I have a problem related to a formula and I need help from all of you. It may use some statistics formula to solve the problem. Your help would be appreciated.
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We have following values (Percentile and respected values)
3% :——————: 7.7999999999999998
15% :——————: 8.5999999999999996
50% :——————: 9.5999999999999996
85% :——————: 10.800000000000001
97% :——————: 11.800000000000001
Now we need to find the percentile of given value with respect to above values
Suppose Given Value -> 8.2 Then what will be the percentile of 8.5 (It should be less than 15% and greater than 3%)?
Then what will be the percentile of 10.0 (It should be less than 85% and greater than 50%)?
Best Answer
The intervals you suggest are reasonable. Unless you have additional information or want to make assumptions, they are the best you can do.
I suppose the information you have is from a sample and you're asking for approximate sample percentiles. Are those supposed to serve as estimates of population percentiles? I assume you don't have access to the entire sample.
You could use the information given in your question to make a plot of the estimated CDF (as below). If it looks like the CDF of a known distribution, perhaps try to fit that. Otherwise, use linear interpolation.
In this case, the population may be that of a normal distribution. If so, you can estimate the mean as about 9.65 (median is 9.6, but other quantiles suggest a little higher) and from the quantiles you can deduce that the SD is about 1.03 or 1.04. Superimposing the CDF of $\mathsf{Norm}(\mu=9.65,\,\sigma=1.035)$ [heavy blue curve] on the estimated CDF shows a pretty good match.
If you believe data may have come from this normal distribution, then the required percentile estimates are 8th percentile for 8.2 and 63rd percentile for 10. [Computations in R below. In R,
pnorm
is a normal CDF.]These estimates are in the intervals you suggested. Perhaps they are more useful than what you would have gotten by linear interpolation using the estimated (broken line) CDF. I will leave the linear interpolation to you. Results should be about the same.