Intuition and the math behind normalization

Question

What exactly is the purpose of normalization. From what I read, it is to adjust two different sets of values so you can compare them, but I don't understand why, nor the math behind it. Could anyone answer this by also giving a concrete example? Thanks!

Answer 1

Note: I'm assuming you are talking about statistical normalization.

Let's use an example. So colleges accept the SAT scores and ACT scores for their admissions processes. How do you compare the SAT scores to the ACT scores, when the two tests use two different scales? The answer is to normalize the data. That is, we're interested in averages and distances from the averages.

So the SAT has a mean score of say, 1700 (out of 2400) with a standard deviation of 150 points. The ACT has a mean score of 25 with a standard deviation of 1.3. Student A scores a 2000 on the SAT while student B scores a 27 on the ACT. Who is the better candidate for admissions?

Hence, we normalize. We take a z-score: $z_{SAT} = \dfrac{2000 - 1700}{150}$, and $z_{ACT} = \dfrac{27 - 25}{1.3}$. This allows us to see where the two points fall on a standard normal distribution $N(0, 1)$ (a mean of $0$ and standard deviation of $1$). We see that the student taking the SAT is two standard deviations above the mean, while the person who took the ACT is $1.54$ standard deviations above the mean. And so, assuming the two tests are equally valid measures of qualifications, then Student A would be the better candidate for admissions.

Does that make sense?