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Overviews descriptive and graphical approaches to analysis of univariate data.

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Semester 1, 2017, University of Canberra, ACT, Australia

James T. Neill

http://www.slideshare.net/jtneill/descriptives-graphing

http://en.wikiversity.org/wiki/Survey_research_and_design_in_psychology/Lectures/Descriptives_%26_graphing

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Description: Overviews descriptive statistics and graphical approaches to analysis of univariate data.

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Ordinal Median e.g.,

See also http://www.quickmba.com/stats/centralten/

OVERHEAD p.84 Bryman & Duncan (1997)

Note that if you want to test whether one frequency is significantly higher than another, then use binomial test or a contingency test (chi-square).

Also note that nominal variables can be used as IV’s in tests of mean differences, but not in parametric tests of association such as correlations. To use in parametric tests of association, nominal data can be dummy coded (i.e., converted into a series of dichotomous variables).

Crosstabs (contingency table) is the bivariate equivalent of frequencies

By Trevor Blake http://www.flickr.com/photos/trevorblake/

License: CC-by-SA 2.0 http://creativecommons.org/licenses/by-sa/2.0/deed.en

Karl Pearson in his 1893 letter to Nature suggested that the moments about the mean could be used to measure the deviations of empirical distributions from the normal distribution

Moments around the mean:

http://www.visualstatistics.net/Visual%20Statistics%20Multimedia/normalization.htm

If SD = 1 one for cms, it would be 10 for ms and 1000 for km

So, don’t assume SD = 50 is big or SD = .1 is small – it all depends on what scale is used.

Be aware that the lower the N, the lower the SD –&gt; large samples reduce the SD

N-1 is the formula when generalising from a sample to a population; otherwise use N if its the SD for the sample.

The kurtosis reflects the extent to which the density of the empirical distribution differs from the probability densities of the normal curve.

Mesokurtic = 0

http://www.visualstatistics.net/Visual%20Statistics%20Multimedia/normalization.htm

Image author: Dan Kernler, https://commons.wikimedia.org/w/index.php?title=User:Mathprofdk

Image license: Creative Commons Attribution-Share Alike 4.0 International license, https://creativecommons.org/licenses/by-sa/4.0/deed.en

The significance tests for skewness / kurtosis are subject to sample size, so with a small size sample they are less likely to be significant than with a large sample size.

Roughly normal, with positive skew

Bimodal

Bimodal, with positive skew

At what age do you think you will die?

There is an outlier near zero which is minimising the positive skew; the data is also leptokurtic.

This distribution is bi-modal. It should not be treated as normal.

In fact, if one looks more closely, it would sense to break down the distribution by gender.

From the Quick Fun Survey data in Tutorial 1.

This is the distribution for males; it has a ceiling effect, with ‘very feminine’ not being selected at all (and not shown on the graph – it should be). It is negatively skewed and leptokurtic. Note though that because ‘Very feminine’ has no cases and is not shown, the population data would probably be even more skewed than this sample indicates. It is probably leptokurtic.

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Cleveland, William S., Elements of Graphing Data, 1985

License: Unknown

Cleveland (1984) conducted experiments to measure people&apos;s accuracy in interpreting graphs, with findings as follows (Robbins):

Position along a common scale

Position along non aligned scales

Length

Angle-slope

Area

Volume

Color hue - saturation - density

Cleveland, William S., Elements of Graphing Data, 1985

License: Unknown

Cleveland (1984) conducted experiments to measure people&apos;s accuracy in interpreting graphs, with findings as follows (Robbins):

Position along a common scale

Position along non aligned scales

Length

Angle-slope

Area

Volume

Color hue - saturation - density

DV = height (ratio)

IV = Gender (categorical)

A bit of a plug and plea for stem & leaf plots – they are underused. They are powerful because they are:

Efficient – e.g., they contain all the data succinctly – others could use the data in a stem & leaf plot to do further analysis

Visual and mathematical: As well as containing all the data, the stem & leaf plot presents a powerful, recognizable visual of the data, akin to a bar graph. Turning a stem & leaf plot 90 degrees counter-clockiwse is recommend – this makes the visual display more conventional and is easy to recognise, and the numbers are are less obvious, hence emphasizing the visual histogram shape.

This is a univariate precursor to a scatterplot (a plot of a ratio by ratio variable).

It works if there is a small amount of data; otherwise use a histogram to indicate the frequency within equal interval ranges.

From: http://www.physics.csbsju.edu/stats/display.distribution.html

Image source: Unknown.

Karl Pearson in his 1893 letter to Nature suggested that the moments about the mean could be used to measure the deviations of empirical distributions from the normal distribution

Moments around the mean:

http://www.visualstatistics.net/Visual%20Statistics%20Multimedia/normalization.htm

Image source: James Neill, 2007, Creative Commons Attribution 2.5 Australia.

Histogram: At what age do you think you will die?

There is an outlier near zero which is minimising the positive skew; the data is also quite strongly leptokurtic.

- 1. Lecture 3 Survey Research & Design in Psychology James Neill, 2017 Creative Commons Attribution 4.0 Descriptives & Graphing Image source: http://commons.wikimedia.org/wiki/File:3D_Bar_Graph_Meeting.jpg
- 2. 2 Overview: Descriptives & Graphing 1. Getting to know a data set 2. LOM & types of statistics 3. Descriptive statistics 4. Normal distribution 5. Non-normal distributions 6. Effect of skew on central tendency 7. Principles of graphing 8. Univariate graphical techniques
- 3. 3 Getting to know a data-set (how to approach data)
- 4. 4 Play with the data – get to know it.Image source: http://www.flickr.com/photos/analytik/1356366068/
- 5. 5 Don't be afraid - you can't break data!Image source: http://www.flickr.com/photos/rnddave/5094020069
- 6. 6 Check & screen the data – keep signal, reduce noise Image source: https://commons.wikimedia.org/wiki/File:Nasir-al_molk_-1.jpg
- 7. 7 Data checking: One person reads the survey responses aloud to another person who checks the electronic data file. For large studies, check a proportion of the surveys and declare the error-rate in the research report. Image source: http://maxpixel.freegreatpicture.com/Business-Team-Two-People-Meeting-Computers-Office-1209640
- 8. 8 Data screening: Carefully 'screening' a data file helps to remove errors and maximise validity. For example, screen for: Out of range values Mis-entered data Missing cases Duplicate cases Missing data Image source: https://commons.wikimedia.org/wiki/File:Archaeology_dirt_screening.jpg
- 9. 9 Explore the data Image source: https://commons.wikimedia.org/wiki/File:Kazimierz_Nowak_in_jungle_2.jpgI
- 10. 10 Get intimate with the data Image source: http://www.flickr.com/photos/elmoalves/2932572231/
- 11. 11 Describe the data's main features find a meaningful, accurate way to depict the ‘true story’ of the data Image source: http://www.flickr.com/photos/lloydm/2429991235/
- 12. 12 Test hypotheses to answer research questions Image source: https://pixabay.com/en/light-bulb-current-light-glow-1042480/
- 13. 13 Level of measurement & types of statistics Image source: http://www.flickr.com/photos/peanutlen/2228077524/
- 14. 14 Golden rule of data analysis A variable's level of measurement determines the type of statistics that can be used, including types of: • descriptive statistics • graphs • inferential statistics
- 15. 15 Levels of measurement and non-parametric vs. parametric Categorical & ordinal data DV → non-parametric (Does not assume a normal distribution) Interval & ratio data DV → parametric (Assumes a normal distribution) → non-parametric (If distribution is non-normal) DVs = dependent variables
- 16. 16 Parametric statistics • Statistics which estimate parameters of a population, based on the normal distribution –Univariate: mean, standard deviation, skewness, kurtosis, t-tests, ANOVAs –Bivariate: correlation, linear regression –Multivariate: multiple linear regression
- 17. 17 • More powerful (more sensitive) • More assumptions (population is normally distributed) • Vulnerable to violations of assumptions (less robust) Parametric statistics
- 18. 18 Non-parametric statistics • Statistics which do not assume sampling from a population which is normally distributed –There are non-parametric alternatives for many parametric statistics –e.g., sign test, chi-squared, Mann- Whitney U test, Wilcoxon matched-pairs signed-ranks test.
- 19. 19 Non-parametric statistics • Less powerful (less sensitive) • Fewer assumptions (do not assume a normal distribution) • Less vulnerable to assumption violation (more robust)
- 20. 20 Univariate descriptive statistics
- 21. 21 Number of variables Univariate = one variable Bivariate = two variables Multivariate = more than two variables mean, median, mode, histogram, bar chart correlation, t-test, scatterplot, clustered bar chart reliability analysis, factor analysis, multiple linear regression
- 22. 22 What do we want to describe? The distributional properties of variables, based on: ● Central tendency(ies): e.g., frequencies, mode, median, mean ● Shape: e.g., skewness, kurtosis ● Spread (dispersion): min., max., range, IQR, percentiles, variance, standard deviation
- 23. 23 Measures of central tendency Statistics which represent the ‘centre’ of a frequency distribution: –Mode (most frequent) –Median (50th percentile) –Mean (average) Which ones to use depends on: –Type of data (level of measurement) –Shape of distribution (esp. skewness) Reporting more than one may be appropriate.
- 24. 24 Measures of central tendency √√If meaningfulRatio √√√Interval √Ordinal √Nominal MeanMedianMode / Freq. /%s If meaningful x x x
- 25. 25 Measures of distribution • Measures of shape, spread, dispersion, and deviation from the central tendency Non-parametric: • Min. and max. • Range • Percentiles Parametric: • SD • Skewness • Kurtosis
- 26. 26 √√√Ratio √√Interval √Ordinal Nominal Var / SDPercentileMin / Max, Range Measures of spread / dispersion / deviation If meaningful √ x x x x
- 27. 27 Descriptives for nominal data • Nominal LOM = Labelled categories • Descriptive statistics: –Most frequent? (Mode – e.g., females) –Least frequent? (e.g., Males) –Frequencies (e.g., 20 females, 10 males) –Percentages (e.g. 67% females, 33% males) –Cumulative percentages –Ratios (e.g., twice as many females as males)
- 28. 28 Descriptives for ordinal data • Ordinal LOM = Conveys order but not distance (e.g., ranks) • Descriptives approach is as for nominal (frequencies, mode etc.) • Plus percentiles (including median) may be useful
- 29. 29 Descriptives for interval data • Interval LOM = order and distance, but no true 0 (0 is arbitrary). • Central tendency (mode, median, mean) • Shape/Spread (min., max., range, SD, skewness, kurtosis) Interval data is discrete, but is often treated as ratio/continuous (especially for > 5 intervals)
- 30. 30 Descriptives for ratio data • Ratio = Numbers convey order and distance, meaningful 0 point • As for interval, use median, mean, SD, skewness etc. • Can also use ratios (e.g., Category A is twice as large as Category B)
- 31. 31 Mode (Mo) • Most common score - highest point in a frequency distribution – a real score – the most common response • Suitable for all levels of data, but may not be appropriate for ratio (continuous) • Not affected by outliers • Check frequencies and bar graph to see whether it is an accurate and useful statistic
- 32. 32 Frequencies (f) and percentages (%) • # of responses in each category • % of responses in each category • Frequency table • Visualise using a bar or pie chart
- 33. 33 Median (Mdn) • Mid-point of distribution (Quartile 2, 50th percentile) • Not badly affected by outliers • May not represent the central tendency in skewed data • If the Median is useful, then consider what other percentiles may also be worth reporting
- 34. 34 Summary: Descriptive statistics • Level of measurement and normality determines whether data can be treated as parametric • Describe the central tendency –Frequencies, Percentages –Mode, Median, Mean • Describe the variability: –Min., Max., Range, Quartiles –Standard Deviation, Variance
- 35. 35 Properties of the normal distribution Image source: http://www.flickr.com/photos/trevorblake/3200899889/
- 36. 36 Four moments of a normal distribution Row 1 Row 2 Row 3 Row 4 0 2 4 6 8 10 12 Column 1 Column 2 Column 3 Mean ←SD→ -ve Skew +ve Skew ←Kurtosis→
- 37. 37 Four moments of a normal distribution Four mathematical qualities (parameters) can describe a continuous distribution which at least roughly follows a bell curve shape: • 1st = mean (central tendency) • 2nd = SD (dispersion) • 3rd = skewness (lean / tail) • 4th = kurtosis (peakedness / flattness)
- 38. 38 Mean (1st moment ) • Average score Mean = Σ X / N • For normally distributed ratio or interval (if treating it as continuous) data. • Influenced by extreme scores (outliers)
- 39. 39 Beware inappropriate averaging... With your head in an oven and your feet in ice you would feel, on average, just fine The majority of people have more than the average number of legs (M = 1.9999).
- 40. 40 Standard deviation (2nd moment) • SD = square root of the variance = Σ (X - X)2 N – 1 • For normally distributed interval or ratio data • Affected by outliers • Can also derive the Standard Error (SE) = SD / square root of N
- 41. 41 Skewness (3rd moment ) • Lean of distribution – +ve = tail to right – -ve = tail to left • Can be caused by an outlier, or ceiling or floor effects • Can be accurate (e.g., cars owned per person would have a skewed distribution)
- 42. 42 Skewness (3rd moment) (with ceiling and floor effects) ● Negative skew ● Ceiling effect ● Positive skew ● Floor effect Image source http://www.visualstatistics.net/Visual%20Statistics%20Multimedia/normalization.htm
- 43. 43 Kurtosis (4th moment ) • Flatness or peakedness of distribution +ve = peaked -ve = flattened • By altering the X &/or Y axis, any distribution can be made to look more peaked or flat – add a normal curve to help judge kurtosis visually.
- 44. 44 Kurtosis (4th moment ) Image source: https://classconnection.s3.amazonaws.com/65/flashcards/2185065/jpg/kurtosis-142C1127AF2178FB244.jpg
- 45. 45 Judging severity of skewness & kurtosis • View histogram with normal curve • Deal with outliers • Rule of thumb: Skewness and kurtosis > -1 or < 1 is generally considered to sufficiently normal for meeting the assumptions of parametric inferential statistics • Significance tests of skewness: Tend to be overly sensitive (therefore avoid using)
- 46. 46 Areas under the normal curve If distribution is normal (bell-shaped - or close): ~68% of scores within +/- 1 SD of M ~95% of scores within +/- 2 SD of M ~99.7% of scores within +/- 3 SD of M
- 47. 47 Areas under the normal curve Image source: https://commons.wikimedia.org/wiki/File:Empirical_Rule.PNG
- 48. 48 Non-normal distributions
- 49. 49 Types of non-normal distribution • Modality –Uni-modal (one peak) –Bi-modal (two peaks) –Multi-modal (more than two peaks) • Skewness –Positive (tail to right) –Negative (tail to left) • Kurtosis –Platykurtic (Flat) –Leptokurtic (Peaked)
- 50. 50 Non-normal distributions
- 51. 51 Histogram of people's weight WEIGHT 110.0100.090.080.070.060.050.040.0 Histogram Frequency 8 6 4 2 0 Std. Dev = 17.10 Mean = 69.6 N = 20.00
- 52. 52 Histogram of daily calorie intake N = 75
- 53. 53 Histogram of fertility
- 54. 54 Example ‘normal’ distribution 1 140120100806040200 Die 60 50 40 30 20 10 0 Frequency Mean =81.21 Std. Dev. =18.228 N =188 At what age do you think you will die?
- 55. 55 Example ‘normal’ distribution 2 Very masculineFairly masculineAndrogynousFairly feminineVery feminine Femininity-Masculinity 60 40 20 0 Count This bimodal graph actually consists of two different, underlying normal distributions.
- 56. 56 Very masculineFairly masculineAndrogynousFairly feminineVery feminine Femininity-Masculinity 60 40 20 0 Count Very masculineFairly masculineAndrogynousFairly feminine Femininity-Masculinity 50 40 30 20 10 0 Count Gender: male Distribution for females Distribution for males Very masculineFairly masculineAndrogynousFairly feminineVery feminine Femininity-Masculinity 60 40 20 0 Count Gender: female
- 57. 57 Non-normal distribution: Use non-parametric descriptive statistics • Min. & Max. • Range = Max. - Min. • Percentiles • Quartiles –Q1 –Mdn (Q2) –Q3 –IQR (Q3-Q1)
- 58. 58 Effects of skew on measures of central tendency +vely skewed distributions mode < median < mean symmetrical (normal) distributions mean = median = mode -vely skewed distributions mean < median < mode
- 59. 59 Effects of skew on measures of central tendency
- 60. 60 Transformations • Converts data using various formulae to achieve normality and allow more powerful tests • Loses original metric • Complicates interpretation
- 61. 61 1. If a survey question produces a ‘floor effect’, where will the mean, median and mode lie in relation to one another? Review questions
- 62. 62 2. Would the mean # of cars owned in Australia to exceed the median? Review questions
- 63. 63 3. Would the mean score on an easy test exceed the median performance? Review questions
- 64. 64 Graphical techniques Image source: http://www.flickr.com/photos/pagedooley/2121472112/
- 65. 65 Visualisation “Visualization is any technique for creating images, diagrams, or animations to communicate a message.” - Wikipedia Image source: http://en.wikipedia.org/wiki/File:FAE_visualization.jpg
- 66. 66 Science is beautiful (Nature Video) (Youtube – 5:30 mins) Image source:http://commons.wikimedia.org/wiki/File:Parodyfilm.png
- 67. 67 Is Pivot a turning point for web exploration? (Gary Flake) (TED talk - 6 min.) Image source:http://commons.wikimedia.org/wiki/File:Parodyfilm.png
- 68. 68 Principles of graphing • Clear purpose • Maximise clarity • Minimise clutter • Allow visual comparison
- 69. 69 Graphs (Edward Tufte) • Visualise data • Reveal data – Describe – Explore – Tabulate – Decorate • Communicate complex ideas with clarity, precision, and efficiency
- 70. 70 Graphing steps 1. Identify purpose of the graph (make large amounts of data coherent; present many #s in small space; encourage the eye to make comparisons) 2. Select type of graph to use 3. Draw and modify graph to be clear, non-distorting, and well- labelled (maximise clarity, minimise clarity; show the data; avoid distortion; reveal data at several levels/layers)
- 71. 71 Software for data visualisation (graphing) 1. Statistical packages ● e.g., SPSS Graphs or via Analyses 2. Spreadsheet packages ● e.g., MS Excel 3. Word-processors ● e.g., MS Word – Insert – Object – Micrograph Graph Chart
- 72. 72 Cleveland’s hierarchyImage source:http://www.processtrends.com/TOC_data_visualization.htm
- 73. 73 Cleveland’s hierarchyImage source:https://priceonomics.com/how-william-cleveland-turned-data-visualization/
- 74. 74 Univariate graphs • Bar graph • Pie chart • Histogram • Stem & leaf plot • Data plot / Error bar • Box plot Non-parametric i.e., nominal or ordinal } } Parametric i.e., normally distributed interval or ratio
- 75. 75 Bar chart (Bar graph) AREA Biology Anthropology Information Technolo Psychology Sociology Count 13 12 12 11 11 10 10 9 9 AREA Biology Anthropology Information Technolo Psychology Sociology Count 12 11 10 9 8 7 6 5 4 3 2 1 0 • Allows comparison of heights of bars • X-axis: Collapse if too many categories • Y-axis: Count/Frequency or % - truncation exaggerates differences • Can add data labels (data values for each bar) Note truncated Y-axis
- 76. 76 Biology Anthropology Information Technolo Psychology Sociology • Use a bar chart instead • Hard to read –Difficult to show • Small values • Small differences –Rotation of chart and position of slices influences perception Pie chart
- 77. 77 Pie chart → Use bar chart instead Image source: https://priceonomics.com/how-william-cleveland-turned-data-visualization/
- 78. 78 Histogram Participant Age 62.552.542.532.522.512.5 3000 2000 1000 0 Std. Dev = 9.16 Mean = 24.0 N = 5575.00 Participant Age 63.0 58.0 53.0 48.0 43.0 38.0 33.0 28.0 23.0 18.0 13.0 8.0 600 500 400 300 200 100 0 Std. Dev = 9.16 Mean = 24.0 N = 5575.00 Participant Age 65 61 57 53 49 45 41 37 33 29 25 21 17 13 9 1000 800 600 400 200 0 Std. Dev = 9.16 Mean = 24 N = 5575.00 • For continuous data (Likert?, Ratio) • X-axis needs a happy medium for # of categories • Y-axis matters (can exaggerate)
- 79. 79 Histogram of male & female heights Wild & Seber (2000) Image source: Wild, C. J., & Seber, G. A. F. (2000). Chance encounters: A first course in data analysis and inference. New York: Wiley.
- 80. 80 Stem & leaf plot ● Use for ordinal, interval and ratio data (if rounded) ● May look confusing to unfamiliar reader
- 81. 81 • Contains actual data • Collapses tails • Underused alternative to histogram Stem & leaf plot Frequency Stem & Leaf 7.00 1 . & 192.00 1 . 22223333333 541.00 1 . 444444444444444455555555555555 610.00 1 . 6666666666666677777777777777777777 849.00 1 . 88888888888888888888888888899999999999999999999 614.00 2 . 0000000000000000111111111111111111 602.00 2 . 222222222222222233333333333333333 447.00 2 . 4444444444444455555555555 291.00 2 . 66666666677777777 240.00 2 . 88888889999999 167.00 3 . 000001111 146.00 3 . 22223333 153.00 3 . 44445555 118.00 3 . 666777 99.00 3 . 888999 106.00 4 . 000111 54.00 4 . 222 339.00 Extremes (>=43)
- 82. 82 Box plot (Box & whisker) ● Useful for interval and ratio data ● Represents min., max, median, quartiles, & outliers
- 83. 83 • Alternative to histogram • Useful for screening • Useful for comparing variables • Can get messy - too much info • Confusing to unfamiliar reader Box plot (Box & whisker) Participant Gender FemaleMaleMissing 10 8 6 4 2 0 Time Management-T1 Self-Confidence-T1 44954162578259628414042044353275182341862330517623006559128211495 3201419358828475475400198324512898200336473 52157129504268724318255928345427211669040523444 4423423635403519067273946893137 3562338330403962312229 12255255545 2385410773323584004 552433515563 28294482267253154120226228451504231939983902646355221793020527435314997364541416412902548168628144167196326144171955174443826882822262617931747148 218736735510399522434250553623594998649620510638344230032962562527 35644317149302843626902101233519693009296541539905538229314216883634 27433593251521081985531655582138303424526783352317 2480296024926454284316542285186 419324766472662291 6084308 17 2699 3556334 1503275241623466255243493045 304032431371222596415943511907247380 4028 18082659 197862231372721142861 226520672270403852527688296021515564300430321938532836535506271835192336608405435012183292849986302224518624385114882241 27806412743294423212570661146542792576430229232476 231214932334 4308292014254307 569 5491
- 84. 84 Data plot & error bar Data plot Error bar
- 85. 85 • Alternative to histogram • Implies continuity e.g., time • Can show multiple lines Line graph OVERALL SCALES-T3 OVERALL SCALES-T2 OVERALL SCALES-T1 OVERALL SCALES-T0 Mean 8.0 7.5 7.0 6.5 6.0 5.5 5.0
- 86. 86 Graphical integrity (part of academic integrity)
- 87. 87 "Like good writing, good graphical displays of data communicate ideas with clarity, precision, and efficiency. Like poor writing, bad graphical displays distort or obscure the data, make it harder to understand or compare, or otherwise thwart the communicative effect which the graph should convey." Michael Friendly – Gallery of Data Visualisation
- 88. 88 Tufte’s graphical integrity • Some lapses intentional, some not • Lie Factor = size of effect in graph size of effect in data • Misleading uses of area • Misleading uses of perspective • Leaving out important context • Lack of taste and aesthetics
- 89. 89 Review exercise: Fill in the cells in this table Level Properties Examples Descriptive Statistics Graphs Nominal /Categorical Ordinal / Rank Interval Ratio Answers: http://goo.gl/Ln9e1
- 90. 90 References 1. Chambers, J., Cleveland, B., Kleiner, B., & Tukey, P. (1983). Graphical methods for data analysis. Boston, MA: Duxbury Press. 2. Cleveland, W. S. (1985). The elements of graphing data. Monterey, CA: Wadsworth. 3. Jones, G. E. (2006). How to lie with charts. Santa Monica, CA: LaPuerta. 4. Tufte, E. R. (1983). The visual display of quantitative information. Cheshire, CT: Graphics Press. 5. Tufte. E. R. (2001). Visualizing quantitative data. Cheshire, CT: Graphics Press. 6. Tukey J. (1977). Exploratory data analysis. Addison-Wesley. 7. Wild, C. J., & Seber, G. A. F. (2000). Chance encounters: A first course in data analysis and inference. New York: Wiley.
- 91. 91 Open Office Impress ● This presentation was made using Open Office Impress. ● Free and open source software. ● http://www.openoffice.org/product/impress.html

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