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Take the following data set of eight observations:

data = 1.22, 0.93, 1.03, 1.45, 1.07, 1.09, 1.17, 1.20

To compute a confidence interval on this data, I use the following formula:

CI = mean(data) +/- t.crit x (sd(data)/sqrt(df))

... and the results are as follows:

mean(data) = 1.14

sd(data) = 0.155

df = 7

t.crit = 3.499 (.01 in two tails).

CI = 0.94 to 1.35

The strange thing here is that two of the eight observations in the sample (0.93 and 1.45) fall outside of this confidence interval. That seems counter-intuitive to me; shouldn't the presence of these data in the sample have increased the standard deviation enough such that they would fall within the confidence interval?

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    $\begingroup$ Find out what happens when you perform the same computation by replicating the data many times. For instance, replace data by rep(data, 100). Then visit some of our higher-voted threads on interpreting confidence intervals, such as stats.stackexchange.com/questions/26450/…. $\endgroup$
    – whuber ♦
    Commented Apr 20, 2020 at 19:32
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    $\begingroup$ 1) Why would you expect a $99\%$ anything to cover $100\%$? 2) The confidence interval concerns the mean of your data, so if you had many thousands of observations, your confidence would be quite narrow and exclude most of your observed values. $\endgroup$
    – Dave
    Commented Apr 20, 2020 at 19:32
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    $\begingroup$ You're right, it seems I needed a refresher on what a confidence interval actually represents - an estimate of the population mean, not the likelihood of a value that had occurred in the sample would occur again. Thanks $\endgroup$
    – identic0n
    Commented Apr 20, 2020 at 21:33

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Confidence intervals describe uncertainty around the mean, they are not intended to capture the range where most of the data fall. The more data you have, the narrower your confidence intervals will be (all else being equal), and so a higher proportion of data will fall outside the confidence interval.

You could instead use quantiles to describe summarise the data range.

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