Let's give a quick example with some simulated data.

Suppose we have exam scores from 9 students.

The mean of this distribution is simply the arithmetic average of these scores.

The mode is the most frequently observed value.

In this case, we have two students who scored 88, so the mode is 88.

However, we can see that with continuous distributions,

it may be very unlikely to observe the same exact value multiple times.

Therefore, the mode of a distribution is not always a very useful measure.

The median is defined as the midpoint or the 50th percentile of the distribution.

In order to calculate the median,

we need to first sort the data in increasing order.

The we find the mid-point of the ordered data which in this case happens to be 87.

But what if we didn't have an exact midpoint of the distribution?

Say we have one more student who scored 100.

Now the sample size is 10 and with an even number of observations

there isn't a simple value that divides the data in half.

In these cases,

the median as defined as the average of the middle of the two observations.

Here, we have 87 and 88 at the middle of our distribution so

the median based on this new data set would be 87.5.

Learning how these values are calculated by hand can be important for

also understanding the concepts, but

we should note that calculations like these are rarely done by hand.

Instead, we often rely on computation, which makes life much easier for

working with data with a larger number of observations.