The arithmetic mean, often just called the average, is the most common way to find the center of a dataset. It gives us a single value that represents the typical or central value of all the observations. Think of it as the balancing point of your data.

How we calculate the mean depends on how the data is presented. We'll look at two main scenarios: ungrouped data and grouped data.

For ungrouped data, we have a simple list of individual values. Imagine you have the test scores of five students: 70, 85, 90, 65, and 80.

To find the mean, we use the direct method. This is the one you probably know already.

For our student scores, the calculation would be:

For more complex datasets, the short-cut method can save time. It involves choosing an 'assumed mean' ($A$) and then calculating the average of the deviations from that assumed value.

Let's use the same scores and pick an assumed mean, $A = 80$. We create a table to stay organised.

When data is grouped into a frequency distribution, we don't have individual values anymore. Instead, we have classes or groups and the number of observations (frequency, $f$) in each. For calculations, we use the mid-point ($x$) of each class interval to represent all the values in that class.

Let's say we have data on the daily wages of 50 workers.

The direct method for grouped data is similar to the one for ungrouped data, but we must account for the frequency of each mid-point.

For our example, the sum of $fx$ would be $(122 \times 3) + (131 \times 5) + ... + (176 \times 2) = 5860$. The total frequency is 40. So, $\bar{X} = 5860 / 40 = 146.5$. The average daily wage is $146.50.

The coding method, or step-deviation method, simplifies the maths even more, especially when class intervals are equal. We pick an assumed mean ($A$), usually the mid-point of the class with the highest frequency, and calculate a new variable, $u$.

Let's use our wage data. The highest frequency is 12, so we'll choose $A = 149$. The class width, $h$, is 9. We can now build a table to find $\sum fu$.

Now, we just plug our totals into the formula.

The arithmetic mean has some unique mathematical properties that make it very useful in statistics.

First, the sum of the deviations of all values from their arithmetic mean is always zero. This is its balancing property.

Second, the sum of the squared deviations of the values from the mean is a minimum. This means if you were to pick any other number and calculate the sum of squared differences from it, the result would be larger than the one you get using the mean. This is called the principle of least squares and is a foundation for more advanced statistics.

Finally, we have the Weighted Arithmetic Mean. This is used when different data points have different levels of importance. Each value ($X$) is assigned a weight ($w$), and values with higher weights contribute more to the final mean.

Understanding these different methods for calculating the mean and its core properties is fundamental for statistical analysis.