Standard Deviation, Standard Error, and Confidence Intervals
What is Standard Deviation?
 The spread around the center of a normal distribution
 The amount of variation in a population
 The point where the normal curve changes from concave down to concave up
What is the Sample Standard Deviation?
 The variance around the mean of a sample from the population
\[\begin{aligned}
s &= \sqrt{\frac{(x\bar{x})^2}{n1}}
\end{aligned} \]
What is the Empirical Rule?
 68% of the data falls within 1 Standard Deviation of the center
 95% of the data falls within 2 Standard Deviations of the center
 99.7% of the data falls within 3 Standard Deviations of the center
 If the percentages don't match the data, the distribution isn't normal
What is the Standard Error?

The amount of variance the measure of central tendency (e.g. the mean) has:\[ \begin{aligned}
SE &= \frac{\sigma}{\sqrt{n}}
\end{aligned}\]
 Sample Standard Deviation is the variance within a sample, Standard Error is how close the Sample mean is to the Population Mean
What is a Margin Of Error?
 A multiple of the Standard Error
 The multiple is based on the Confidence Interval you want
 For example: 68% Confidence uses a multiple of 1 (see the Empirical Rule)
MOE &= multiple \times StandardError
\end{aligned}\]
What is a Confidence Interval?
 Level of Confidence based on the percent of the distribution your Margin of Error covers

If you have a small data set or you don't know that the distribution is normal use the tdistribution:\[\begin{aligned}
ConfidenceInterval &= \bar{x} \pm t_{n1} \frac{s}{\sqrt{n1}}
\end{aligned}\]
 n is the size of the sample
 s is the Sample Standard Deviation
 If you know that the distribution is normal and/or your sample is large, use the zscore instead.