# Can the standard deviation be larger than the mean

Hi there, this is the second time I calculate the conventional deviation value and the number is higher than the average. I wonder if this case is normal and what is the easiest explacountry to such result? Thank you in development...

You watching: Can the standard deviation be larger than the mean Get ago to basics. Remember the standardized normal distribution has actually expect zero and SD 1 (thus SD > mean). Additionally, including or subtracting a consistent worth (say the mean vlaue of a variable) will certainly not adjust its typical deviation. As such measurement scales (e.g., likert scales) based upon 0 to 6 (such as degree of agreement) have the right to be simply also transformed to -3 to +3 and also therefore most likely to yield reduced intend values closer to zero. This transformation will certainly not influence the spreview of the circulation and have the right to hence the SD is even more likely to be higher than the 'mean' value.
Of course if you are capturing indevelopment such as the respondents' annual earnings wbelow you suppose each worth to be big, one would not suppose the SD for this measure to be better than the mean. correct Dr. Peter, I check out what you intend. i forobtain around the negative worths. actually my data collection does not have actually negative values however i think the wide array between my big data set is the factor. your explacountry was so comprehensive. i appreciate your aid.

Get earlier to basics. Remember the standardized normal distribution has suppose zero and also SD 1 (thus SD > mean). Furthermore, adding or subtracting a constant value (say the mean vlaue of a variable) will certainly not change its typical deviation. Therefore measurement scales (e.g., likert scales) based on 0 to 6 (such as degree of agreement) can be simply as well transdeveloped to -3 to +3 and also therefore most likely to yield lower suppose values closer to zero. This transdevelopment will certainly not affect the spreview of the distribution and can thus the SD is more likely to be better than the 'mean' worth.
Of course if you are recording information such as the respondents' yearly revenue wright here you suppose each worth to be huge, one would certainly not expect the SD for this meacertain to be higher than the intend.
Assuming your affiliation is correct, then below can be an example of some even more "useful relevance":
Consider you count fishes of a certain species in a certain location of a reef. Throughout the counting procedure, each of the fishes approximately this reef have to have actually the very same and also constant probability to be counted. You carry out not count all the fishes that live in the reef, only those that swim with your counting-location within a solved time period.
Each time you repeat this experiment you will count a much more or less different number of fishes. From the very same and constant probabilities for each individual fish to be counted follows what we have the right to mean around such counts. This expextation is mathematically expressed by a distribution design, and the design here is the Poisson circulation.
Given the probabilities are frequencies (the probability of observing a fish is approximated by the family member frequency with what the fish is counted) then the Poiskid circulation describes the meant relative frequencies (or proportions) of counts in a long series of such experiments.
The Poischild circulation has one parameter, lambda, what equals both the intend and also the variance of the counts. The typical deviation is the square-root of the variance. This alone demonstrates that for a mean counts below 1 the standard deviation will be larger than the mean. I attached a number mirroring the Poisboy circulation for a mean count of 0.5 fishes (per experiment). The distribution shows that most frequently (in 60% of such experiments) tbelow will be not a single fish counted (k=0), in 30% of such experiments one fish will be watched, and also in 10% 2 or more fishes will certainly be seen.
The mean is 0.5 and also the variance is 0.5, too. The standard deviation is sqrt(0.5) = 0.707 what is bigger than the expect.
Now you may not be interested in repeating the counting in the very same area all the moment. This area may be special and might not be representative for the reef. So it is much better to repeat the counting in different locations of the reef to obtain a better (= more typical for the whole reef) estimate for the average number of fishes. But this can cause a violation of the presumption that the probabilities of seeing a fish are equivalent and also constant; tright here may be locations where the fishes are rare and others wright here the fishes are regular. As such the Poiskid model becomes inproper. Your counts will certainly vary also even more that intended from the Poisson model. This is referred to as "overdispersion": the variance will be bigger than the mean. It should be obvious from the previous explacountries that for such overdistributed information the standard deviation can be bigger than the males also for higher suppose counts.
The second figure mirrors an example. This is essentially a mixture of two distributions. You can imagine that the 25% of the reef is grvery own with a particular form of coral, and also in these areas the mean number is 100, and in the various other areas the mean number is 1. If you count many such areas, then the distribution reflects what you will certainly get: most of the moment you will watch zero or one fish, however you will additionally regularly have actually counts between 90 and also 110. The expect is 34 and also the conventional deviation is 47.
So you watch that it is possible for genuine information that the standard deviation is bigger than the mean. In this certain instance it is even more interesting that the variance is larger than the expect. If you have actually a sample of count data and you uncover that for your information the (sample) variance is bigger than the (sample) mean, than this suggests that the Poisson distribution may not be correct to medel the abundance, and this in turn suggests that the locations are not as homogeneous as you thought. This inhomogeneity can mean that tright here are kinds of "preferred places" or "avoided places" of the fishes, and it might be exciting to search a feasible factor for this to better understand also the biology of these fishes.