# Same mean, different standard deviation in data sets

How would a data set containing the values of a variable with a mean of 50 and a standard deviation of 3 compare with another data set containing the same variable, but a mean of 50 and a standard deviation of 12?

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The individual observations would not typically be as far from $50$ in the former data set. –  Michael Hardy Mar 11 '14 at 22:26
normal distributions? that would mean a larger spread of data in the latter. If the data is not normal, then it also means that you have you a larger spread in data, but it might not be symmetric. –  Chinny84 Mar 11 '14 at 22:26

The first is more precise.

Given the question, that's the only difference in the data. Basically, the data in the second set is less precise, or more spread out.

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With the given information you know that

1. The two data sets have the same central tendency as it is expressed by the mean
2. The two data sets have different dispersion as it is expressed by the standard deviation. In the first data set, the observations are located more closely around the mean (50) compared to the second data set, where they are more dispersed. For example, values above 60 or less than 40 would be very exceptional in the first data set in contrast to the second data set where such values seem to be more common.
3. However, a good tool to compare the difference in the variability (or homogeneity) between two data sets is the Coefficient of Variation (CV), which is given by $$\mathrm {CV}=\frac{s}{\bar x}$$ For the first data set we find that $$\mathrm {CV_A}=\frac{3}{50}=0.06$$ or 6% which denotes a homogenous data set (empirical cutoff is 10%) and for the second data set we find that $$\mathrm {CV_B}=\frac{12}{50}=0.24$$ or 24% which denotes a non-homogenous data set (or a data set with high variability).
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