28 Sep, 2016

Quiz 1 - Averages, Medians, and Standard Deviations

Here is recap on how to compute averages, medians and standard deviations.

In each of these examples, suppose a sample

\[(x_1,x_2,...,x_n)\]

of size $n$ is observed.

Definitions

Average
\[\bar{\mathbf x} = \ds\frac{x_1 + x_2 + ... + x_n}{n}\]

= sum of observations divided by number of observations.

Standard Deviation (SD)
\[SD\p{\mathbf x} = \ds\sqrt\frac{(x_1-\bar{x})^2 + (x_2-\bar{x})^2 + ... + (x_n-\bar{x})^2}{n}\]

where $\bar{x}$ is the average. (From here on, standard deviation and the abbreviation SD will be used interchangeably.)

Median
The observation in the middle after sorting.

Say $(x_{(1)},…,x_{(n)})$ is a sorted sample such that $x_{(1)}$ is the minumum and $x_{(n)}$ is the maximum, the median of the sample is

\[\begin{cases} x_{(\frac{n+1}{2})} & \text{, if $n$ is odd} \\ \ds\frac{x_{(\frac{n}{2})} + x_{(\frac{n+1}{2})}}{2}& \text{, if $n$ is even} \end{cases}\]

Examples:

We will now concretize what we’ve defined with examples. Say we have observed the following sample (of size 5) which represent the quiz scores of 5 students in AMS5:

\[(3,2,4,5,1)\]

Compute the

  1. average of the sample
  2. sd of the sample
  3. median of the sample

Solutions:

  1. First, note that $n=5$. So, the average = $\frac{3+2+4+5+1}{5} = \mathbf 3$.

  2. We have already computed the average to be 3. So, substitute $\bar{x}$ with 3. Then,

    \[\begin{array}{rcl} SD &=& \ds\sqrt\frac{(3-3)^2 + (2-3)^2 + (4-3)^2 + (5-3)^2 + (1-3)^2}{5} \\ &=& \ds\sqrt\frac{0^2 + (-1)^2 + (1)^2 + (2)^2 + (-2)^2}{5} \\ &=& \ds\sqrt\frac{0 + 1 + 1 + 4 + 4}{5} \\ &=& \ds\sqrt\frac{10}{5} \\ &=& \sqrt 2 \\ &\approx& 1.414 \end{array}\]

  3. To compute the median, first order the data:

    \[(3,2,4,5,1) \rightarrow (1,2,3,4,5)\]

    Then, the median is sorted observation number $\frac{5+1}{2} = 3$. The third observation (sorted) also happens to be 3. So 3 is the median.

    • Here is a second example for medians. Say the sample was instead

      \[(3,5,7,2,6,9)\]

      Now we have an even number of observations. First, sort the data. The sorted observations are

      \[(2,3,5,6,7,9)\]

      The median is now the average of the $3^{rd}$ and $4^{th}$ observations which is $\frac{5+6}{2} = \mathbf{5.5}$