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I asked people how many times they visited their local pub in a 'normal' week.

The result can be zero, one, two, three, four, and five and more.

  • The mean is 2 and the standard deviation is 1.3.
  • So two standard deviations above the mean is 4.6.
  • However, two standard deviation below the mean is -0.6.

Is this negative figure an error? How do I interpret it?

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In general, if you have any data with a positive mean and nonzero SD, any sufficiently large number of SDs below the mean will be negative. Given that this is a mathematical certainty, an appropriate reaction would be "so what?" My question is, why isn't this the appropriate reaction to your question? Why is -2 SDs special to you? – whuber Oct 20 '11 at 23:20
@whuber I can't answer your question (not yet) but very grateful for prompting me to start thinking on this issue. – Adhesh Josh Oct 21 '11 at 2:50
up vote 7 down vote accepted

The short answer, is no, it is not an error.

As @whuber notes, there is nothing surprising (at least to a statistician) about the fact that two standard deviations below the mean of a count variable could be a negative value. Thus, to answer your question, perhaps it would be more useful to ponder why you might find the result surprising.

Why you might be surprised

  • Many introductory statistics textbooks show how you can use the mean, standard deviation, and the normal distribution to make claims like approximately 2.5% of the sample is expected to score below two standard deviations below the mean. You may have generalised this idea to a variable where the assumptions of such a procedure are invalid.
  • If you did this, you would be saying to yourself: "this is strange, how is it possible for 2.5% of the data to have counts below -0.6".

Estimating percentiles for counts

  • Your variable is not normally distributed, it is a count variable. It is discrete; it is a non-negative integer. Thus, in order to estimate the percentage that is greater than or equal to a given value, you need an approach suited to counts. A basic approach would involve using the sample data to estimate such percentiles. More sophisticated approaches could involve developing a model of the distribution suited to counts, justified by the data and knowledge of the phenomena, and estimated using the sample data.
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Thanks. I think I have learnt what is very obvious to others, that is, count variable is discrete and not normally distributed. I understand the logic here (i.e there is no upper limit to a count variable). Should I just report the mean, median and mode for count variables? – Adhesh Josh Oct 21 '11 at 2:21
I will ask a similar question on composite scores. – Adhesh Josh Oct 21 '11 at 2:51
@Adhesh The standard deviation is still valid for counts. In general when reporting descriptive statistics, you may want to indicate something about the distributional properties of the variable (e.g., skewness, kurtosis, presence of outliers, nature of distribution, etc.) – Jeromy Anglim Oct 21 '11 at 3:13

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