How can I get the percentile rank of a given score if I know the scores at standard percentiles? For a distribution, let's say I know the scores at the 10th, 30th, 50th, 70th and 90th percentiles. I also know the interquartile range and range. Is it possible to get the exact percentile rank of a given score? I'll be using R if that makes things any easier.
As a more concrete example, I have the following information:
Lowest: 5.10,  Highest: 6.00, iqr=0.12
10% <= 5.25
30% <= 5.71
50% <= 5.84
70% <= 5.95
90% <= 5.96
Let's say I want to know the percentile rank of the score 5.33. I can see that it's somewhere between 10 and 30. Is there a way to calculate this outright, or maybe somehow I can recreate (estimate) the distribution based on those percentiles and estimate it that way?
 A: No way to know, unless you know something about the distribution. Think about the plot of the distribution function, $F$ (continuous for our sake), with range in $[0,1]$. The percentiles you know correspond to the x-values satisfying $F(5.25) = .10$, $F(5.71) = .30$, etc. But the function $F$ could do anything continuously and monotonically increasingly between those two x-values. It could be very flat near $x=5.25$, so that your percentile will be close to .30, or it could rise quickly then flatten near $x=5.71$, in which case your percentile will be close to .10. No way to know unless you know a functional family for $F$.
There are inequalities for percentiles in terms of moments of the distribution, Chebyshev-like inequalites, but those don't seem to help here.
A: You can interpolate it using a monotone spline.  R example code and results appended below.

# Input data
x <- c(5.10, 5.25, 5.71, 5.84, 5.95, 5.96, 6.00)
pct <- c(0, 10, 30, 50, 70, 90, 100)

# Calculate the monotone spline curve
f <- splinefun(x,pct, method="monoH.FC")

# Draw the monotone spline curve
curve(f(x), 5.10, 6.00, col="blue", lwd = 2, xlab="Score", ylab="Percentile")

# Draw a selected point on the spline curve
points(5.33, f(5.33), col="red", pch=16)

