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I am not a math wizard, so please keep your response simple enough. I need to complete a statistics screening exam for a methods course later on today, and I am hung up on one topic that came up during the practice test. The data set I got was in reference to the number of homicides that have occurred in a number of cities. The range of this data is 0-5. When I am putting together confidence intervals and calculating out as much as two standard deviations from the mean I am getting low values that are negative. Obviously you cannot have a negative number of homicides. When calculating the confidence intervals out to two standard deviations from the mean should I present the low value at ZERO or should I actually present the negative number? For example, if a 95% CI caused the calculation to be -1.5 to 3, would I present that or would I present 0 to 3? Thanks.

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This issue is common when dealing with positive numbers. To the best of my knowledge, both are accepted depending on point of view. However, what people most often do is to construct a confidence interval for the logarithm of the parameter ($\in \mathbb{R}$) and then exponentiate the confidence interval. – ocram Jan 17 '13 at 8:14
Thanks for the help. Logs aren't part of the practice test material so I am going to assume that they will not expect that of me at this point. Perhaps it will be covered during the course itself. – Rick Jan 17 '13 at 8:17
Does the question require you to calculate two standard deviations of the data out from the mean of the data? Or does it require you to give a confidence interval for the mean? Although the confidence interval for the mean involves standard deviations, these are two very different calculations. The second one makes more sense for a stat exam to me, and it is much less likely to go beneath zero. If you do need to calculate the confidence interval for the mean, you will need to calculate the standard error of the mean and use that instead of the population standard deviation. – Stephan Kolassa Jan 17 '13 at 8:53
Great point @StephanKolassa. Why would we be interested in two standard deviations from the mean in a distribution such as this which is unlikely to be normal (which is unlikely to be symmetrical). But an estimate of the mean homicides, and plus or minus two standard deviations of the estimate of the mean, would be of interest as estimate of the mean is roughly normally distributed (as sample size gets bigger) under the central limit theorem. – Peter Ellis Jan 17 '13 at 9:54
@StephanKolassa - Suggest you combine your 2 comments into an answer field. – rolando2 Feb 11 '13 at 14:29

It appears unlikely to me that the question would require you to calculate two standard deviations of the data out from the mean - especially given that your data are unlikely to be even symmetric, much less normally distributed (since they are discrete). I see no interesting question that could really be answered by this calculation.

It appears more likely that you are asked to give a confidence interval for the mean. This also involves calculating the standard deviations of the data, but then you calculate the standard error of the mean from this standard deviation by dividing by the square of the sample size and finally construct the confidence interval based on the standard error. This confidence interval is therefore much less likely to go beneath zero (and if it did, you should indeed truncate at zero). Note that the sampling distribution of the mean will be roughly normally distributed as sample size increases, which is why this interval actually answers an interesting question, namely where we expect the actual mean to lie.

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Hat tip to @PeterEllis, whose comment on the original answer I shamelessly included in this answer. – Stephan Kolassa Feb 12 '13 at 13:40

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