Multiple grammatical issues, but I've heard raters are more flexible on grammar for non-native speakers.

Let's look at structure. From reading your intro here are your three reasons:

1. Number of schools surveyed
2. "Place"
3. Year the article was published

#1 seems quite relevant--though it says the survey is "nationwide" there is no mention of the number of surveyed schools. I'm not sure what "place" means but location and distribution of the schools may also affect the survey results. Year published seems irrelevant since the argument pre-empts your response by claiming that it's "recent." You may be right in pointing out the ambiguous time marker but it's not the strongest refutation you could make.

I like the topic sentence in your first paragraph, but a single sentence does not make a paragraph. Expand on your point with another sentence or two. Point out a possible counterexample or specific items that are missing from the survey. In your introduction you mentioned sample size. So point out that the sample size may be miniscule.

The second paragraph was grammatically nonsensical, though I understood fully what you were trying to say. More clearly, some public schools may have extraordinarily low on-time graduation rates. But this does not refute the argument! The argument claims that in fact, fewer students graduate on time from public schools. So you fail to make a strong claim here.

I like the idea in your third paragraph. It may well be that public schools offer a better and more difficult educational experience, and thus fewer students graduate. On the other hand, private schools may have more lenient graduation requirements possibly as a result of their private administration. BUT you failed to read the question accurately. The prompt stated that parents who wanted their children to graduate on time should send their children to private school. It NEVER says "parents who want a better education for their children." Do not be confused--the prompt asks you to analyze whether parents should send their children to private school if they want their children to graduate on time. You don't refute the argument.

Your conclusion is good if a bit formulaic. It is also rife with redundancy--"strengthened" and "bolstered" mean the same thing, whereas "persuasive" and "convincing" also mean the same thing. You should aim for more words without wordiness.

Overall I think there were serious flaws, but with a little work you can improve your AWA a lot. I encourage you to look at the AWA 6.0 guide how-to-get-6-0-awa-my-guide-64327.html.

Here are some arguments you could have made, and which apply to most AWA Analysis prompts.
"Generalizing from particulars" -- It may be true that GENERALLY private schoolers graduate on time, but perhaps the private schools are so terrible in Bedford Falls that in Bedford Falls the public high school graduates more students on time.
"Faulty causal reasoning" aka "Confounding correlation with causation" -- It may be true that GENERALLY private schoolers graduate on time more than public schoolers, but that does not mean private school is the CAUSE of graduating on time. It may be that wealthy children are more liable to graduate on time, and wealthy parents usually send their wealthy children to private school. In fact, wealthy public schoolers are more likely to graduate high school on time than low-income private schoolers. In other words, there may be an alternate causal explanation at hand.
You did well to question the detailed methods of the survey as you did in the first paragraph.

Hope this helps.