Confidence interval

Question

I have a sample of 160. Mean value is 262, standard deviation is 165. The normal distribution curve comes nicely, from approximately -200 (mean -3*deviation) to near 800 (mean + 3*deviation). On the next step I'm looking at 95% confidence interval. For this example I used function CONFIDENCE in excel and it says that this inteval is [236,287].

My question is - why is this range so small? Each book states and shows 95% confidence interval as a approximately +/- 2*st.dev., which is in my case far from it. What could be the explanation for this? Such small range can be quite deceiving.

Thanks for your comments Gasper

Answer 1

You are conflating 2 things:

• with a normal distribution 95% of values are in the range mean +/- 2 standard deviations

• given the sample mean, what is a confidence interval for the true mean. Excel is telling you that there is a 95% chance that the true mean is in the range 236 to 287. This confidence interval will get tighter as the number of samples increases (for more detail on this, lookup the central limit theorem)

The excel documentation for the confidence function has some further explanation

Answer 2

Sampling data that is individually normally distributed with N(u,s²), the distribution of the average of n samples is N(u,s²/n). That is, the average of the samples has standard deviation s/sqrt(n).

The excel function assumes that it is given the distribution parameters of the single observation and computes the expected distribution of the average of n samples, so it computes s/sqrt(n)=165/sqrt(160) approx 13 and with that the 2-sigma-interval.

Answer 3

The confidence interval is computed based on the standard error, which is not the familiar standard deviation (well, not directly anyhow). The formula for computing the standard error varies depending on the metric being measured (e.g the mean in this case, perhaps Cohen's d in another), a common formula is "standard deviation divided by the square root of the sample size". This is referenced above in the previous answer, but it is referred to as the standard deviation.

I think technically, standard error is simply the name given to the square root of the variance of an estimator, so whether or not calling it standard deviation is wrong or not is beyond my expertise, but if you are trying to compute confidence intervals, you will most frequently find them defined in terms of standard error. I ran into this problem myself recently because I was using the formula given above for the standard error but it is not the correct formula for the standard error of Cohen's d...