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{"sheet":"Intro1","index":0,"n_rows":13,"n_cols":5,"rows_kept":13,"cut":false,"hidden":false,"merged":[[12,4,20,7]]}
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[null,null,null,null,"Growth, reproduction,   inheritance…. Inherently all averaging processes"]

{"sheet":"Intro2","index":1,"n_rows":0,"n_cols":0,"rows_kept":0,"cut":false,"hidden":false,"merged":[[4,4,10,7]]}

{"sheet":"Central","index":2,"n_rows":20001,"n_cols":21,"rows_kept":20001,"cut":false,"hidden":false,"merged":[]}
["Replications  < 1000",null,"Realisations (RxS)","Underlying data","Reps","Xbar",null,"GDIB - 2009 xl 2007",null,null,null,"Range of underlying data","Frequency",null,null,"Range of Xbar","Frequency","Range of Xbar","Frequency"]
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["Sample Size (S) < 500",null,2,4.4667915435968339,2,4.8807376474208377,null,null,null,null,null,0.8,1575,null,null,3.8931226760198534,0,0.4,0,{"$f":"NORMDIST(P3,$A$42,$A$43,TRUE)","v":2.324080094853187E-4},9.6446284477758049E-2]
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["(Underlying) Distribution",null,4,16.738797405394667,4,5.1678304132544097,null,null,null,null,null,2.4,2373,null,null,4.0901839938409754,0,1.2,0,{"$f":"NORMDIST(P5,$A$42,$A$43,TRUE)","v":2.0068002366986359E-3},0.62888386586965606]
["c",null,5,3.1964582463171149,5,5.1401652166814662,null,null,null,null,null,3.2,2403,null,null,4.1887146527515355,2,1.6,0,5.1512195660469162E-3,1.3898244381486125]
["Mean of Xbars",null,6,3.4634719552630537,6,4.7901315519430376,null,null,null,null,null,4,2197,null,null,4.2872453116620965,4,2,0,1.2100341756789978E-2,2.7894432176621464]
[{"$f":"Mean","v":5.0262252534913037},null,7,9.8730341258954404,7,5.0178462248418851,null,null,null,null,null,4.8,1888,null,null,4.3857759705726576,1,2.4,0,2.6047557845100711E-2,5.0844667631476437]
["Stdev of Xbars",null,8,7.5362953189194215,8,4.9771145195238349,null,null,null,null,null,5.6,1519,null,null,4.4843066294832186,8,2.8,0,5.1469891660838929E-2,8.4167714518578194]
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["U - Uniform [0 , 10]",null,15,6.0495887637707648,15,4.8244244225195665,null,null,null,null,null,11.2,263,null,null,5.1740212418571447,22,5.6,7,0.70894438301833929,19.336031266066467]
["N - Normal (5 , 4)",null,16,2.680549770008315,16,5.0352198568720761,null,null,null,null,null,12,205,null,null,5.2725519007677057,16,6,0,0.80562453934867162,14.814197312833066]
["E - Exponential (5)",null,17,6.089282189898432,17,5.0475398736648494,null,null,null,null,null,12.8,133,null,null,5.3710825596782668,11,6.4,0,0.87969552591283695,10.307767724145833]
["C - Chi-squared (5)",null,18,1.4841022912523456,18,5.5102270398658844,null,null,null,null,null,13.6,114,null,null,5.4696132185888278,4,6.8,0,0.93123436453356612,6.5136684594855776]
["yes",null,19,4.6355480379633738,19,5.383700810624517,null,null,null,null,null,14.4,64,null,null,5.5681438774993888,5,7.2,0,0.963802706830994,3.7381757061284659]
["reached here",null,20,10.523631059112045,20,5.5706374086168591,null,null,null,null,null,15.2,58,null,null,5.6666745364099498,2,7.6,0,0.98249358536163633,1.9483412081188867]
["reached here",null,21,6.9119313252434438,21,4.9625489451438742,null,null,null,null,null,16,40,null,null,5.7652051953205099,3,8,0,0.99223529140223077,0.92223050467659107]
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["500 reps and SS=100",null,35,12.357478517134263,35,4.9099004028396021]
["gives xbar pretty close to Normal",null,36,15.024435810411619,36,5.394960588936442]
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[null,null,19994,4.1146432845728764]
[null,null,19995,0.53328927460747055]
[null,null,19996,2.1704771902145699]
[null,null,19997,5.2445019759373963]
[null,null,19998,4.7586506646996405]
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{"sheet":"Notes","index":3,"n_rows":20,"n_cols":5,"rows_kept":20,"cut":false,"hidden":false,"merged":[]}
[]
[null,null,null,"Aim:"]
[]
[null,null,null,1,"Particularly in the case of first year this simulation should make one important point and should not be too technical"]
[null,null,null,null,"The important point is that when one averages fixed size samples from distributions which may be asymmetrical "]
[null,null,null,null,", then as the size of the sample increases the distribution of these averages tends to the normal, independently of the shape of the distribution of the underlying data"]
[]
[null,null,null,2,"I got my Honours students to effectively produce this sim and discuss it in their exam"]
[null,null,null,null,"Not a single student properly understood the CLT"]
[null,null,null,null,"They got confused between the size of the sample (over which one computes the average) and the number of reps in the sim"]
[null,null,null,null,"So, let's not get too ambitious at first year level!!!"]
[]
[]
[null,null,null,"Exercises"]
[]
[null,null,null,1,"fix n and consider increasing the number of reps (note the case of n=1 is particularly instructive)!!"]
[null,null,null,null,"The key point is that increasing the #reps takes to closer to the true position (theoretical distribution of the xbars)"]
[null,null,null,2,"The size of n is what drives the distribution of xbar to normality. Students should experiment with n=1, n=10, n=30 and n=100"]
[null,null,null,3,"The key exercise is to try different distributions. Thus in the case of the chi-squared the important point is how the data starts off asymmetric, but the xbars form a normal distribution"]
[null,null,null,4,"Establish that as n changes the variance of the xbars changes according to sigma^2/n"]