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Here is a more basic definition of Means.

The Arithmetic Mean of the sequence of numbers $a_1,a_2,\ldots a_n$ is defined as $$\dfrac{a_1+a_2+\cdots +a_n}{n}$$

where the sum of the set of numbers is divided by the amount of numbers.

Now for a harder one:

Now that we are done with the arithmetic mean, there is another type of mean to discuss. This one is called the Geometric Mean. The Geometric Mean of a set of numbers $a_1,a_2,\ldots a_n$ is defined as $$\sqrt[n]{a_1a_2\cdots a_n}$$

where the product of the set of numbers is taken to the $n$th root, where $n$ is the amount of numbers.

Bonus: AM-GM

One last important thing to remember about the arithmetic mean and the geometric mean is the $\textbf{Arithmetic Mean}-\textbf{Geometric Mean Inequality}$, or $\text{AM}-\text{GM}$ for short. It states that for any non-negative real numbers $a_1,a_2,\ldots a_n$, that $$\dfrac{a_1+a_2+\cdots +a_n}{n}\ge \sqrt[n]{a_1a_2\cdots a_n}$$

You can test this out to see if it's true. Let's make $a_1=1$, $a_2=2$, $a_3=3$, $a_4=4$, and $a_5=5$. We have that $$\dfrac{1+2+3+4+5}{5}=3$$ and $$\sqrt[n]{1\cdot 2\cdot 3\cdot 4\cdot 5}\approx 2.605$$

Indeed, $3 \ge 2.605$.

For now, just knowing this inequality will suffice. It may become much more useful later on.

An average is a way of summarizing a group of data. Typically an average is a single number that somehow characterizes the data from which it is derived. Common averages include the mode (most common data point), median (middle data point), and arithmetic mean (or mean). The mean is a commonly used average in calculus and is generally more widely utilized than either of the mode or median, and therefore merits further discussion.

In the future, you will also learn about the weighted average. The expected value is a weighted average of all possible outcomes.

The mean, $\overline{x}$, of a set of numbers, $S$, is defined as

$$\overline{x} = \frac{1}{|S|}\sum_{s \in S}s$$

the sum of the values of the elements of $S$ divided by the number of elements in $S$. Note that this is equivalent to the sum of $\frac{1}{|S|}$ of the value of each of the elements of $S$.

Sometimes not all data points should be considered equally.

Consider a group of ten people, nine of whom have $\$1$ and one of whom has $\$11$. The mean average of their incomes ($\$2$) would indicate that each person makes about $\$2$, but this is misleading, as most people in the group make only half that. To correct for this misrepresentation it is often desirable to give certain points more influence on the final average.

Recall that when finding the mean of a set of data we scale each data point to $\frac{1}{|S|}$ its original value and then sum. This scale is called the weight of the point. Because each data point is scaled by the same amount, each has the same weight. We can make some points more influential than others by shifting their weights, so long as the weights still sum to $1$. Then the weighted average of a set $S$ of points becomes

$$\sum_{i}w_{i}s_i$$

where $w_{i} \in W$ and $W$ is the set of weights of corresponding elements of $S$.

The concept of weighted averages is especially useful in the calculation of center of mass, for which each point of a body is weighted by its density.

Another important average for sets is the geometric mean. Where the arithmetic mean sums the values of each of the data points of a set and then divides by the number of elements in that set, the geometric mean multiplies the vales of each of the data points and then takes the $|S|^{th}$ root.

Thus the geometric mean $G$ of a set $S$ is

$$G = (\prod_{s \in S}s)^{\frac{1}{|S|}}$$

One application of geometric mean is in finance. Perhaps you have a bank account that during each of the past three years earned $7\%$ interest, $6\%$ interest, and $2\%$ interest. Your current investment is $1.07 \times 1.06 \times 1.02$ what it was three years ago. On average your account grows at a rate of $1.0498=(1.07 \times 1.06 \times 1.02)^{\frac{1}{3}}$ per year. Your annual interest rate is then roughly $4.98\%$.

The mean of a some numbers is just the average of the terms. We can take the mean of any collection of numbers