Why is the center of data important?

I have recently begun learning statistics along with Python with the Anaconda IDE and started a Machine Learning course by Andrew Ng. I understand the mathematics behind what I am doing as the textbooks usually explain this part very well. However, whether it is the mean, median, mode, or any other technique used to find the center of data, I am wondering why we do this and what real-world results are after we apply it in a professional setting. Thank you.

Actually it has nothing to do with being "in the center". In fact, those statistics can sometimes be far from what many people consider as "center", e.g. for skewed distribution. The point is rather that they provide a single-value summary of the distribution: mean is the value where probability mass concentrates, it is the outcome that we "expect on average", median is the value that divides your data in two halves (50% lies on the right of it, 50% lies on the left), while mode is the "most common" value. Each of those statistics gives you some simple summary of the data.

• Tim there might be a typo in your answer. You defined "mode" twice. The first "mode" was probably meant to be median. – Stefan Mar 1 '18 at 21:59
• @Tim From your experience, why are these simple summaries of data important? – StatisticallyDead Mar 1 '18 at 22:45
• @StatisticallyDead say that you have dataset of 10000 cases by 1000 variables, would you look at each of the 10000x1000 values individually to get a feeling about the data? Summary statistics summarize the data to make understanding it easier. – Tim Mar 1 '18 at 23:00
• @Tim Thank you, I guess I have to keep in mind the big picture of statistics each step of the way as I continue to learn it. Often times we cannot have the entire population in a data set so we have to rely on a mean of a sample to help us make an inference to learn what the mean of the population may be. – StatisticallyDead Mar 1 '18 at 23:07

I think the question is too broad to be answered. But one example would be : assume $A$ is your data matrix,

If you center columns (variables) of $\bf A$, then $\bf A'A$ is the scatter (or co-scatter, if to be rigorous) matrix and $\mathbf {A'A}/(n-1)$ is the covariance matrix.

Details can be found in

Is there an intuitive interpretation of $A^TA$ for a data matrix $A$?

If no centering, such nice property will disappear. A related post here, explains why it matters in the application of PCA

How does centering make a difference in PCA (for SVD and eigen decomposition)?