What is the intuition behind the $N$ in the denominator in the definition of variance? There are many questions and answers on why $n-1$ is used instead of $n$ in sample vs population variance.
However, I wonder why the population variance definition even has $n$ in the first place. The definition is given as
$$\sigma^2 = \frac{\sum_{i=1}^n(x_i - \mu)^2}{n}.$$
What is the intuitive purpose of the $n$? Is it just a normalizing term? If so, why was a different normalizing term not chosen?

As said in the comments, variance is defined as
$$
\mathbb E[X-\mathbb EX]^2
$$
Expected value of a random value can be estimated from the sample by using the arithmetic average. Variance is just an average squared deviation of a random variable from its mean. We use average instead of sum because otherwise, it would grow with sample size and be harder to interpret.
But why do we care about average squared deviation? This is nicely answered in Why square the difference instead of taking the absolute value in standard deviation?, but also in Why is the squared difference so commonly used?, and What makes mean square error so good?.
See also Understanding "variance" intuitively.
