# Can I reasonably estimate the population mean and standard deviation from a large sample all taken at a single percentile?

I am currently looking at a dataset of Fair Market Rents which are determined at different percentiles over the years - for example, nationally in 1983 they were all set at the 40th percentile, and in 2005 you had some areas set at 40 and some at 50. I would ideally like to try to standardize the data for one particular state in some way to allow for easier comparison (e.g. transform all the 40th and 45th percentiles to the 50th percentile) over time.

As reasonably pointed out in this thread, fundamentally having only a single percentile point does not allow estimation of two other distinct values. However, does this extend to if one has a range of percentile values? This doesn't fit with a sampling distribution approach, since all the data is again taken at a single percentile already, though it's a bit difficult to tell from the documentation how often this adjustment is done from raw data (e.g. phone surveys) vs. predictive methods and models (e.g. local CPI adjustment) - though this is likely a moot point given the raw data is not released to my knowledge.

I do have access to a wide range of rent values often calculated at that same singular percentile which has its mean and variance in various areas of the country, with the above caveats, which is not a single point. Plus, a 40th percentile value is definitionally part of a normal distribution of rents.

Is this ever sufficient information to make a reasonable guess about the overall mean or variance, thus allowing at least a somewhat acceptable standardization to the mean using z-tables? Or would I still just be guessing with only a "single" data point?

Means

I would not be comfortable estimating mean rents from data at 40th and 50th percentiles. There are too many possible changes at the extremes of the distribution which can affect the mean and standard deviation without showing up in the middle percentiles.

For example, imagine two datasets of rents, one with and one without summer vacation rentals. These datasets might have similar 40th and 50th percentiles, but the one with summer vacation rentals would have a higher mean, and the 40th and 50th percentile data can't tell you which dataset you have.

Medians

I'd be more comfortable standardizing to the 50th percentile, using difference-in-differences.

For example, consider a model where each state at each time has a lognormal distribution of rents $$LN(\mu_{s,t},\sigma_t)$$, with auto-correlation across time and positive correlation between states. In this model the dispersion of rents varies by time but is constant across states.

Let $$L$$ be the states with 40th-percentile data in 2005, and $$M$$ be the states with 50th-percentile data in 2005. Let $$\mu_{L,t}$$, $$\mu_{M,t}$$ be the average $$\mu$$'s for those groups of states at some time.

Then we can estimate \begin{align} A&:=\mu_{L,2004}-0.25\sigma_{2004}\simeq\text{mean-log of 2004 data for L's}\\ B&:=\mu_{M,2004}-0.25\sigma_{2004}\simeq\text{mean-log of 2004 data for M’s}\\ C&:=\mu_{L,2005}-0.25\sigma_{2005}\simeq\text{mean-log of 2005 data for L's}\\ D&:=\mu_{M,2005}\phantom{-0.25\sigma_{2005}}\simeq\text{mean-log of 2005 data for M's} \end{align} The difference in differences is $$(A-B)-(C-D)=\mu_{L,2004}-\mu_{M,2004}-\mu_{L,2005}+\mu_{M,2005}-0.25\sigma_{2005}$$ If the states in the two groups changed similarly between 2004 and 2005, then $$(A-B)-(C-D)\simeq -0.25\sigma_{2005}$$ which gives the factor for standardizing 2005 figures from 50th to 40th percentiles or vice versa, relying only on the model being accurate in the middle percentiles.