Mean and Median — Formula Sheet

Mean of ungrouped data: \bar{x}=\frac{x_{1}+x_{2}+x_{3}+\ldots \ldots \ldots+x_{n}}{n}=\frac{1}{n} \cdot \sum_{i=1}^{n} x_{i}

\Sigma x=x_{1}+x_{2}+x_{3}+\ldots \ldots+x_{n}

Sum of deviations (Property 1): \Sigma(x-\bar{x})=0

Mean after adding constant (Property 2): mean of x1+a, x2+a, ..., xn+a = \bar{x}+a

Mean after subtracting constant (Property 3): mean of x1-a, x2-a, ..., xn-a = \bar{x}-a

Mean after multiplying by constant (Property 4): mean of ax1, ax2, ..., axn = a\bar{x}

Mean after dividing by constant (Property 5): mean of x1/a, x2/a, ..., xn/a = \bar{x}/a

Correcting sum after misread observation: Correct value of \Sigma x = Incorrect value of \Sigma x - Incorrect observation + correct observation

Median for odd n: Median = \left(\frac{n+1}{2}\right)^{\text{th}} term

Median for even n: Median = \frac{1}{2}\left[\left(\frac{n}{2}\right)^{\text{th}} term + \left(\frac{n}{2}+1\right)^{\text{th}} term\right]

Property 1

Σ(x - x̄) = 0.

Property 2

mean of x1 + a, x2 + a, ... , xn + a = x̄ + a.

The sum of deviations of all observations from their mean is zero: Σ(x−x̄)=0.

Mean of x1+a, x2+a, ..., xn+a = x̄+a.

Mean of x1−a, x2−a, ..., xn−a = x̄−a.