# Confidence intervals and range of possible values

1. Sep 6, 2010

### striphe

When i did a business statistics course some time ago, I was able to calculate confidence intervals, but i didn't understand ‘why’ they were calculated in the way they were.

I considered that the size of a confidence interval is based on the number of observations and ‘the range of possible values’ that the those observations may result in. When calculating confidence intervals for the mean of a population, you use standard deviation rather than the range of possible vales.

The reason i considered the range of possible values rather than the variance or standard deviation, is because they are a statistic of their own and could have a confidence interval applied to them. Something suggests to me that the confidence interval of variance can not be based the number of observations alone and that if the confidence interval of variance is dependent on another factor, then the confidence interval of the variance or standard deviation would affect the confidence interval of the mean.

So where have I gone wrong here?

2. Sep 11, 2010

### Mapes

A confidence interval does not describe a "range of possible values." (This sounds more like a prediction interval.) A confidence interval at the x% level for a certain parameter will contain the true value of that parameter x% of the time. For example, a confidence interval for the mean of a sample can be calculated from the sample standard deviation and the number of observations (it's approximately $1.96\mathrm{SD}/\sqrt{N}$ for a Gaussian distribution, a 95% CI, and large $N$), but it's not the same as the sample standard deviation. Also, there is a confidence interval for the sample standard deviation itself (and this confidence interval may or may not contain the true population standard deviation), but that's a side issue that doesn't enter into the question of how to estimate the true population mean. Does this answer your question?

3. Sep 11, 2010

### striphe

when we did our calculations, i think we were given the population SD.

The smaller your population SD the smallar the confidence interval (not percentage size, but value size). I considered the fact that there is uncertainty with regards to the population SD that one should use the maximum possible SD for a population. This maximum would result if all the observations were evenly the highest possible and lowest possible observation. As it is assumed that the population is unlimited any sample results will be outweighed entirely by the possible highest and lowest observations.

So on second thought my question is how does one estimate the population mean, with only sample statistics?

4. Sep 12, 2010

### Mapes

With the average of the sample values.

5. Sep 12, 2010

### striphe

And give it a confidence interval

6. Sep 13, 2010

### Mapes

Is the data Gaussian? If so, the confidence interval is the mean plus or minus the t statistic times the standard error. (For large N, the 95% t statistic is 1.96, as I wrote above.) More http://en.wikipedia.org/wiki/Confidence_interval" [Broken].

Last edited by a moderator: May 4, 2017