What makes one "Mean" value more important than another?

Question

As of TODAY, let's say that my data has a 5-day Mean of 10 with a Standard Deviation of 2 and thus 2 Standard Deviations of 4. Statistically speaking, with approximately 95% certainty, my data should fall between 14 and 6. Or should it?--hence my question. For example, what if as of TODAY my data ALSO has a 10-day Mean of 14 with a Standard Deviation of 3 and thus 2 Standard Deviations of 6. This tells me that my data should fall between 20 and 8 AS OPPOSED to 14 and 6. Is there a way to measure which one of these Means is more important for purposes of defining outer boundaries for my data TOMORROW. Is it some kind of "NORMAL DISTRIBUTION" test that I should perform.

In any case, without using Mathematical symbols (because I'm not completely familiar with them), could you please explain to me, step by step, how to determine which Mean is more important?

Answer 1

Your 5-day mean, while it has a smaller interval, is less accurate. Your 10-day mean, while it has a larger interval, is more accurate. To better understand this without math, its best to understand that as your number of samples goes up, your data will "normalize", or approach a consistent distribution. While it is impossible to say within what finite range, with 100% accuracy, your mean will fall, your interval will become more and more accurate as your number of samples increases. So, as per predicting the results after tomorrow's data is entered, your 10-day mean is more accurate. Understand also, though, that neither mean is a tool for predicting the results of tomorrow ALONE, rather they are tools of predicting the interval after tomorrow's data is ADDED to the data ALREADY COLLECTED.

Steve

Answer 2

That depends entirely on what you're measuring. If you're talking about a variable whose probability distribution can be expected not to vary with time, the 10-day mean is more representative of the true mean, simply because it involves a larger sample. On the other hand, if the probability distribution does change with time, then the 5-day set is probably more representative, because it is derived from a probabilty distribution closer to whatever the probability distribution is now. Of course, that assumes the probability distribution changes slowly and is roughly monotone - if it changes chaotically with time, there's no guantee that the average distribution over the last 5 days is closer to the current distribution than the distribution over the last 10 days, and so you may get a better value from the last 10 days, since it is closer to the average probabilty distribution over time.

An interesting point of note here is that standard deviation of the values over the last 5 days is LESS than the SD over the last 10 days. This suggests that either:

whatever you're measuring has (temporarily) become more stable, or

much of the variance in the first 5 days was due to measurement error, and your methods have since improved.

In the second case, the 5-day mean is definitely more representative of what you can expect tomorrow, because your measurement methods tomorrow will resemble those used over the last 5 days more than they will resemble those used over the last 10 days. Plus, since they are presumably better, this value is also more representative of the underlying phenomenon being measured.

In the first case, it could go either way. Ask yourself whether the phenomenon is of such a kind that the decreased variance is likely to be more than a coincidence (and is likely to continue) - if so, go with the 5-day mean, if not, go with the 10-day.

And if you still have trouble deciding, go with the 10-day. After all, you are less likely to be wrong if you pick a strictly wider range, no matter what your data.

Answer 3

It depends on a lot of things. Do you expect that the underlying distribution of the observed values is unchanging? If so, then a confidence interval built with more data is better. If you expect the distribution type remains the same, but its parameters are changing, then you have a problem. Do you use the most recent data (to better estimate the changed parameters), or do you use more data (to have a more reliable estimate of the parameters) and attempt to measure the change in the parameters by using the data as two separate samples? It's not a simple question.

In other words, there is no simple answer to your question without understanding the nature of the observed data.

You may want to look at the field of Bayesian statistics, which can also help you to understand the value of a priori knowledge vs. a posteriori observations.

Answer 4

The way to determine which mean is more significant is to know how many data samples were taken. The more data that is used to compute the mean and standard deviation, the better the accuracy. Obviously collecting a million data points for a particular experiment would be more accurate than collecting just ten data points.

Answer 5

The 10-day mean is more important, because more data is being sampled. Suppose you are measuring the average height of the people living on the planet of age 15. It is impractical to measure every 15 year old on the planet. Suppose you measure all the 15 year olds in your school; that would be a start. But maybe your school has overly tall or overly short 15 year olds; your mean would not be as good as taking the measure of 15 year olds from your school, as well as a few others. Thus, the more data, the closer to true your mean will be.

Answer 6

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