formula could tell you that the likelihood of this happening was .90 - very likely. Maybe these students are all super-bright 12 year olds in a special college that only accepts 12 year old students. On the other hand the Z formula could tell you that the likelihood of this result happening was only .01 - meaning you need to work on your sampling methods. Somehow you managed to get every single short person in the entire class.

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