Download - Interpreting Performance Data
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Interpreting Performance Data
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Expected Outcomes Understand the terms mean, median, mode, standard deviation
Use these terms to interpret performance data supplied by EAU
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Measures of Central Tendency Mean the average score
Median the value that lies in the middle after ranking all the scores
Mode the most frequently occurring score
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Which measure of Central Tendency should be used?
Measures of Central Tendency
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Measures of Central Tendency The measure you choose should give you a good indication of the typical score in the sample or population.
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Measures of Central Tendency Mean the most frequently used but is sensitive to extreme scorese.g. 1 2 3 4 5 6 7 8 9 10Mean = 5.5 (median = 5.5)e.g. 1 2 3 4 5 6 7 8 9 20Mean = 6.5 (median = 5.5)e.g. 1 2 3 4 5 6 7 8 9 100Mean = 14.5 (median = 5.5)
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Measures of Central Tendency Median
is not sensitive to extreme scores
use it when you are unable to use the mean because of extreme scores
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Measures of Central Tendency Mode
does not involve any calculation or ordering of data
use it when you have categories (e.g. occupation)
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A Distribution CurveMean: 54Median: 56Mode: 63
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The Normal Distribution Curve In everyday life many variables such as height, weight, shoe size and exam marks all tend to be normally distributed, that is, they all tend to look like the following curve.
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The Normal Distribution CurveIt is bell-shaped and symmetrical about the meanThe mean, median and mode are equalMean, Median, ModeIt is a function of the mean and the standard deviation
Chart1
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WELCOME SCREEN
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Welcome to DISCUS 5 !
Workbook for Continuous Distributions
copyright Neville Hunt & Sidney Tyrrell 1995
INTRODUCTION
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DISCUS 5 is a workbook about continuous distributions.
It is designed to help you understand more about : the Normal distribution the Exponential distribution the Student's t distribution the Beta distribution the Chi-squared distribution the Gamma distribution the F distribution the Weibull distribution and to show comparisons between them.
Self-teaching notes and questions are provided for each topic on separate workcards which are designed to be USED with the workbook.
There is a spare spreadsheet at the end of the workbook which can be used in the usual way for making notes, recording results, or doing calculations.
NORMAL DISTRIBUTION HELP SCREEN
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USING the Normal distribution spreadsheet
Aim: To see how the Normal distribution alters with different means and standard deviations.
Three plots are given. Change the means and standard deviations, highlighted in blue, to compare different normal distributions.
Use workcard 5.1 to learn more. DISCUS 5.1
DISCUS 5.1
NORMAL DISTRIBUTION
Plot 1Plot 2Plot 3
Mean5000
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lowerupperincr
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-10.00026069700
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3.080.000520379700
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5.1 The Normal distribution
NORMAL DISTRIBUTION
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t DISTRIBUTION HELP SCREEN
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USING the Student's t Distribution spreadsheet
Aim: To see how the shape of the Student's t distribution changes as the number of degrees of freedom is altered.
Three plots are given. Change the values of the degrees of freedom, highlighted in blue, to compare different t distributions.
Use workcard 5.2 to learn more. DISCUS 5.2
DISCUS 5.2
STUDENT'S t DISTRIBUTION
Plot 1Plot 2Plot 3
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0.480.25870439380.32600755330.3432900439
0.560.24231873180.31051427450.3283326163
0.640.22581575350.29388894940.3120206661
0.720.20963506730.27651554960.2946777433
0.80.1940913940.25875353680.2766251323
0.880.17939015220.24092631740.2581716602
0.960.16564835870.22331374850.2396051355
1.040.15291597140.20614844110.2211856788
1.120.1411949460.18961531780.2031410449
1.20.13045487140.17385372360.1856638936
1.280.12064504480.15896134910.1689108518
1.360.1117033570.1449992720.1530031216
1.440.10356256060.13199752250.1380283445
1.520.09615450890.1199607090.1240434061
1.60.08941288940.10887336540.1110778773
1.680.0832748760.0987048060.0991378108
1.760.077682030.08941336630.0882096527
1.840.07258069280.08094998810.0782640742
1.920.06792204810.07326116030.0692595764
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2.080.05976079270.05998436550.0538662433
2.160.05618290850.0542856010.0473610686
2.240.05289648470.04914211320.0415688165
2.320.04987307070.04450372730.0364282272
2.40.04708726130.0403233590.0318794937
2.480.04451637480.03655723610.0278652236
2.560.04214015650.03316497480.0243311186
2.640.03994050970.03010955020.0212264203
2.720.03790125340.02735719160.0185041601
2.80.03600790570.02487722820.0161212574
2.880.03424749160.02264190160.0140384982
2.960.03260837220.02062616210.0122204244
3.040.03108009360.01880745730.0106351614
3.120.02965325370.01716552210.0092542023
3.20.02831938490.01568217420.0080521674
3.280.02707085030.01434111950.0070065507
3.360.02590075240.01312776930.0060974652
3.440.02480285240.01202907060.005307392
3.520.02377149940.01103334920.0046209408
3.60.02280156780.01013016750.0040246232
3.680.02188840120.00931019380.003506643
3.760.02102776440.00856508540.0030567021
3.840.02021579910.00788738210.0026658251
3.920.01944898610.00727041170.0023261997
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5.2 The Student's t distribution
STUDENT'S t DISTRIBUTION
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CHI-SQUARED HELP SCREEN
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USING the Chi-squared distribution spreadsheet
Aim : To understand how the chi-squared distribution alters as the number of degrees of freedom change.
Up to three plots can be drawn for comparison.
Change the number of degrees of freedom, highlighted in blue, to see how the chi-squared distribution alters.
Use workcard 5.3 to learn more. DISCUS 5.3
CHI-SQUARED DISTRIBUTION
Plot 1Plot 2Plot 3
n4
mode200
s.d.2.82840.00000.0000
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lowerupperincr
0.028426620713.27669855890.1324827194
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0.02842662070.007006360500
0.16090934010.037117630400
0.29339205950.063340150100
0.42587477880.086048716500
0.55835749820.105586056800
0.69084021760.122265400200
0.8233229370.136372850800
0.95580565640.14816957800
1.08828837580.157893837800
1.22077109510.165762837100
1.35325381450.171974452900
1.48573653390.176708818100
1.61821925330.180129783300
1.75070197270.182386263700
1.8831846920.183613480900
2.01566741140.183934106100
2.14815013080.183459313900
2.28063285020.182289751700
2.41311556960.180516432600
2.5455982890.178221557200
2.67808100830.175479268900
2.81056372770.172356349400
2.94304644710.168912857500
3.07552916650.16520271700
3.20801188590.161274256100
3.34049460520.157170703300
3.47297732460.152930642300
3.6054600440.148588430300
3.73794276340.14417458100
3.87042548280.13971611600
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5.3 The Chi-squared distribution
CHI-SQUARED DISTRIBUTION
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DISTRIBUTIONS HELP SCREEN
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USING the continuous distributions spreadsheets.
Aim: To get a feel for how each of a variety of continuous distributions changes as their parameters are altered.
The distributions looked at are: the F distribution, the Exponential distribution, the Beta distribution, the Gamma distribution and the Weibull distribution.
The next five screens show each of the above distributions in turn. The final screen enables you to compare up to 4 of these distributions.
Change the numbers highlighted in blue to see how the shape of each distribution changes and to make comparisons for different values.
Use workcard 5.4 to learn more. DISCUS 5.4
F DISTRIBUTION
Plot 1Plot 2Plot 3
Numerator df3
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xf1f2f3
0.010.252859836400
0.04990.493274464400
0.08980.590398206800
0.12970.63647189600
0.16960.655576024900
0.20950.658708826400
0.24940.651924542700
0.28930.638884157200
0.32920.621925751700
0.36910.602594038400
0.4090.581931436500
0.44890.560650072300
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0.52870.51803269900
0.56860.497258887300
0.60850.477068394300
0.64840.457557909800
0.68830.438785275400
0.72820.420780522400
0.76810.403553827700
0.8080.38710129100
0.84790.371409153200
0.88780.356456887600
0.92770.342219470400
0.96760.328669015900
1.00750.315776116400
1.04740.303510596300
1.08730.291842147800
1.12720.280740951300
1.16710.270177866700
1.2070.260124693100
1.24690.250554314100
1.28680.241440786300
1.32670.232759385400
1.36660.224486621400
1.40650.216600230700
1.44640.209079152700
1.48630.201903494200
1.52620.195054487300
1.56610.18851444100
1.6060.182266691300
1.64590.176295548800
1.68580.170586247300
1.72570.165124892100
1.76560.159898410300
1.80550.154894502200
1.84540.150101595200
1.88530.145508799600
1.92520.141105866200
1.96510.13688314700
2.0050.132831557400
2.04490.128942540800
2.08480.125208035600
2.12470.121620443900
2.16460.118172602300
2.20450.114857756800
2.24440.111669528200
2.28430.10860190200
2.32420.105649196700
2.36410.102806044900
2.4040.100067374500
2.44390.097428389800
2.48380.094884555100
2.52370.092431578300
2.56360.090065396300
2.60350.087782161400
2.64340.085578227800
2.68330.0834501400
2.72320.081394621300
2.76310.079408563200
2.8030.077489015600
2.84290.075633177600
2.88280.073838388900
2.92270.072102121500
2.96260.070421972500
3.00250.068795656500
3.04240.067220999500
3.08230.065695932300
3.12220.064218484900
3.16210.062786780800
3.2020.061399032200
3.24190.060053534800
3.28180.058748663600
3.32170.057482868500
3.36160.056254670200
3.40150.055062656700
3.44140.053905479600
3.48130.052781850900
3.52120.051690539500
3.56110.05063036900
3.6010.049600214200
3.64090.048598998700
3.68080.047625692700
3.72070.046679310400
3.76060.045758907900
3.80050.044863581200
3.84040.043992463900
3.88030.04314472600
3.92020.042319571500
3.96010.041516237200
40.04073399100
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5.4 The F distribution
F DISTRIBUTION
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EXPONENTIAL DISTRIBUTION
Plot 1Plot 2Plot 3
Mean1
S.d.100
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lowerupperincr
040.04
xf1f2f3
0100
0.040.960789439200
0.080.923116346400
0.120.886920436700
0.160.85214378900
0.20.818730753100
0.240.786627861100
0.280.755783741500
0.320.726149037100
0.360.697676326100
0.40.67032004600
0.440.644036421100
0.480.618783391800
0.520.59452054800
0.560.571209063800
0.60.548811636100
0.640.52729242400
0.680.506616992400
0.720.48675225600
0.760.46766642700
0.80.449328964100
0.840.431710523400
0.880.414782911700
0.920.398519041100
0.960.38289288600
10.367879441200
1.040.35345468200
1.080.339595525600
1.120.326279794600
1.160.313486180900
1.20.301194211900
1.240.289384217900
1.280.278037300500
1.320.26713530200
1.360.25666077700
1.40.246596963900
1.440.236927758700
1.480.227637688400
1.520.21871188700
1.560.210136071200
1.60.20189651800
1.640.193980042300
1.680.18637397600
1.720.179066147900
1.760.172044863800
1.80.165298888200
1.840.158817426100
1.880.152590105800
1.920.146606962100
1.960.140858420900
20.135335283200
2.040.130028710900
2.080.124930212200
2.120.120031628500
2.160.11532512100
2.20.110803158400
2.240.106458504400
2.280.102284206700
2.320.098273585600
2.360.094420223200
2.40.090717953300
2.440.087160851500
2.480.083743225600
2.520.080459606700
2.560.077304740400
2.60.074273578200
2.640.071361269600
2.680.068563154200
2.720.065874754400
2.760.063291768400
2.80.060810062600
2.840.05842566600
2.880.056134762800
2.920.053933687300
2.960.051818917200
30.049787068400
3.040.047834889500
3.080.045959256600
3.120.044157168400
3.160.042425741100
3.20.04076220400
3.240.039163895100
3.280.037628256800
3.320.036152831800
3.360.034735258900
3.40.0333732700
3.440.032064685300
3.480.03080741100
3.520.029599435200
3.560.028438824700
3.60.027323722400
3.640.02625234400
3.680.025222974800
3.720.024233967800
3.760.023283740400
3.80.022370771900
3.840.021493601300
3.880.020650825200
3.920.019841094700
3.960.019063114300
40.018315638900
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5.5 The Exponential distribution
EXPONENTIAL DISTRIBUTION
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BETA DISTRIBUTION
Plot 1Plot 22
a2
b2
0
lowerupperincr
0.0010.9990.00998
xf1f2f3
0.0010.00599400
0.010980.065156637600
0.020960.123124070400
0.030940.179896298400
0.040920.235473321600
0.05090.2898551400
0.060880.343041753600
0.070860.395033162400
0.080840.445829366400
0.090820.495430365600
0.10080.5438361600
0.110780.591046749600
0.120760.637062134400
0.130740.681882314400
0.140720.725507289600
0.15070.7679370600
0.160680.809171625600
0.170660.849210986400
0.180640.888055142400
0.190620.925704093600
0.20060.9621578400
0.210580.997416381600
0.220561.031479718400
0.230541.064347850400
0.240521.096020777600
0.25051.126498500
0.260481.155781017600
0.270461.183868330400
0.280441.210760438400
0.290421.236457341600
0.30041.2609590400
0.310381.284265533600
0.320361.306376822400
0.330341.327292906400
0.340321.347013785600
0.35031.3655394600
0.360281.382869929600
0.370261.399005194400
0.380241.413945254400
0.390221.427690109600
0.40021.4402397600
0.410181.451594205600
0.420161.461753446400
0.430141.470717482400
0.440121.478486313600
0.45011.4850599400
0.460081.490438361600
0.470061.494621578400
0.480041.497609590400
0.490021.499402397600
0.51.500
0.509981.499402397600
0.519961.497609590400
0.529941.494621578400
0.539921.490438361600
0.54991.4850599400
0.559881.478486313600
0.569861.470717482400
0.579841.461753446400
0.589821.451594205600
0.59981.4402397600
0.609781.427690109600
0.619761.413945254400
0.629741.399005194400
0.639721.382869929600
0.64971.3655394600
0.659681.347013785600
0.669661.327292906400
0.679641.306376822400
0.689621.284265533600
0.69961.2609590400
0.709581.236457341600
0.719561.210760438400
0.729541.183868330400
0.739521.155781017600
0.74951.126498500
0.759481.096020777600
0.769461.064347850400
0.779441.031479718400
0.789420.997416381600
0.79940.9621578400
0.809380.925704093600
0.819360.888055142400
0.829340.849210986400
0.839320.809171625600
0.84930.7679370600
0.859280.725507289600
0.869260.681882314400
0.879240.637062134400
0.889220.591046749600
0.89920.5438361600
0.909180.495430365600
0.919160.445829366400
0.929140.395033162400
0.939120.343041753600
0.94910.2898551400
0.959080.235473321600
0.969060.179896298400
0.979040.123124070400
0.989020.065156637600
0.9990.00599400
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5.6 The Beta distribution
BETA DISTRIBUTION
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GAMMA DISTRIBUTION
Plot 1Plot 22
a2
b4
0
lowerupperincr
0.0127.79898987320.2778898987
xf1f2f3
0.010.000623439500
0.28788989870.016743613700
0.56577979750.030697194900
0.84366969620.042702391100
1.12155959490.052957879500
1.39944949370.061644445200
1.67733939240.068926488400
1.95522929110.074953408800
2.23311918990.079860878900
2.51100908860.083772013500
2.78889898730.086798444400
3.06678888610.089041306900
3.34467878480.090592146100
3.62256868350.091533748600
3.90045858230.091940905800
4.1783484810.091881114200
4.45623837970.091415217600
4.73412827840.090597996100
5.01201817720.089478706100
5.28990807590.088101574700
5.56779797460.086506253200
5.84568787340.084728231900
6.12357777210.082799220100
6.40146767080.080747493900
6.67935756960.078598213900
6.95724746830.076373716100
7.2351373670.07409377800
7.51302726580.071775861100
7.79091716450.069435332700
8.06880706320.067085668600
8.3466969620.064738637600
8.62458686070.062404469700
8.90247675940.060092010300
9.18036665820.057808859200
9.45825655690.055561498400
9.73614645560.05335540800
10.01403635440.051195171100
10.29192625310.049084569700
10.56981615180.047026671800
10.84770605060.045023909800
11.12559594930.043078152400
11.4034858480.041190769100
11.68137574680.039362688600
11.95926564550.037594452200
12.23715554420.035886261400
12.5150454430.03423802100
12.79293534170.032649378300
13.07082524040.031119757700
13.34871513910.029648392500
13.62660503790.028234353100
13.90449493660.026876572300
14.18238483530.025573867700
14.46027473410.024324962600
14.73816463280.023128503200
15.01605453150.021983075300
15.29394443030.020887218400
15.5718343290.019839437800
15.84972422770.018838216500
16.12761412650.017882024500
16.40550402520.016969327300
16.68339392390.016098593800
16.96128382270.015268302200
17.23917372140.014476946100
17.51706362010.013723039300
17.79495351890.013005119500
18.07284341760.012321751900
18.35073331630.011671532600
18.62862321510.011053090100
18.90651311380.010465087800
19.18440301250.009906225600
19.46229291130.009375240500
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20.01807270870.00839204200
20.29596260750.007937496200
20.57385250620.007506162900
20.85174240490.007096973800
21.12963230360.006708899400
21.40752220240.006340948600
21.68541210110.005992168200
21.96330199980.005661642400
22.24119189860.00534849200
22.51908179730.005051873400
22.7969716960.004770978100
23.07486159480.004505031800
23.35275149350.00425329300
23.63064139220.004015052600
23.9085312910.003789632600
24.18642118970.003576385400
24.46431108840.003374692300
24.74220098720.00318396300
25.02009088590.003003634300
25.29798078460.002833169200
25.57587068340.002672056100
25.85376058210.002519807400
26.13165048080.002375959100
26.40954037960.002240069300
26.68743027830.002111717600
26.9653201770.001990504500
27.24321007580.001876049700
27.52109997450.001767992200
27.79898987320.001665988700
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5.7 The Gamma distribution
GAMMA DISTRIBUTION
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WEIBULL DISTRIBUTION
Plot 1Plot 22
a2
b4
0
mean1mean2mean3
1.772453850811
var1var2var3
3.942070181511
upper1upper2upper3
9.714312758155
lower1lower2lower3
-4.1839403297-2-2
lowerupperincr
0.019.71431275810.0970431276
xf1f2f3
0.010.001249992200
0.10704312760.013370812200
0.20408625520.025444458600
0.30112938270.037428447400
0.39817251030.04928082100
0.49521563790.060960392500
0.59225876550.072426984800
0.68930189310.083641660400
0.78634502060.094566941200
0.88338814820.105167017300
0.98043127580.115407941400
1.07747440340.125257808500
1.1745175310.1346869200
1.27156065860.143667929500
1.36860378610.152175970700
1.46564691370.160188765900
1.56269004130.167686715300
1.65973316890.17465296500
1.75677629650.181073455800
1.8538194240.186936950700
1.95086255160.192235042500
2.04790567920.196962141300
2.14494880680.20111544300
2.24199193440.204694878900
2.33903506190.20770304700
2.43607818950.210145127400
2.53312131710.212028781300
2.63016444470.213364035800
2.72720757230.214163155700
2.82425069980.214440503500
2.92129382740.214212389100
3.0183369550.213496911500
3.11538008260.212313792300
3.21242321020.21068420500
3.30946633780.208630599400
3.40650946530.206176524300
3.50355259290.203346448900
3.60059572050.200165585300
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3.79468197570.192855005100
3.89172510320.188777863900
3.98876823080.184454755600
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4.27989761360.170272075300
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4.66807012390.1494750600
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4.95919950660.133280288200
5.05624263420.127879451400
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5.3473720170.111918367900
5.44441514450.106729816300
5.54145827210.101629763800
5.63850139970.096629920700
5.73554452730.091740780100
5.83258765490.086971638300
5.92963078240.082330621500
6.026673910.077824718900
6.12371703760.073459821600
6.22076016520.069240765400
6.31780329280.065171379100
6.41484642030.061254534700
6.51188954790.057492201500
6.60893267550.05388550200
6.70597580310.050434769600
6.80301893070.047139606500
6.90006205820.043998943300
6.99710518580.041011097200
7.09414831340.038173831100
7.1911914410.035484410400
7.28823456860.032939659500
7.38527769620.03053601600
7.48232082370.028269583700
7.57936395130.026136182400
7.67640707890.024131396300
7.77345020650.022250619200
7.87049333410.020489097200
7.96753646160.018841968600
8.06457958920.017304300800
8.16162271680.015871124900
8.25866584440.014537466200
8.3557089720.013298373400
8.45275209950.012148943200
8.54979522710.011084344100
8.64683835470.010099835500
8.74388148230.009190786100
8.84092460990.008352688400
8.93796773740.007581171800
9.0350108650.006872012700
9.13205399260.006221143600
9.22909712020.005624659200
9.32614024780.005078821400
9.42318337540.004580062500
9.52022650290.004124986900
9.61726963050.003710371300
9.71431275810.003333164200
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5.8 The Weibull distribution
WEIBULL DISTRIBUTION
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COMPARING DISTRIBUTIONS
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COMPARING DISTRIBUTIONS
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COMPARE MACRO
5.9 Comparing distributions
Plot 1
Plot 2
Plot 3
Plot 4
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Item
x
y
width
height
text
init/result
Plot
Cancel
Delete
Newplot
Plot1
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Initialise
Norm
Beta
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Variation or Spread of DistributionsMeasures that indicate the spread of scores:
Range
Standard Deviation
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Variation or Spread of DistributionsRangeIt compares the minimum score with the maximum scoreMax score Min score = RangeIt is a crude indication of the spread of the scores because it does not tell us much about the shape of the distribution and how much the scores vary from the mean
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Variation or Spread of DistributionsStandard DeviationIt tells us what is happening between the minimum and maximum scoresIt tells us how much the scores in the data set vary around the meanIt is useful when we need to compare groups using the same scale
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Calculating a Mean and a Standard Deviation
WELCOME SCREEN
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Welcome to DISCUS 1 !
Workbook for Descriptive Statistics
copyright Neville Hunt & Sidney Tyrrell 1995
INTRODUCTION
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DISCUS 1 is a workbook about descriptive statistics to help you understand:
different averages - the mean and median measures of spread - the standard deviation and IQR statistical diagrams - histograms and boxplots
Self-teaching notes and questions are provided for each topic on separate workcards which are designed to be USED with the workbook.
There is a spare spreadsheet at the end of the workbook which can be used in the usual way for making notes, recording results, and doing calculations.
MEAN & S.D. HELP SCREEN
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USING the mean and standard deviation spreadsheet
Aim: To understand how a mean and a standard deviation are calculated and, in particular, how the standard deviation measures spread.
Five numbers are given, highlighted in pale blue.
The mean, variance and standard deviation have been calculated, showing each step in the calculations.
Change the five numbers to see how the mean and standard deviation alter.
Use workcard 1.1 to learn more. DISCUS 1.1
THE MEAN & STANDARD DEVIATION
AbsoluteSquared
DataDeviationDeviationDeviation
xx - Mean|x - Mean|(x-Mean)
10-2020400
20-1010100
30000
401010100
502020400
Sums1500601000
Means30012200
Variance
14.1421356237
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Standard deviation = Variance
1.1 Calculating a mean and standard deviation
MEAN AND MEDIAN HELP SCREEN
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USING the means and median spreadsheet
Aim: To understand what the mean and median each measure, and the difference between them.
A frequency distribution of some students' examination marks is given, for which the mean and the median have been calculated. The distribution is displayed as a histogram with the mean and median appropriately marked. Change the frequencies (in pale blue) to see how the shape of the distribution changes and the values of the mean and median alter. Use workcard 1.2 to learn more. DISCUS 1.2
THE MEAN AND MEDIAN
MarkFrequency
0 - 910
10 - 1920
20 - 2930
30 - 3940
40 - 4950
50 - 5960
60 - 6970
70 - 7980
80 - 8990
90 - 99100
Mean65.00
Median69.29
00510503600000100100100
001520300500001010201009090
0102530750480003020301908080
9.991035401400360006030402707070
9.990455022502000010040503406060
9.992055603300600015050604005050
19.982065704550021060704504040
19.98075806000800028070804903030
19.9830859076503600036080905202020
29.973095100950090000450901005401010
29.97055035750330000550100055000
29.9740
39.9640Mean65Half275
39.960Variance600Median69.2857142857
39.9650S.d.24.4948974278Median169.2857142857
49.9550Maxf100Median269.2857142857
49.950
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THE MEAN AND MEDIAN
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Mark
Frequency
Mean
Median
S.D. AND IQR HELP SCREEN
1.2 Measures of location (averages)
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USING the standard deviation and IQR spreadsheet
Aim: To understand what the standard deviation and the interquartile range (IQR) each measure, and the difference between them. The same frequency distribution of students' examination marks is given as in spreadsheet 1.2. This time the standard deviation and IQR have been calculated.
The marks are displayed on a histogram with the standard deviation and the IQR labelled appropriately.
Change the frequencies (in pale blue) to see how the values of the standard deviation and IQR alter. Use workcard 1.3 to learn more. DISCUS 1.3
THE STANDARD DEVIATION AND IQR
MarkFrequency
0 - 94
10 - 196
20 - 2910
30 - 3912
40 - 493
50 - 594
60 - 696
70 - 791
80 - 891
90 - 991
S.d.21.31
IQR30.50
0054204117.361111111100401001
00156902926.041666666741061901
0425102501460.06944444441020102801
9.994351242052.08333333332030123706
9.990453135188.0208333333324039604
9.9965542201284.02777777783550413503
19.9866563904676.041666666739606164012
19.980751751437.673611111145701283010
19.9810851852296.00694444444680138206
29.9710951953354.34027777784790144104
29.970Sumf48178021791.66666666674810004800
29.9712
39.9612Mean37.0833333333Half24Quart12Quart336
39.960Variance453.9930555556Median33.3333333333Q122Q352.5
39.963S.d.21.3071127926Median133.3333333333Q1122Q3152.5
49.953Maxf12Median233.3333333333Q1222Q3252.5
49.950IQR30.5
49.954
59.944
59.940
59.946
69.936
69.930
69.931
79.921
79.920
79.921
89.911
89.910
89.911
99.91
99.90
99.90
37.083333333312.84
37.083333333313.56
37.083333333313.2
58.390446125913.2
55.390446125912.84
58.390446125913.2
55.390446125913.56
2514.64
2215
2515.36
2215
33.333333333315
33.333333333314.64
33.333333333315.36
33.333333333315
52.515
49.515.36
52.515
49.514.64
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THE STANDARD DEVIATION AND IQR
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Mark
Frequency
S.d.
IQR
Median
BOXPLOTS HELP SCREEN
neville
1.3 Measures of dispersion
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USING the boxplots spreadsheet
Aim: To introduce the boxplot as a means of showing the main features of a set of data.
A boxplot shows the median, quartiles and IQR on a single diagram. It also indicates possible outliers.
A data set of 48 numbers is given, highlighted in pale blue. The median, quartiles, IQR and fences have been calculated and the boxplot is displayed with two possible outliers marked.
Change or delete some of the data to see the effect on the boxplot and accompanying statistics. Use workcard 1.4 to learn more. DISCUS 1.4
BOXPLOTS
D81012141921242529303237
a414347495052535354545455
t555556565656575758585960
a606162636570788599140151182
Median55Lower quartile40Inner fences1090
IQR20Upper quartile60Outer fences-20120
0
DataWhiskersProbablesOutliersValuePlotLOF-20
801001010LIF10
4141004010LW10
5555004015LQ40
6060006015Median55
1010006010UQ60
4343008510UW85
5555006010UIF90
616100605UOF120
121200555
4747005515
565600555
626200405
1414004010
494900
565600
636300
191900
505000
565600
656500
212100
525200
565600
707000
242400
535300
575700
787800
252500
535300
575700
858500
292900
545400
585800
990100
303000
545400
585800
1400010
323200
545400
595900
1510010
373700
555500
606000
1820010
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1.4 Drawing boxplots
BOXPLOTS
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HISTOGRAMS HELP SCREEN
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USING the histograms spreadsheet
Aim: To understand the concept of a histogram for representing continuous data. Histograms are NOT bar charts and this spreadsheet helps in understanding the differences between the two types of chart.
A data set of 48 numbers is given which have been tallied into 7 classes of unequal width and the histogram constructed.
Change the data values and upper limits of the classes to see how the histogram alters. Use the spare spreadsheet at the end of the workbook to draw the corresponding bar chart and note the difference. Use workcard 1.5 to learn more. DISCUS 1.5
HISTOGRAMS
D102022324344462132203231
a113231424565662526712925
t12232472354255108521614
a151221314229308972661999
UpperFreq.Freq.
LimitsTallyDens.
70
40310.9393939394
5070.7
6020.2
7030.3
8030.3
9920.1052631579
0
0
0
700
700
700.9393939394
40310.9393939394
4000
4000.7
5070.7
5000
5000.2
6020.2
6000
6000.3
7030.3
7000
7000.3
8030.3
8000
8000.1052631579
9920.1052631579
9900
9900
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HISTOGRAMS
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SPARE SPREADSHEET
1.5 Drawing histograms
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-
Interpreting DistributionsMean = 50Std Dev = 1534%14%2%34%14%
Chart1
0.000102818600
0.000126731600
0.00015558100
0.00019023300
0.000231671800
0.000281007900
0.000339486100
0.000408491800
0.000489556400
0.000584359400
0.00069472900
0.00082263800
0.000970197400
0.001139644700
0.001333327400
0.001553682100
0.001803206600
0.002084427700
0.002399861400
0.002751968700
0.003143104400
0.003575461700
0.004051011300
0.004571437300
0.005138070700
0.005751821300
0.006413109900
0.007121803500
0.00787715300
0.008677738800
0.009521423100
0.01040531300
0.01132573600
0.012278228500
0.013257541300
0.014257660200
0.015271845800
0.016292690100
0.017312191900
0.018321849700
0.019312770200
0.020275791900
0.021201621300
0.022080978700
0.022904750200
0.02366414400
0.024350844800
0.024957165800
0.025476190800
0.025901905700
0.026229314400
0.026454536500
0.026574883600
0.026588912900
0.02649645600
0.026298622100
0.025997775800
0.025597490600
0.025102477500
0.024518493200
0.023852228600
0.023111181400
0.022303515700
0.02143791300
0.020523417400
0.0195692800
0.018584805200
0.017579202800
0.016561449200
0.015540160200
0.014523478500
0.013518975900
0.012533574300
0.011573483100
0.010644155800
0.009750263600
0.008895686900
0.008083521900
0.007316101900
0.006595031400
0.005921230700
0.005294989500
0.004716026200
0.004183553400
0.003696344400
0.003252801200
0.002851022400
0.002488867200
0.002164017700
0.001874035900
0.001616415400
0.001388628500
0.001188166100
0.001012572700
0.000859475100
0.00072660500
0.000611816800
0.000513100500
0.000428589500
0.000356564900
0.000295456600
WELCOME SCREEN
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Welcome to DISCUS 5 !
Workbook for Continuous Distributions
copyright Neville Hunt & Sidney Tyrrell 1995
INTRODUCTION
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DISCUS 5 is a workbook about continuous distributions.
It is designed to help you understand more about : the Normal distribution the Exponential distribution the Student's t distribution the Beta distribution the Chi-squared distribution the Gamma distribution the F distribution the Weibull distribution and to show comparisons between them.
Self-teaching notes and questions are provided for each topic on separate workcards which are designed to be USED with the workbook.
There is a spare spreadsheet at the end of the workbook which can be used in the usual way for making notes, recording results, or doing calculations.
NORMAL DISTRIBUTION HELP SCREEN
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USING the Normal distribution spreadsheet
Aim: To see how the Normal distribution alters with different means and standard deviations.
Three plots are given. Change the means and standard deviations, highlighted in blue, to compare different normal distributions.
Use workcard 5.1 to learn more. DISCUS 5.1
DISCUS 5.1
NORMAL DISTRIBUTION
Plot 1Plot 2Plot 3
Mean5000
S.d.1500
0
lowerupperincr
0950.95
xf1f2f3
00.000102818600
0.950.000126731600
1.90.00015558100
2.850.00019023300
3.80.000231671800
4.750.000281007900
5.70.000339486100
6.650.000408491800
7.60.000489556400
8.550.000584359400
9.50.00069472900
10.450.00082263800
11.40.000970197400
12.350.001139644700
13.30.001333327400
14.250.001553682100
15.20.001803206600
16.150.002084427700
17.10.002399861400
18.050.002751968700
190.003143104400
19.950.003575461700
20.90.004051011300
21.850.004571437300
22.80.005138070700
23.750.005751821300
24.70.006413109900
25.650.007121803500
26.60.00787715300
27.550.008677738800
28.50.009521423100
29.450.01040531300
30.40.01132573600
31.350.012278228500
32.30.013257541300
33.250.014257660200
34.20.015271845800
35.150.016292690100
36.10.017312191900
37.050.018321849700
380.019312770200
38.950.020275791900
39.90.021201621300
40.850.022080978700
41.80.022904750200
42.750.02366414400
43.70.024350844800
44.650.024957165800
45.60.025476190800
46.550.025901905700
47.50.026229314400
48.450.026454536500
49.40.026574883600
50.350.026588912900
51.30.02649645600
52.250.026298622100
53.20.025997775800
54.150.025597490600
55.10.025102477500
56.050.024518493200
570.023852228600
57.950.023111181400
58.90.022303515700
59.850.02143791300
60.80.020523417400
61.750.0195692800
62.70.018584805200
63.650.017579202800
64.60.016561449200
65.550.015540160200
66.50.014523478500
67.450.013518975900
68.40.012533574300
69.350.011573483100
70.30.010644155800
71.250.009750263600
72.20.008895686900
73.150.008083521900
74.10.007316101900
75.050.006595031400
760.005921230700
76.950.005294989500
77.90.004716026200
78.850.004183553400
79.80.003696344400
80.750.003252801200
81.70.002851022400
82.650.002488867200
83.60.002164017700
84.550.001874035900
85.50.001616415400
86.450.001388628500
87.40.001188166100
88.350.001012572700
89.30.000859475100
90.250.00072660500
91.20.000611816800
92.150.000513100500
93.10.000428589500
94.050.000356564900
950.000295456600
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5.1 The Normal distribution
NORMAL DISTRIBUTION
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t DISTRIBUTION HELP SCREEN
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USING the Student's t Distribution spreadsheet
Aim: To see how the shape of the Student's t distribution changes as the number of degrees of freedom is altered.
Three plots are given. Change the values of the degrees of freedom, highlighted in blue, to compare different t distributions.
Use workcard 5.2 to learn more. DISCUS 5.2
DISCUS 5.2
STUDENT'S t DISTRIBUTION
Plot 1Plot 2Plot 3
n1410
s.d.0.00001.41421.1180
0
lowerupperincr
-440.08
xf1f2f3
-40.01872411090.00670820390.0020310339
-3.920.01944898610.00727041170.0023261997
-3.840.02021579910.00788738210.0026658251
-3.760.02102776440.00856508540.0030567021
-3.680.02188840120.00931019380.003506643
-3.60.02280156780.01013016750.0040246232
-3.520.02377149940.01103334920.0046209408
-3.440.02480285240.01202907060.005307392
-3.360.02590075240.01312776930.0060974652
-3.280.02707085030.01434111950.0070065507
-3.20.02831938490.01568217420.0080521674
-3.120.02965325370.01716552210.0092542023
-3.040.03108009360.01880745730.0106351614
-2.960.03260837220.02062616210.0122204244
-2.880.03424749160.02264190160.0140384982
-2.80.03600790570.02487722820.0161212574
-2.720.03790125340.02735719160.0185041601
-2.640.03994050970.03010955020.0212264203
-2.560.04214015650.03316497480.0243311186
-2.480.04451637480.03655723610.0278652236
-2.40.04708726130.0403233590.0318794937
-2.320.04987307070.04450372730.0364282272
-2.240.05289648470.04914211320.0415688165
-2.160.05618290850.0542856010.0473610686
-2.080.05976079270.05998436550.0538662433
-20.06366197720.06629126070.0611457663
-1.920.06792204810.07326116030.0692595764
-1.840.07258069280.08094998810.0782640742
-1.760.077682030.08941336630.0882096527
-1.680.0832748760.0987048060.0991378108
-1.60.08941288940.10887336540.1110778773
-1.520.09615450890.1199607090.1240434061
-1.440.10356256060.13199752250.1380283445
-1.360.1117033570.1449992720.1530031216
-1.280.12064504480.15896134910.1689108518
-1.20.13045487140.17385372360.1856638936
-1.120.1411949460.18961531780.2031410449
-1.040.15291597140.20614844110.2211856788
-0.960.16564835870.22331374850.2396051355
-0.880.17939015220.24092631740.2581716602
-0.80.1940913940.25875353680.2766251323
-0.720.20963506730.27651554960.2946777433
-0.640.22581575350.29388894940.3120206661
-0.560.24231873180.31051427450.3283326163
-0.480.25870439380.32600755330.3432900439
-0.40.27440507430.33997573350.3565785337
-0.320.28874263980.35203531880.3679048451
-0.240.30097379550.36183299450.3770089082
-0.160.31036455360.36906656050.3836750301
-0.080.31628565790.37350418990.3877415668
00.31830988610.3750.389108384
0.080.31628565790.37350418990.3877415668
0.160.31036455360.36906656050.3836750301
0.240.30097379550.36183299450.3770089082
0.320.28874263980.35203531880.3679048451
0.40.27440507430.33997573350.3565785337
0.480.25870439380.32600755330.3432900439
0.560.24231873180.31051427450.3283326163
0.640.22581575350.29388894940.3120206661
0.720.20963506730.27651554960.2946777433
0.80.1940913940.25875353680.2766251323
0.880.17939015220.24092631740.2581716602
0.960.16564835870.22331374850.2396051355
1.040.15291597140.20614844110.2211856788
1.120.1411949460.18961531780.2031410449
1.20.13045487140.17385372360.1856638936
1.280.12064504480.15896134910.1689108518
1.360.1117033570.1449992720.1530031216
1.440.10356256060.13199752250.1380283445
1.520.09615450890.1199607090.1240434061
1.60.08941288940.10887336540.1110778773
1.680.0832748760.0987048060.0991378108
1.760.077682030.08941336630.0882096527
1.840.07258069280.08094998810.0782640742
1.920.06792204810.07326116030.0692595764
20.06366197720.06629126070.0611457663
2.080.05976079270.05998436550.0538662433
2.160.05618290850.0542856010.0473610686
2.240.05289648470.04914211320.0415688165
2.320.04987307070.04450372730.0364282272
2.40.04708726130.0403233590.0318794937
2.480.04451637480.03655723610.0278652236
2.560.04214015650.03316497480.0243311186
2.640.03994050970.03010955020.0212264203
2.720.03790125340.02735719160.0185041601
2.80.03600790570.02487722820.0161212574
2.880.03424749160.02264190160.0140384982
2.960.03260837220.02062616210.0122204244
3.040.03108009360.01880745730.0106351614
3.120.02965325370.01716552210.0092542023
3.20.02831938490.01568217420.0080521674
3.280.02707085030.01434111950.0070065507
3.360.02590075240.01312776930.0060974652
3.440.02480285240.01202907060.005307392
3.520.02377149940.01103334920.0046209408
3.60.02280156780.01013016750.0040246232
3.680.02188840120.00931019380.003506643
3.760.02102776440.00856508540.0030567021
3.840.02021579910.00788738210.0026658251
3.920.01944898610.00727041170.0023261997
40.01872411090.00670820390.0020310339
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5.2 The Student's t distribution
STUDENT'S t DISTRIBUTION
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CHI-SQUARED HELP SCREEN
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USING the Chi-squared distribution spreadsheet
Aim : To understand how the chi-squared distribution alters as the number of degrees of freedom change.
Up to three plots can be drawn for comparison.
Change the number of degrees of freedom, highlighted in blue, to see how the chi-squared distribution alters.
Use workcard 5.3 to learn more. DISCUS 5.3
CHI-SQUARED DISTRIBUTION
Plot 1Plot 2Plot 3
n4
mode200
s.d.2.82840.00000.0000
0
lowerupperincr
0.028426620713.27669855890.1324827194
xf1f2f3
0.02842662070.007006360500
0.16090934010.037117630400
0.29339205950.063340150100
0.42587477880.086048716500
0.55835749820.105586056800
0.69084021760.122265400200
0.8233229370.136372850800
0.95580565640.14816957800
1.08828837580.157893837800
1.22077109510.165762837100
1.35325381450.171974452900
1.48573653390.176708818100
1.61821925330.180129783300
1.75070197270.182386263700
1.8831846920.183613480900
2.01566741140.183934106100
2.14815013080.183459313900
2.28063285020.182289751700
2.41311556960.180516432600
2.5455982890.178221557200
2.67808100830.175479268900
2.81056372770.172356349400
2.94304644710.168912857500
3.07552916650.16520271700
3.20801188590.161274256100
3.34049460520.157170703300
3.47297732460.152930642300
3.6054600440.148588430300
3.73794276340.14417458100
3.87042548280.13971611600
4.00290820210.135236887700
4.13539092150.130757873700
4.26787364090.126297448100
4.40035636030.121871628600
4.53283907970.117494303300
4.66532179910.113177437700
4.79780451840.108931264200
4.93028723780.104764455400
5.06276995720.100684281500
5.19525267660.096696754700
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