# Compare Log-Normal Distributions

I am looking at distributions for the fuel consumption of vehicles in the US and I have two sets of data:

Data Set 1 - This is a dataset for the fuel consumption of all vehicles made available for sale from 1990 to 2010 (not sales weighted), which follows a log-normal distribution. Thus, I am able to define the mean and standard deviation for the entire population of vehicles available for sale each year.

Data Set 2 - This just has the annual average sales weighted fuel consumption of vehicles in the US from 1990 to 2010. Thus, I am only able to define the mean (no standard deviation) of sales weighted fuel consumption each year.

Is there any way for me to infer a distribution for the sales-weighted fuel consumption with the data in Data Set 1? Put another way, am I able to establish the standard deviation for Data Set 2 using the data that I have available in Data Set 1?

Any advice would be greatly appreciated and many thanks!

• My control systems engineer self smells a transfer function. :) Sep 9 '14 at 23:13

## 1 Answer

These data sets appear to have two fundamentally different types of data.

Data Set 1 contains a distribution of fuel consumptions for vehicles in each year. This distribution has a mean and standard deviation.

Data Set 2 contains one number for each year - the sales-weighted average fuel consumption. This number is not a distribution. It exists only as an aggregate measure of the data in Data Set 1. The distribution you would have from Data Set 2 is the distribution of sales-weighted average fuel consumption in different years.

• Thanks for the response. You are correct when you say that Data Set 1 is a distribution and Data Set 2 is just a number. But there is a difference in that Data Set 1 contains data for all vehicles available for sale. Data Set 2 is just the average of sales weighted vehicles (not vehicles available for sale). Thus, it's not fair to say that "[Data Set 2] exists only as an aggregate measure of the data in Data Set 1." So, to clarify, you don't think there's a way to establish the standard deviation of Data Set 2 from Data Set 1? May 8 '14 at 22:49
• In the precise use of the language, a data set does not have a mean and standard deviation - a distribution which is part of a data set has a mean and standard deviation. You can find the standard deviation of the distribution of sales-weighted average fuel consumptions over the years (i.e. how much the average varies over the years), but there's no distribution for the sales-weighted average for any one year. May 9 '14 at 0:11
• Thanks @PSchnell. I guess the best I can do is to put a triangular distribution on the sales-weighted data with the max and min values from Data Set 1 (Available vehicles) and the mean from Data Set 2. May 9 '14 at 0:18
• @hoof That's far from the best you can do, because positing a triangular distribution is just inventing information where none exists. Any subsequent results of your analysis would therefore have this (potentially huge) element of arbitrariness to them, which is rarely a good thing.
– whuber
May 9 '14 at 14:05
• Yes: the main thrust of such (negative) considerations is to suggest there is value in collecting more information. There's loads of data available about fuel consumption in most major countries. Even if you can't find exactly what you're looking for, you at least should be able to find enough information to make some defensible adjustments or assumptions about the SD of the sales-weighted fuel consumption.
– whuber
May 9 '14 at 14:12