Confidence Interval

Chirantana · November 13, 2014, 1:23am

Hello Everyone,

Please help how 95th percentile point = 1.65 differ from when we talk about 90% confidence interval is 1.65 standard deviation from mean .

Both talk about % of the population then what difference they make?

Thanks

Ron_Swanson · November 14, 2014, 2:19am

A 90% confidence interval uses the 95th percentile in calculations (Z-score of 1.65). This means 5% are below this level and 5% are above. This corresponds to below the 5th percentile or above the 95th percentile. Note: this comment has been edited (see below)

Chirantana · November 13, 2014, 2:11am

Hello Ron,

I agree with your point.

Let say I have to calculate range of portfolio return that is earned 90 % of the times

Confidence interval = Mean+(-) Z score * Std deviation

in this case Y i am using Z score=1.645 which is 95th percentile - tells P(Z lesser than 1.645) what about that below 5th percentile?

Why not is 1.285 for 9oth percentile

thanks

Ron_Swanson · November 13, 2014, 3:19am

You would only do that if you wanted a one sided confidence interval, to determine a retun that is only achieved 10% of the time. The Z table lists percentiles, which is why the value for a Z score of 0 is .5000, this means that 50% of the values are below that value, not that 50% of values are exactly there. If we wanted to find the range that 50% of values fell in we would use the 75th percentile (so that everything between the 25th percentile and 75th percentile is included) . Same thing here, with the 95th percentile we have 5% on each side, 10% total, and 90% inside the confidence interval.

tickersu · November 13, 2014, 7:16pm

Ron_Swanson:

A 90% confidence interval means we are 90% sure the true mean of a data set is within +/- 1.65 standard deviations of what?.

This is not what it means- it does not mean “sure”, and it is standard errors, not standard deviations (review the calculation and ideas for a confidence interval). A confidence level comes from a theoretical idea.

For example, for a 90% confidence level: In repeated sampling, 90% of all possible similarly constructed intervals will capture the true parameter, and 10% will not capture it.

A 90% confidence interval can be interpreted that we are 90% confident the true population parameter for all experimental units is contained within the calculated interval. There is a .90 probability of randomly selecting a confidence interval that contains the true parameter.

Same thing here, with the 95th percentile we have 5% on each side, 10% total, and 90% inside the confidence interval. This isn’t very clear. 1.645 is the 95th percentile, yes. Since we want an INTERVAL (lower, upper bounds) we split the 10% and see we should have 5% in each tail, beyond our interval. Then we see this corresponds to the 5th and 95th percentiles and z-scores of +/-1.645. Within these z-scores is a probability of .9 (90%).

Chirantana · November 13, 2014, 11:36am

Hi Tickersu,

But in case if you have normal distrbuted popoulation parameters with you in that case you can replace std error with std dev .

I think what I was missing is

90% 2 Tail = 95% 1 Tail = 1.96

Thanks

tickersu · November 13, 2014, 6:41pm

This is incorrect. A population parameter (true mean, true proportion, etc) is a single value and does NOT have a distribution. There is a sampling distribution of the estimators for the parameter, though. When you do a confidence interval for a population parameter, you are using the standard error which is like the standard deviation of the sampling distribution, but a standard error is NOT the same as standard deviation.

If you are using a confidence interval for a SINGLE OBSERVATION (not parameter) then you can use the standard deviation to create the interval. If you are marking a confidence interval for a population parameter, you use the standard error from the sampling distribution.

In your example, if you are looking for a 90% confidence interval for a single observation or where 90% of the data will be located within, then you can use the standard deviation (your main question, I believe).

But this is NOT a population parameter. I suggest you review this terminology and see the differences between a confidence interval for a single observation (again, the problem you were asking) and a confidence interval for a population parameter (the reply of Swanson).

Chirantana · November 13, 2014, 9:26pm

Hi Tickersu,

Well it was typing mistake

I want to say when population is normally distributed and population parameters- variance or std deviation is given, in that case I can replace std deviation of sample with std deviation of population while calculating std error.

Ya std error is not adjusted std deviation but not exact deviation.

I agree with statement for confidence interval .

I must say that you are very much through with your concept.

thanks

Ron_Swanson · November 14, 2014, 2:22am

Well I think at least the 95th percentile issue has been cleared up, and some review is always helpful. Sorry if anyone was confused, my first comment certainly was imprecise (I also have the bad habit of thinking about every Z-score as a standard deviation), and has been changed. Given the question was about the disconnect between confidence intervals and Z-score percentiles the comments referencing an example have been removed (the interval and percentile logic works in a number of situations, and and the question was about that single input so in retrospect I probably shouldn’t have used an example the first time around, and it wasn’t really a full blown example anyway). Original comment below for reference. A 90% confidence interval means we are 90% sure the true mean of a data set is within +/- 1.65 standard deviations. This means there is a 5% chance it is below it and a 5% chance it is above it, and therefore any data sets that have means below the 5th percentile or above the 95th percentile will fall outside of our 90% confidence interval.