Find the mean, median, and mode of the data set 6, 7, 3, 4, 9, 5.

Step-by-step solution

1. 1

   Add the values and divide by how many there are. The mean is the total shared equally among all data points:

   $6+7+3+4+9+5=34$

   There are $6$ values, so

   $\text{mean}=\frac{34}{6}=\frac{17}{3}\approx 5.67$

   Keeping $\tfrac{17}{3}$ as a fraction avoids rounding error if the mean feeds into a later calculation.

2. 2

   Sort the data before touching the median. The median is defined by position, so the list must be in order first:

   $3,\ 4,\ 5,\ 6,\ 7,\ 9$

3. 3

   Average the two middle values. With an even count of $6$, there is no single middle entry; the median is the mean of the $3$rd and $4$th values, which are $5$ and $6$:

   $\text{median}=\frac{5+6}{2}=\frac{11}{2}=5.5$

   Notice the median need not be a member of the data set.

4. 4

   Count repetitions for the mode. Each of $3,4,5,6,7,9$ appears exactly once, so no value occurs more often than any other. The set therefore has no mode - reporting "all of them" or "$0$" would both be wrong.

5. 5

   Sanity-check the three summaries. The mean $5.67$ and median $5.5$ both sit inside the range $[3,9]$ and close to each other, which is what you expect when the data has no strong skew: $34/6=5.6\overline{6}$ and $(5+6)/2=5.5$ differ by only $0.1\overline{6}$.