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
Calculation:
\begin{aligned} s &= \sqrt{\frac{(x-\bar{x})^2}{n-1}} \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)
\begin{aligned} 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 t-distribution:
\begin{aligned} ConfidenceInterval &= \bar{x} \pm t_{n-1} \frac{s}{\sqrt{n-1}} \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 z-score instead.
Source: Statistic For Dummies