Basic Statistics

There are a few standard numbers used to describe a data set. The mean is the average: add up all the values, then divide by how many there are. The median is the middle value once the data is sorted. The mode is the value that occurs most often.

Take the set $12, 15, 15, 17, 20$.
Mean: $\frac{12 + 15 + 15 + 17 + 20}{5} = \frac{79}{5} = 15.8$.
Median: the middle value is $15$.
Mode: $15$ appears twice, more than any other, so the mode is $15$.

When there is an even number of values there is no single middle one, so the median is the average of the two middle values.
For $4, 8, 10, 14$ the two middle values are $8$ and $10$, so the median is $\frac{8 + 10}{2} = 9$.

A data set can also have more than one mode. In $2, 2, 5, 7, 7$ both $2$ and $7$ appear twice, so the set has two modes. If every value appears equally often, there is no mode at all.

Two more numbers describe how spread out the data is. The range is the largest value minus the smallest. The interquartile range is the third quartile minus the first quartile, which is the spread of the middle half of the data.

A box plot shows these at a glance. The median splits the data in half; the first quartile (Q1) is the middle of the lower half and the third quartile (Q3) is the middle of the upper half. The box runs from Q1 to Q3, the line inside is the median, and the whiskers reach the smallest and largest values.

Find the range of $20, 14, 32, 48, 23, 35, 40$.
The largest value is $48$ and the smallest is $14$.
Range $= 48 - 14 = 34$.

Find the mode of $12, 15, 15, 17, 20$.
The value $15$ appears twice, and every other value appears once.
So the mode is $15$.

Find the median of the $15$ sorted ages $20, 25, 28, 32, 45, 50, 53, 55, 60, 70, 72, 77, 78, 79, 85$.
With $15$ values the middle one is the $8$th.
Counting to the $8$th value gives $55$.

Find the median of $4, 8, 10, 14$.
There are $4$ values, an even number, so average the two middle ones, $8$ and $10$.
Median $= \frac{8 + 10}{2} = 9$.

Find the mode of $2, 2, 5, 7, 7$.
Both $2$ and $7$ appear twice, more than any other value.
So there are two modes: $2$ and $7$.