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Statistics for Managers using Microsoft Excel 3rd Edition Kafli 1 Inngangur og gagnaöflun © 2002 Prentice-Hall, Inc. Chap 1-1 Efni kaflans Hvers vegna þurfa stjórnendur á tölfræði að halda Vöxtur og viðgangur nútíma tölfræði Helstu skilgreiningar Lýsandi- og ályktannatölfræði Guðmundur Ólafsson lektor Kafli 1-2 Efni kaflans Til hvers eru gögn Tegundir gagna og uppruni Hönnun rannsókna Úrtaksaðferðir Kannanaskekkjur og villur Guðmundur Ólafsson lektor (framhald) Kafli 1-3 Hvað þurfa stórnendur að vita um tölfræði Að geta sett fram upplýsingar á viðeigandi hátt Að vita hvernig draga á ályktanir um safn (Þýði) út frá úrtaki Að geta bætt vinnuferla Að geta gert spár sem má treysta Guðmundur Ólafsson lektor Kafli 1-4 Vöxtur og viðgangur nútíma tölfræði Þörf hins opinbera fyrir gögn um einstaklinga Þróun líkindastærðfræði Tölvur koma til sögunnar Guðmundur Ólafsson lektor Kafli 1-5 Lykilhugtök Þýði (úrtaksrúm, population) er safn þeirra tilvika sem til greina koma Úrtak (sample) er hluti af þýði, sem tekið er til rannsóknar, hugsanlega þýðið allt (hlutimengi í þýði) Stiki (parameter) er einkennistala sem reiknuð er út til þess að lýsa þýði Lýsitala (statistic, reiknihending) er einkennistala sem reiknuð er út til þess að lýsa úrtaki Guðmundur Ólafsson lektor Kafli 1-6 Þýði og úrtak Þýði Úrtak Lýsitölur lýsa einkennum úrtaks Stikar lýsa einkennum þýðis Ályktanir um þýði út frá úrtaki Guðmundur Ólafsson lektor Kafli 1-7 Tölfræðilegar aðferðir Lýsandi tölfræði Söfnun og lýsing gagna Ályktunartölfræði Draga ályktanir um þýði, sem byggðar eru á vinnslu úrtaka úr þýðinu Guðmundur Ólafsson lektor Kafli 1-8 Lýsandi tölfræði Safna gögnum Setja gögn fram t.d. með skoðanakönnun t.d. með töflum og línuritum Lýsa einkennum gagna t.d. með úrtaksmeðaltali = Guðmundur Ólafsson lektor X i n Kafli 1-9 Ályktanatöfræði Mat T.d..: Meta meðalþunga þýðis út frá meðalþunga úrtaks Tilgátuprófun T.d.: Kanna þá fullyrðingu að meðalþungi í þýði sé 60 kg Draga ályktanir og/eða taka ákvarðanir varðandi þýði, sem byggðar eru á niðurstöðum úr úrtakskönnun. Guðmundur Ólafsson lektor Kafli 1-10 Til hvers eru gögn? Þau eru notuð í könnunum Þau eru nauðsinleg í rannsóknum Til þess að meta frammistöðu í þjónustu eða framleiðslu Til þess að meta hvort farið sé eftir stöðluðum reglum Til þess að hjálpa til við að finna aðrar lausnir Til þess að svala forvitni Guðmundur Ólafsson lektor Kafli 1-11 Uppruni gagna Frumgögn Afleidd gögn Gagnaöflun Gagnasöfnun Athugun Könnun Prentuð eða rafræn gögn Tilraunir Guðmundur Ólafsson lektor Kafli 1-12 Tegundir gagna Gögn Töluleg gögn Flokkunargögn Strjál Guðmundur Ólafsson lektor Samfelld Kafli 1-13 Uppsetning kannanna Veljið viðeigandi svarform Frumform sem hægt er að treysta einstaklingsviðtöl símaviðtöl póstkannanir Síður áreiðanleg form, sjálfvalið (gagnast ekki til ályktana um þýðið í heild) sjónvarpskannanir internetkannanir Prentaðar kannanir í blöðum og tímaritum Vöru- eða þjónustukannanir Guðmundur Ólafsson lektor Kafli 1-14 Uppsetning kannana Finnið almenn atriði Gerið fullkomin lista yfir atriði sem ekki skarast og snerta hið kannaða Gerið nákvæmar spurningar (framhald) Spurningar skulu vera skýrar og ótvíræðar Notið almennt viðurkennd og vel skilgreind orð Prófið könnunina Gerið tilraun með könnunina á litlum hópi til að hún verði skýrari og til þess að afmarka lengd Guðmundur Ólafsson lektor Kafli 1-15 Uppsetning kannana (framhald) Skrifið vandað kynningarbréf Setið fram markmið og hlutverk könnunarinnar Útskýrið mikilvægi svörunar Tryggið nafnleynd svarenda Bjóðið hugsanlega upp á hvatningu, til dæmis gjafir Guðmundur Ólafsson lektor Kafli 1-16 Ástæður fyrir úrtöku Tekur skemmri tíma en að kanna heildina Kostar minna en að kanna heildina Einfaldara að framkvæma úrtakskönnun en að taka heildina fyrir Guðmundur Ólafsson lektor Kafli 1-17 Úrtökuaðferðir Úrtök Ekki líkindaúrtök Líkindaúrtök Einfalt tilviljana Mat Chunk Kvóti Guðmundur Ólafsson lektor Lagskipt Hópaúrtak Kerfisbundið Kafli 1-18 Líkindaúrtök Valið samkvæmt þekktum líkum Líkindaúrtök Einföld tilviljana Kerfisbundin Guðmundur Ólafsson lektor Lagskipt Hópa Kafli 1-19 Einföld tilviljanaúrtök Sérhvert stak í þýði er jafn líklegt til að lenda í úrtakinu Val á staki getur verið endurtekið eða ekki Úrtök gerð samkvæmt tilviljanatöflum eða tilviljanaforritum í tölvu Guðmundur Ólafsson lektor Kafli 1-20 Kerfisbundin úrtök Ákveðið úrtaksstærð: n Skiptið N stökum upp í k hópa: k=n/n Veljið eitthvert stak úr fyrsta hópi tilviljanakent Veljið síðan næsta stak k sætum neðar N = 64 n=8 k=8 Guðmundur Ólafsson lektor Fyrsti hópur Kafli 1-21 Lagskipt úrtök Þýði skipt upp í tvo eða fleirri hópa samkvæmt almennum einkennum Einfallt tilviljanaúrtak valið úr hverjum hópi Tvö eða fleirri úrtök eru síðan samaeinuð Guðmundur Ólafsson lektor Kafli 1-22 Hópaúrtök Þýðinu skipt upp í hópa sem einkenna það Einfallt tilviljanaúrtak tekið úr hverjum hópi Úrtökin sameinuð í eitt Þýði skipt upp í 4 hópa. Guðmundur Ólafsson lektor Kafli 1-23 Kostir og gallar Einföld tilviljanaúrtök og kerfisbundin Stratified sample Einföld í notkun Tiltekin einkenni þýðis kunna að vera vanmetin Tryggja að tillit sé tekið til allra hópa í þýði Cluster sample Ódýrari Ekki eins nákvæm (úrtök þurfa stundum að vera stærri til þess að ná tiltekinni nákvæmni) Guðmundur Ólafsson lektor Kafli 1-24 Mat á gæðum könnunar Hvert er markmiðið? Er könnun byggð á líkindaúrtaki? Er um bjögun að ræða, ef ekki næst í alla Er reynt að kanna þá sem ekki taka afstöðu Mælingaskekkjur? Eru villur í framkvæmd könnunar Guðmundur Ólafsson lektor Kafli 1-25 Könnunarskekkjur Ekki næst í alla Svara ekki Könnunarskekkjur Mælingaskekkjur Ekki með í úrtaki. Eftirfylgni, aukaspurningar Ekki spurt um að sama alstaðar. Vondar spurningar! Guðmundur Ólafsson lektor Kafli 1-26 Yfirlit Fjallað um hvað stjórnendur þurfa að vita um tölfræði Fjallað um vöxt og viðgang nútíma tölfræði Lýsandi og ályktanatölfræði kynntar Mikilvægi gagna tekið fyrir Guðmundur Ólafsson lektor Kafli 1-27 Yfirlit (framhald) Mismunandi uppruni og gerð gagna skilgreind og skýrð Fjallað um gerð kannana Úrtaksaðferðir reifaðar. Helstu kannanavillur kynntar Guðmundur Ólafsson lektor Kafli 1-28 Statistics for Managers using Microsoft Excel 3rd Edition Kafli 2 Lýsing gagna í töflum og myndrænt © 2002 Prentice-Hall, Inc. Chap 1-29 Chapter Topics Organizing numerical data Tabulating and graphing Univariate numerical data The ordered array and stem-leaf display Frequency distributions: tables, histograms, polygons Cumulative distributions: tables, the Ogive Graphing Bivariate numerical data Guðmundur Ólafsson lektor Kafli 1-30 Chapter Topics Tabulating and graphing Univariate categorical data The summary table Bar and pie charts, the Pareto diagram (continued) Tabulating and graphing Bivariate categorical data Contingency tables Side by side bar charts Graphical excellence and common errors in presenting data Guðmundur Ólafsson lektor Kafli 1-31 Organizing Numerical Data Numerical Data Ordered Array 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 Stem and Leaf Display 2 144677 3 028 4 1 Guðmundur Ólafsson lektor 41, 24, 32, 26, 27, 27, 30, 24, 38, 21 Frequency Distributions Cumulative Distributions Histograms Tables Ogive Polygons Kafli 1-32 Organizing Numerical Data (continued) Data in raw form (as collected): 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Data in ordered array from smallest to largest: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 Stem-and-leaf display: 2 144677 3 028 4 1 Guðmundur Ólafsson lektor Kafli 1-33 Tabulating and Graphing Numerical Data Numerical Data Ordered Array 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 41, 24, 32, 26, 27, 27, 30, 24, 38, 21 Frequency Distributions Cumulative Distributions O g ive 120 100 80 60 40 20 0 10 Stem and Leaf Display 2 144677 3 028 4 1 Histograms 30 40 50 60 Ogive 7 6 5 4 Tables Polygons 3 2 1 0 10 Guðmundur Ólafsson lektor 20 20 30 40 50 60 Kafli 1-34 Tabulating Numerical Data: Frequency Distributions Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find range: 58 - 12 = 46 Select number of classes: 5 (usually between 5 and 15) Compute class interval (width): 10 (46/5 then round up) Determine class boundaries (limits): Compute class midpoints: Count observations & assign to classes Guðmundur Ólafsson lektor 10, 20, 30, 40, 50, 60 15, 25, 35, 45, 55 Kafli 1-35 Frequency Distributions, Relative Frequency Distributions and Percentage Distributions Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Class 10 but under 20 20 but under 30 30 but under 40 40 but under 50 50 but under 60 Total Guðmundur Ólafsson lektor Relative Frequency Frequency Percentage 3 6 5 4 2 20 .15 .30 .25 .20 .10 1 15 30 25 20 10 100 Kafli 1-36 Graphing Numerical Data: The Histogram Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Frequency Histogram 7 6 5 4 3 2 1 0 6 5 3 2 0 5 Class Boundaries Guðmundur Ólafsson lektor No Gaps Between Bars 4 0 15 25 36 45 Class Midpoints 55 More Kafli 1-37 Graphing Numerical Data: The Frequency Polygon Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Frequenc y 7 6 5 4 3 2 1 0 5 Guðmundur Ólafsson lektor 15 25 36 45 55 Class Midpoints M ore Kafli 1-38 Tabulating Numerical Data: Cumulative Frequency Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Class 10 but under 20 20 but under 30 30 but under 40 40 but under 50 50 but under 60 Guðmundur Ólafsson lektor Cumulative Frequency 3 9 14 18 20 Cumulative % Frequency 15 45 70 90 100 Kafli 1-39 Graphing Numerical Data: The Ogive (Cumulative % Polygon) Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Ogive 100 80 60 40 20 0 10 20 30 40 50 60 Class Boundaries (Not Midpoints) Guðmundur Ólafsson lektor Kafli 1-40 Graphing Bivariate Numerical Data (Scatter Plot) Mutual Funds Scatter Plot Total Year to Date Return (%) 40 30 20 10 0 0 Guðmundur Ólafsson lektor 10 20 30 Net Asset Values 40 Kafli 1-41 Tabulating and Graphing Categorical Data:Univariate Data Categorical Data Tabulating Data The Summary Table Graphing Data Pie Charts Bar Charts Guðmundur Ólafsson lektor Pareto Diagram Kafli 1-42 Summary Table (for an Investor’s Portfolio) Investment Category Amount Percentage (in thousands $) Stocks Bonds CD Savings Total 46.5 32 15.5 16 110 42.27 29.09 14.09 14.55 100 Variables are Categorical Guðmundur Ólafsson lektor Kafli 1-43 Graphing Categorical Data: Univariate Data Categorical Data Graphing Data Tabulating Data The Summary Table Pie Charts CD Pareto Diagram S a vi n g s Bar Charts B onds S to c k s 0 10 20 30 40 50 45 120 40 100 35 30 80 25 60 20 15 40 10 20 5 0 0 S to c k s Guðmundur Ólafsson lektor B onds S a vi n g s CD Kafli 1-44 Bar Chart (for an Investor’s Portfolio) Investor's Portfolio Savings CD Bonds Stocks 0 10 20 30 40 50 Amount in K$ Guðmundur Ólafsson lektor Kafli 1-45 Pie Chart (for an Investor’s Portfolio) Amount Invested in K$ Savings 15% Stocks 42% CD 14% Bonds 29% Guðmundur Ólafsson lektor Percentages are rounded to the nearest percent. Kafli 1-46 Pareto Diagram Axis for bar chart shows % invested in each category 45% 100% 40% 90% 80% 35% 70% 30% 60% 25% 50% 20% 40% 15% 30% 10% 20% 5% 10% 0% 0% Stocks Guðmundur Ólafsson lektor Bonds Savings Axis for line graph shows cumulative % invested CD Kafli 1-47 Tabulating and Graphing Bivariate Categorical Data Contingency tables: Investment Category Investor A Stocks Bonds CD Savings 46.5 32 15.5 16 Total 110 Guðmundur Ólafsson lektor investment in thousands of dollars Investor B Investor C Total 55 44 20 28 27.5 19 13.5 7 129 95 49 51 147 67 324 Kafli 1-48 Tabulating and Graphing Bivariate Categorical Data Side by side charts C o m p arin g In vesto rs S avings CD B onds S toc k s 0 10 Inves tor A Guðmundur Ólafsson lektor 20 30 Inves tor B 40 50 60 Inves tor C Kafli 1-49 Principles of Graphical Excellence Presents data in a way that provides substance, statistics and design Communicates complex ideas with clarity, precision and efficiency Gives the largest number of ideas in the most efficient manner Almost always involves several dimensions Tells the truth about the data Guðmundur Ólafsson lektor Kafli 1-50 Errors in Presenting Data Using “chart junk” Failing to provide a relative basis in comparing data between groups Compressing the vertical axis Providing no zero point on the vertical axis Guðmundur Ólafsson lektor Kafli 1-51 “Chart Junk” Bad Presentation Good Presentation Minimum Wage 1960: $1.00 Minimum Wage 4 $ 1970: $1.60 2 1980: $3.10 0 1990: $3.80 Guðmundur Ólafsson lektor 1960 1970 1980 1990 Kafli 1-52 No Relative Basis Bad Presentation Good Presentation A’s received by Freq. students. 300 200 30 % 10 0 FR SO JR SR A’s received by students. FR SO JR SR FR = Freshmen, SO = Sophomore, JR = Junior, SR = Senior Guðmundur Ólafsson lektor Kafli 1-53 Compressing Vertical Axis Bad Presentation Good Presentation Quarterly Sales 200 $ Quarterly Sales 50 100 25 0 0 Q1 Q2 Q3 Q4 Guðmundur Ólafsson lektor $ Q1 Q2 Q3 Q4 Kafli 1-54 No Zero Point on Vertical Axis Bad Presentation Good Presentation Monthly Sales 45 $ Monthly Sales 45 42 39 42 39 $ 36 36 J F M A M J 0 J F M A M J Graphing the first six months of sales. Guðmundur Ólafsson lektor Kafli 1-55 Chapter Summary Organized numerical data Tabulated and graphed univariate numerical data The ordered array and stem-leaf display Frequency distributions: tables, histograms, polygon Cumulative distributions: tables and the Ogive Graphed bivariate numerical data Guðmundur Ólafsson lektor Kafli 1-56 Chapter Summary Tabulated and graphed univariate categorical data The summary table Bar and pie charts, the Pareto diagram Tabulated and graphed bivariate categorical data (continued) Contingency tables Side by side charts Discussed graphical excellence and common errors in presenting data Guðmundur Ólafsson lektor Kafli 1-57 Statistics for Managers using Microsoft Excel 3rd Edition Chapter 3 Numerical Descriptive Measures © 2002 Prentice-Hall, Inc. Chap 1-58 Chapter Topics Measures of central tendency Mean, median, mode, geometric mean, midrange Quartile Measure of variation Range, interquartile range, variance and standard deviation, coefficient of variation Shape Symmetric, skewed, using box-and-whisker plots Guðmundur Ólafsson lektor Kafli 1-59 Chapter Topics (continued) Coefficient of correlation Pitfalls in numerical descriptive measures and ethical considerations Guðmundur Ólafsson lektor Kafli 1-60 Summary Measures Summary Measures Central Tendency Mean Quartile Mode Median Range Variation Coefficient of Variation Variance Geometric Mean Guðmundur Ólafsson lektor Standard Deviation Kafli 1-61 Measures of Central Tendency Central Tendency Average Median Mode n X X i 1 i X G X1 X 2 n N X i 1 Geometric Mean Xn 1/ n i N Guðmundur Ólafsson lektor Kafli 1-62 Mean (Arithmetic Mean) Mean (arithmetic mean) of data values Sample mean Sample Size n X X i 1 i n X1 X 2 n Xn Population mean Population Size N Guðmundur Ólafsson lektor X i 1 N i X1 X 2 N XN Kafli 1-63 Mean (Arithmetic Mean) (continued) The most common measure of central tendency Affected by extreme values (outliers) 0 1 2 3 4 5 6 7 8 9 10 Mean = 5 Guðmundur Ólafsson lektor 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6 Kafli 1-64 Median Robust measure of central tendency Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 Median = 5 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 In an ordered array, the median is the “middle” number If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers Guðmundur Ólafsson lektor Kafli 1-65 Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may may be no mode There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Guðmundur Ólafsson lektor Mode = 9 0 1 2 3 4 5 6 No Mode Kafli 1-66 Geometric Mean Useful in the measure of rate of change of a variable over time X G X1 X 2 Xn 1/ n Geometric mean rate of return Measures the status of an investment over time RG 1 R1 1 R2 Guðmundur Ólafsson lektor 1 Rn 1/ n 1 Kafli 1-67 Example An investment of $100,000 declined to $50,000 at the end of year one and rebounded to $100,000 at end of year two: X1 $100,000 X 2 $50,000 X 3 $100,000 Average rate of return: (50%) (100%) X 25% 2 Geometric rate of return: RG 1 50% 1 100% 1/ 2 0.50 2 1/ 2 Guðmundur Ólafsson lektor 1 1 1 1 0% 1/ 2 Kafli 1-68 Quartiles Split Ordered Data into 4 Quarters 25% 25% Q1 25% Q2 Position of i-th Quartile 25% Q3 i n 1 Qi 4 Data in Ordered Array: 11 12 13 16 16 17 18 21 22 1 9 1 Position of Q1 2.5 4 Q1 12 13 12.5 2 Q1 and Q3 Are Measures of Noncentral Location Q = Median, A Measure of Central Tendency 2 Guðmundur Ólafsson lektor Kafli 1-69 Measures of Variation Variation Variance Range Population Variance Sample Variance Interquartile Range Guðmundur Ólafsson lektor Standard Deviation Coefficient of Variation Population Standard Deviation Sample Standard Deviation Kafli 1-70 Range Measure of variation Difference between the largest and the smallest observations: Range X Largest X Smallest Ignores the way in which data are distributed Range = 12 - 7 = 5 Range = 12 - 7 = 5 7 8 9 10 11 Guðmundur Ólafsson lektor 12 7 8 9 10 11 12 Kafli 1-71 Interquartile Range Measure of variation Also known as midspread Spread in the middle 50% Difference between the first and third quartiles Data in Ordered Array: 11 12 13 16 16 17 17 18 21 Interquartile Range Q3 Q1 17.5 12.5 5 Not affected by extreme values Guðmundur Ólafsson lektor Kafli 1-72 Variance Important measure of variation Shows variation about the mean Sample variance: n S 2 X i 1 X i 2 n 1 Population variance: N 2 Guðmundur Ólafsson lektor X i 1 i N 2 Kafli 1-73 Standard Deviation Most important measure of variation Shows variation about the mean Has the same units as the original data Sample standard deviation: n S Population standard deviation: Guðmundur Ólafsson lektor X i 1 X i 2 n 1 N X i 1 i 2 N Kafli 1-74 Comparing Standard Deviations Data A 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 3.338 Data B 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = .9258 Data C 11 12 13 14 15 16 17 18 19 20 21 Guðmundur Ólafsson lektor Mean = 15.5 s = 4.57 Kafli 1-75 Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units S CV X Guðmundur Ólafsson lektor 100% Kafli 1-76 Comparing Coefficient of Variation Stock A: Stock B: Average price last year = $50 Standard deviation = $5 Average price last year = $100 Standard deviation = $5 Coefficient of variation: Stock A: Stock B: Guðmundur Ólafsson lektor S CV X $5 100% 100% 10% $50 S CV X $5 100% 100% 5% $100 Kafli 1-77 Shape of a Distribution Describes how data is distributed Measures of shape Symmetric or skewed Left-Skewed Mean < Median < Mode Guðmundur Ólafsson lektor Symmetric Mean = Median =Mode Right-Skewed Mode < Median < Mean Kafli 1-78 Exploratory Data Analysis Box-and-whisker plot Graphical display of data using 5-number summary X smallest Q 1 4 6 Guðmundur Ólafsson lektor Median( Q2) 8 Q3 10 Xlargest 12 Kafli 1-79 Distribution Shape and Box-and-Whisker Plot Left-Skewed Q1 Q2 Q3 Guðmundur Ólafsson lektor Symmetric Q1Q2Q3 Right-Skewed Q1 Q2 Q3 Kafli 1-80 Coefficient of Correlation Measures the strength of the linear relationship between two quantitative variables n r X i 1 n X i 1 Guðmundur Ólafsson lektor i i X Yi Y X 2 n Y Y i 1 2 i Kafli 1-81 Features of Correlation Coefficient Unit free Ranges between –1 and 1 The closer to –1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker any positive linear relationship Guðmundur Ólafsson lektor Kafli 1-82 Scatter Plots of Data with Various Correlation Coefficients Y Y Y X r = -1 X r = -.6 Y X r=0 Y r = .6 Guðmundur Ólafsson lektor X r=1 X Kafli 1-83 Pitfalls in Numerical Descriptive Measures Data analysis is objective Should report the summary measures that best meet the assumptions about the data set Data interpretation is subjective Should be done in fair, neutral and clear manner Guðmundur Ólafsson lektor Kafli 1-84 Ethical Considerations Numerical descriptive measures: Should document both good and bad results Should be presented in a fair, objective and neutral manner Should not use inappropriate summary measures to distort facts Guðmundur Ólafsson lektor Kafli 1-85 Chapter Summary Described measures of central tendency Mean, median, mode, geometric mean, midrange Discussed quartile Described measure of variation Range, interquartile range, variance and standard deviation, coefficient of variation Illustrated shape of distribution Symmetric, skewed, box-and-whisker plots Guðmundur Ólafsson lektor Kafli 1-86 Chapter Summary (continued) Discussed correlation coefficient Addressed pitfalls in numerical descriptive measures and ethical considerations Guðmundur Ólafsson lektor Kafli 1-87