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
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}}
  • 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

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:
    ConfidenceInterval &= \bar{x} \pm t_{n-1} \frac{s}{\sqrt{n-1}}
  • 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