# How can I create a Graph of a Probability Density Function from Percentiles?

I play a game online (Heroes of Newerth) which has a large ladder of players, each player having a couple of different ratings. I've manually gathered the rating data for all percentiles of players but am blanking at how to turn this into a histogram and compute the Mean and Standard Deviation. I'd also like to have a graph to show the curve.

I could easily do so using my TI-83+ and I consider myself very proficient at Excel, but I don't see an obvious way to do this. Below is a link to the data, or if you'd prefer to give me instructions on how to create the distribution I'd be happy to do so myself.

-
Would you include some example data? –  csgillespie Jun 27 '11 at 14:34
Edited the OP and beat you to the punch. -.- As you can see, there are two different ratings, MMR and PSR. Many people dispute a correlation between the two, so I decided to settle the discussion by doing a bit of statistical analysis. This is where I've become stuck. I'd like to compare the Mean, StdDev, and histogram graphs of the two ratings. Thanks. –  Decency Jun 27 '11 at 14:36
@Decency To look for correlations, begin by drawing a scatterplot of the raw (per-person) (MMR, PSR) pairs. Comparing their histograms (and means and SDs) tells you nothing about correlation. –  whuber Jun 27 '11 at 19:40
@Decency Any reason for reverting @mbq's edits about title sentence capitalization that was discussed on Meta (but see other questions having the word "histogram" in their title)? –  chl Jun 27 '11 at 21:03
@Decency (If you want to continue this discussion about editing, please do it on meta or on chat.) Concerning your analytical objectives, looking at the individual curves still tells you nothing about how ratings "align" or are correlated. One way to handle large datasets is with random subsampling. In your case, a scatterplot involving, say, 1000 randomly selected (MMR, PSR) pairs will be informative and accurate. –  whuber Jul 4 '11 at 16:42

Your life is a bit easier because you have values at every integer percentile.

1. To calculate the mean and std deviation just take the mean and standard deviation of the columns.
2. To plot histograms, calculate the difference between the $i^{th}$ and the $(i-1)^{th}$ value. You can now just take histograms of these numbers to get what you want. Personally, I won't bother with a histogram, but would just plot the differences.

When I played about with your data, I got a "U-shaped" distributions. Indicating that there are lots of very good player and lots of players who start and then quit the game.

-
Thanks for taking a look and the instructions. If I told you that players start at 1500 rating and these ladders use a modified Elo system, would that change what you said in the last sentence? –  Decency Jun 27 '11 at 15:01
Yes. If players start at 1500, that implies that people are either good or bad, with very few people in the middle - hence the U shape. –  csgillespie Jun 27 '11 at 15:07
Sorry it's taken so long for me to get back to you, I kept forgetting to look at the data. After playing with it and your solution, I believe I've asked the wrong question. What I'm looking for is a probability density function, or a curve that looks similar to a normal distribution. That explains your confusion at my asking for a histogram. –  Decency Jul 4 '11 at 3:51
You have the probability from the percentiles. What's the probability of getting over the 50th value - 0.5. What's the probability of getting over the 95th value, 0.05. –  csgillespie Jul 4 '11 at 8:55