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<br />. <br /> <br />corresponding to cell count and at a distance <br />from the ordinate axis corresponding to the <br />number of days since the beginning observation. <br />The peaks and troughs, their frequency, together <br />with intimate knowledge of the conditions of <br />the study, might suggest something of biological <br />interest, further statistical analysis, or further <br />field or laboratory work. <br />In summary, carefully prepared tables and <br />graphs may be important and informative steps <br />in data analysis. The added effort is usually <br />small, whereas gains in interpretive insight may <br />be large. Therefore, graphic examination of data <br />is a recommended procedure in the course of <br />most investigations. <br /> <br />300 <br /> <br /> <br />.., <br />I <br />c::> <br /> <br />-;;- 200 <br /> <br />..... <br />:::E <br /> <br />"" <br />..... <br />..... <br />~ 100 <br />..... <br />.... <br /><.!O <br />..... <br />.... <br /> <br />o <br /> <br />10 <br /> <br />20 30 <br />DAYS <br /> <br />40 <br /> <br />Figure 5. An example of a two-dimensional <br />graph plotted from algal-count data in Table l. <br /> <br />4.0 SAMPLE MEAN AND VARIANCE <br /> <br />. <br /> <br />4.1 General Application <br /> <br />Knowledge of certain computations and <br />computational notations is essential to the use <br />of statistical techniques. Some of the more basic <br />of these will be briefly reviewed here. <br />To illustrate the computations, let us assume <br />we have a set of data, i.e., a list of numeric <br />values written down. Each of these values can be <br />labeled by a set of numerals beginning with l. <br />Thus, the first of these values can be called Xl' <br />the second X2, etc., and the last one we call Xn. <br /> <br />BIOMETRICS - SAMPLE MEAN AND VARIANCE <br /> <br />The data values are labeled with consecutive <br />numbers (recall from the definitions that these <br />numeric values are observations), and there are n <br />values in the set of data. A typical observation is <br />Xi> where i may take any value between 1 and n, <br />inclusive, and the subscript indicates which X is <br />being referenced. <br />The sum of the numbers in a data set, such as <br />our sample, is indicated in statistical computa- <br />tions by capital sigma, L. Associated with L are <br />an operand (here, XJ, a subscript (here, i = 1), <br /> <br />n <br />and a superscript (here, n), iEI Xi. The sub- <br /> <br />script i = 1 indicates that the value of the <br />operand X is to be the number labeled Xl in our <br />data set and that this is to be the first observa- <br />tion of the sum. The superscript n indicates that <br />the last number of the summation is to be the <br />value of Xn, the last X in our data set. <br /> <br />Computations for the mean, variance, <br />standard deviation, variance of the mean, and <br />standard deviation of the mean (standard error) <br />are presented below. Note that these are compu- <br />tations for a sample of n observations, Le., they <br />are statistics. <br />Mean eX): <br /> <br />n <br />~ X. <br />i=l I <br /> <br />(11 ) <br /> <br />- <br />x-- <br /> <br />n <br /> <br />Variance (s 2): <br /> <br />n n <br />.~ Xi2 - (.~ Xi) 2 <br />t=l t=l <br /> <br />(12) <br /> <br />S2 = <br /> <br />n <br /> <br />n-l <br /> <br />Note: The Xi'S are squared, then the summation <br />is performed in the first term of the numerator; <br />in the second term, the sum of the Xi'S is first <br />formed, then the sum is squared, as indicated by <br />the parentheses. <br />Standard deviation (s): <br /> <br />s=W <br /> <br />(13) <br /> <br />Variance of the mean (s~ ): <br /> <br />S2 <br />S~=- <br />X n <br /> <br />(14) <br /> <br />9 <br />