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These data on housefly wing lengths provide an excellent example of normally distributed data from the field of biometry.  The normal distribution, one of the most widely used distributions in statistics, is often referred to as the Gaussian or bell-shaped distribution.  In a normal distribution the mean m and the standard deviation s determine the position and shape of the histogram, respectively.   What is meant by "position" is the location of the center of the "bell" along the horizontal axis.  The bell is located at the mean (which in a normal distribution is also the value of the median and mode).   What is meant by "shape" is the spread or width of the bell-shaped curve, which is a measure of the variability in the data.  In other words, the standard deviation s measures how spread out the data are from the center of the data, i.e. the mean.  Because m can assume any real number value, and s can assume any non-negative real value, there are infinitely many unique normal distributions.

In general, a variable or quantity that is the result of many factors that act independently and additively will be normally distributed.  Additionally, all normally distributed data sets share the property that 68.26% of the data fall within 1s of the mean, 95.46% of the data fall within 2s of the mean, and 99.73% of the data fall within 3s of the mean.  One simple method to check the normality of a data set is to see if this rule approximately holds.  For the housefly data, the mean wing length is m = 45.5 and the standard deviation is s = 3.92.  With a bit of simple math we find that 68%, 96%, and 100% of the actual data lie within 1s, 2s, and 3s of  the mean, respectively.  Therefore, we can conclude that the housefly data display a nearly perfect Gaussian or normal distribution of wing length sizes.  An excellent question for the student is this:  What biological or environmental factors, that act independently and additively, contribute to the normalcy of wing lengths?

Source:  Sokal, R.R. and F.J. Rohlf, 1968. Biometry, Freeman Publishing Co., p 109.  Original data from Sokal, R.R. and P.E. Hunter. 1955. A morphometric analysis of DDT-resistant and non-resistant housefly strains Ann. Entomol. Soc. Amer. 48: 499-507.

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 Normally Distributed Housefly Wing Lengths Sokal, R.R., and P.E.Hunter. 1955. length (x.1mm) 36 Bin Frequency 37 36-38 2 38 38-40 4 38 40-42 10 39 42-44 15 39 44-46 19 40 46-48 19 40 48-50 15 40 50-52 10 40 52-54 4 41 54-56 2 41 41 41 41 41 42 42 42 42 42 42 42 43 43 43 43 43 43 43 43 44 44 44 44 44 44 44 44 44 45 45 45 45 45 45 45 45 45 45 46 46 46 46 46 46 46 46 46 46 47 47 47 47 47 47 47 47 47 48 48 48 48 48 48 48 48 49 49 49 49 49 49 49 50 50 50 50 50 50 51 51 51 51 52 52 53 53 54 55
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