standard deviation vs percentiles

No, they are completely different things and you can't convert from one to the other

I guess I don't understand why you say that.

Is it not true that, if a test score is 2 Standard deviations above the average, this score is in the top 2.28% percentile?

Standard deviation is calculated based on the square root of the variance, and the variance is the mean of the data when the dependent variable is squared minus the mean of the data squared.

Percentile is the point on the cumulative distribution function where it equals the percent in question (the cumulative distribution function is the area under the curve of the probability density function)

So...they are calculated from completely different things and show completely different things.

However, if you are looking at data that has a standard normal distribution then they can be approximately related with the empiricial rule.

The empirical rule will tell you that, for instance, 95.44% of the values will be within 2 SDs of the mean. So I suppose from this you could say that if you knew it was greater than 2 SDs above average, it must be at least within the top 2.33% percentile.

If we let
Phi(Z) = cumulative distribution function of the random variable,
z <= standard deviation of data

2( Phi(z) - Phi(-z) ) <=~ percentile

Phi(z) can be looked up in a table for the standard normal distribution, or for an arbitrary normal distribution.

By the way, how did you get 2.28%? It does not follow from your equation:

Percentile = 50 + (2)/2 = 50

percentile vs SD
OK. So what you're saying is that the SD is the theoretical value and the percentile is taken from a group of sampled data. That makes sense to me. But, if the data sample is large enough, aren't all populations normal in distribution? (I seem to remember that from a probability class I took eons ago).

Percentile = 50 + SD/2 ..... well, that's not too clever. I guess I meant the number represented by an SD .. e.g. SD = 2 => 95.44, the area under the curve from SD = -2 to SD = +2. Or better still look it up in the table, as you suggest.

And thank you for your response.

The normal distribution is very common in measurements; it represents data that has some randomness, with natural forces encouraging one stable equilibrium point.

However, it is quite possible to have multiple stable equilibrium points, metastable equilibrium points, unstable equilibrium points, etc...in which case the distribution would not be normal.

Is fetus size distribution normal? Probably, but I don't know. I'm sure it's not standard normal distribution though.

I should probably not say too much more on this because I hate statistics and I don't know much...!

Re: percentile vs SD


I think you're missing the point of the post. Standard Deviation is just a statistical tool to show you what the "extremes" of a normal distribution curve would be within some realm of certanty.

For example, I'll use a manufacturing analogy, one from a direct related statistical "fix" to an everyday design.

Say you have a curtain rod in two pieces. One end of the rod slides into the other end. Say, for example, the inner rod has a diameter of .25" with a tolerance of .05 inches. The outer rod has a diameter of .35" with a tolerance of .05". Theoretically, these two pieces will always fit together, with a very tight fit in the worst condition.

Well, manufacturing facilities use STD DEV to determine their "yield" rate or the number of failures. Say they have 3 failures per 1000 pieces. This corresponds to a 3 sigma level or 99.97% reliability. There is a standard formula to figure out what sigma is, but I know 3 sigma is 99.97.

Now, say they want to get a better yield out of their parts. They have two options, either change the manufacture process, or change the tolerance. The goal is to get 3 failures in 1 million parts (I think this is 6 sigma). In terms of %, it can be described as 99.9997% success.

In this case, they changes the tolerances, the cheap way out. The changed the size and tolerance from .25" tol .05" to .20" tol .10" They in essence make the small part smaller and the large part larger to take up any mismatches. This is how statistics can be used in a detremental form pushed by a corporate mindset. I don't know how it corellates to the doctor problem, but maybe I showed you a little insight as to where the numbers come from and how they extract %s from a standard deviation problem.

Neither standard deviations nor percentiles require a Gaussian distribution, however, one is normally assumed when working with either. For a large enough sample that does have such a distribution, SD (the standard deviation) approaches σ, the parameter in the formula for a Gaussian that determines the width.

By integrating the formula of a Gaussian over the appropriate range, you can calculate the percentile, but it's usually done by looking at a table. The tables generally give z (the distance from the center of the distribution ot the value, measured in units of σ) and A (the normalized area under the Gaussian, integrated from the center to z).

Assuming a normal distribution, and a value greater than the central value, the percentile is:
(0.5 + A)*100%

Assuming a normal distribution, and a value less than the central value, the percentile is:
(0.5-A)*100%


Any statistics textbook or quantitative chmistry textbook will give you this table. I'm sure it is also online in several places, but don't have time to look for it right now.

Turns out I had one on my computer. I've added the percentile calculation results to save you the work.