Numerical Methods

After going over graphical methods in the previous section, we now go over numerical methods.

• Consider a data set \(n\) observations. \(\{x_{1}, x_{2}, ... , x_{n}\}\)

• Sample mean \(\bar{x}\) \[ \bar{x} = \frac{x_{1} + ... + x_{n}}{n} = \frac{1}{n} \sum_{i=1}^{n} x_{i} \]

• Similarly, there exists a population mean \(\mu\)

• Sample median, \(\tilde{x}\) this is just the middle number in the dataset
  If \(n\) is odd: \[ \tilde{x} = (x_{(n + 1)/2}) \] If \(n\) is even: \[ \tilde{x} = \frac{1}{2} (x_{\frac{n}{2}} + x_{\frac{n}{2} + 1}) \]

• Example: \(\{-3, 0, 4 \}\)
  \[ \bar{x} = \frac{1}{3} (-3 + 0 + 4) = \frac{1}{3} \] \[ \tilde{x} = 0 \]

• Sample variance denoted by \(s^{2}\) \[ s^{2} = \frac{(x_{1} - \bar{x})^{2} + (x_{2} - \bar{x})^{2} + ... + (x_{n} - \bar{x})^{2}}{n - 1} \]

• Population variance denoted by \(\sigma^{2}\)

• Sample standard deviation is the square root of variance
  \[ s = \sqrt{s^{2}} \]

• Population standard deviation is denoted by \(\sigma\)

• Sample range
  \[ x_{max} - x_{min} \]

• Example: \(\{0, 1, 5 \}\)

  • Sample mean
    \[ \bar{x} = \frac{0 + 1 + 5}{3} = 2 \]
  • Sample variance
    \[ s^{2} = \frac{(0 - 2)^{2} + (1 - 2)^{2} + (5 - 2)^{2}}{3 - 1} = 7 \]
  • Sample standard deviation
    \[ s = \sqrt{7} \]