a)

The mean is the sum of all the responses ($\sum x$) divided by the number of responses ($n$).

Here, $n = 20$.

$$\begin{align} \sum x &= \var{a[0]} + \var{a[1]} +\var{a[2]} +\var{a[3]} +\var{a[4]} +\var{a[5]} +\var{a[6]} +\var{a[7]} +\var{a[8]} +\var{a[9]} + \var{a[10]} + \var{a[11]} +\var{a[12]} +\var{a[13]} +\var{a[14]} +\var{a[15]} +\var{a[16]} +\var{a[17]} +\var{a[18]} +\var{a[19]} \\ &= \var{sum(a)} \text{.} \end{align}$$

Therefore we calculate the mean

$$\begin{align} \overline{x} &= \frac{\sum x}{n} \\[0.5em] &= \frac{\var{sum(a)}}{20} \\[0.5em] &= \var{mean} \text{.} \end{align}$$

b)

The median is the middle value. We need to sort the list in order:

[Math Processing Error] $$\var{as[0]}, \quad \var{as[1]}, \quad \var{as[2]}, \quad \var{as[3]}, \quad \var{as[4]}, \quad \var{as[5]}, \quad \var{as[6]}, \quad \var{as[7]}, \quad \var{as[8]}, \quad \var{as[9]}, \quad \var{as[10]}, \quad \var{as[11]}, \quad \var{as[12]}, \quad \var{as[13]}, \quad \var{as[14]}, \quad \var{as[15]}, \quad \var{as[16]}, \quad \var{as[17]}, \quad \var{as[18]}, \quad \var{as[19]}$$

There is an even number of responses, so there are two numbers in the middle (10^th and 11^th place). To find the median, we need to find the mean of these two numbers $\var{as[9]}$ and $\var{as[10]}$:

$$\begin{align} \frac{\var{as[9]} + \var{as[10]}}{2} &=  \frac{\var{as[9] + as[10]}}{2} \\ &= \var{median} \text{.}  \end{align}$$

c)

The mode is the value that occurs the most often in the data.

To find a mode, we can look at our sorted list:

$\var{as[0]}, \var{as[1]}, \var{as[2]}, \var{as[3]}, \var{as[4]}, \var{as[5]}, \var{as[6]}, \var{as[7]}, \var{as[8]}, \var{as[9]}, \var{as[10]}, \var{as[11]}, \var{as[12]}, \var{as[13]}, \var{as[14]}, \var{as[15]}, \var{as[16]}, \var{as[17]}, \var{as[18]}, \var{as[19]}$.

We notice that $\var{mode1}$ occurs the most ($\var{modetimes[mode1]}$ times) so $\var{mode1}$ is the mode.

d)

Range is the difference between the highest and the lowest value in the data.

To find this, we subtract the lowest value from the highest value:

$$\var{max(a)} - \var{min(a)} = \var{range} \text{.}$$