What is $\cfrac 1n$ in this expression?

Question

I bought a T-shirt at Kenendy Space Center a while back that had this expression on it:

$$ B > \frac1n\sum_{i=1}^n x_i $$

And below it is captioned "Be greater than average." I can see the "Be greater than" and the "average" in the expression, but I do not understand what the $\cfrac 1n$ is for.

So, what is the purpose of $\cfrac 1n$ in the expression above?

(Sorry, I have absolutely no idea how to tag this)

Answer 1

The term $\dfrac 1n$ is part of the average. If we want to find the average of two numbers $a$ and $b$, the average would be $\dfrac{a + b}{2} = \dfrac 12(a + b)$. The denominator is equal to the amount of variables we have. In this case, we have two variables $a$ and $b$, so we add them together and divide it by two. This means that the average of $a, b$ and $c$ is equal to $\dfrac{a + b + c}{3} = \dfrac 13(a + b + c)$. The average is also known as the arithmetic mean, and the right hand side of the inequality is just notation for the average of $n$ given variables denoted as $x_i$.

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$$\sum_{i=1}^n x_i$$ is read as the sum that goes from $x_1$ to $x_n$. In other words, it is the sum that takes the values of $x_i$ from $i = 1$ to $i = n$. We can now write this as $$\sum_{i=1}^n x_i = x_1 + x_2 +\cdots + x_n$$ and now to find the average of these $n$ different variables, we divide by the total of how many there are, which is $n$. This is the same as multiplying the entire sum by $\dfrac 1n$ which makes the right hand side of the inequality define the average of $n$ variables. Now it is clear that the symbols $$B > \frac 1n \sum_{i=1}^n x_i$$ means be greater than the average.

Answer 2

It's because the (arithmetic) mean (that is, the average) of the numbers $x_1,\ldots,x_n$ is$$\frac{x_1+\cdots+x_n}n=\frac1n\sum_{i=1}^nx_i.$$

Answer 3

The average (arithmetic mean) is the total of all entries over the number of them.

Answer 4

That formula literally means

BE(B) GREATER($>$) THAN THE AVERAGE($\frac{1}{n}\sum_{i=i}^{n} x_i$)

Which mathematically is written as $$\large{B>\frac{1}{n}\sum_{i=i}^{n} x_i}$$

So the sum divided by $n$ represent the average here