Steps with data (how to approach data)

Descriptives & Graphing Lecture 3 Survey Research & Design in Psychology James Neill, 216 Creative Commons Attribution 4. Overview: Descriptives & Graphing 1. Steps with data 2. Level of measurement & 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 2 Check and screen data Steps with data Steps with data (how to approach data) Explore, describe, & graph Test hypotheses 3 4 Data checking Have one one person read 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. 6

Data screening Carefully 'screening' a data file helps to minimise errors and maximise validity. For example, screen for: Out of range values (min. and max.) Mis-entered data Missing cases Duplicate cases Missing data 7 Level of measurement & types of statistics Golden rule of data analysis Level of measurement determines type of descriptive statistics and graphs Level of measurement determines which types of descriptive statistics and which types of graphs are appropriate. 11 12

Levels of measurement and non-parametric vs. parametric Categorical & ordinal DVs non-parametric (Does not assume a normal distribution) Interval & ratio DVs parametric (Assumes a normal distribution) non-parametric (If distribution is non-normal) Parametric statistics Procedures which estimate parameters of a population, usually based on the normal distribution Key parametric statistics: Univariate: M, SD, skewness, kurtosis t-tests, ANOVAs Bi/multivariate: r linear regression, multiple linear regression DVs = dependent variables 13 14 Parametric statistics More powerful (more sensitive) More assumptions (normal distribution) More vulnerable to violations of assumptions Non-parametric statistics (Distribution-free tests) Fewer assumptions (do not assume a normal distribution) Common non-parametric statistics: Frequency sign test, chi-squared Rank order Mann-Whitney U test, Wilcoxon matched-pairs signed-ranks test 15 16 Univariate descriptive statistics 17

What do we want to describe? The distributional properties of underlying variables, based on: Central tendency(ies): Frequencies, Mode, Median, Mean Shape: Skewness, Kurtosis Spread (dispersion): Min., Max., Range, IQR, Percentiles, Var/SD for sampled data. 19 Measures of central tendency Statistics which represent the centre of a frequency distribution: Mode (most frequent) Median (5 th 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. 2 Measures of central tendency Nominal Ordinal Interval Ratio Mode / Freq. /%s If meaningful Median x If meaningful Mean x x 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 21 22 Nominal Ordinal Interval Ratio Measures of spread / dispersion / deviation Min / Max, Range Percentile Var / SD x x x If meaningful x Descriptives for nominal data Nominal LOM = Labelled categories Descriptive statistics: Most frequent? (Mode e.g., females) Least frequent? (e.g., Males) Frequencies (e.g., 2 females, 1 males) Percentages (e.g. 67% females, 33% males) Cumulative percentages Ratios (e.g., twice as many females as males) 23 24

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 25 Descriptives for interval data Interval LOM = order and distance, but no true ( 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) 26 Descriptives for ratio data Ratio = Numbers convey order and distance, meaningful 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) 27 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 28 Frequencies (f) and percentages (%) # of responses in each category % of responses in each category Frequency table Visualise using a bar or pie chart 29 Median (Mdn) Mid-point of distribution (Quartile 2, 5 th 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 3

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 31 Properties of the normal distribution 32 Four moments of a normal distribution 12 1 8 Mean SD 6 4 -ve Skew 2 Row 1 Row 2 Row 3 Row 4 Kurt Column 1 Column 2 Column 3 +ve Skew 33 Four moments of a normal distribution Four mathematical qualities (parameters) can describe a continuous distribution which as least roughly follows a bell curve shape: 1 st = mean (central tendency) 2 nd = SD (dispersion) 3 rd = skewness (lean / tail) 4 th = kurtosis (peakedness / flattness) 34 Average score Mean (1st moment ) Mean = Σ X / N For normally distributed ratio or interval (if treating it as continuous) data. Influenced by extreme scores (outliers) 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). 35 36

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 37 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) 38 Skewness (3rd moment) (with ceiling and floor effects) Negative skew Ceiling effect Positive skew 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. Floor effect 39 4 Kurtosis (4th moment ) Red = Positive (leptokurtic) Blue = Negative (platykurtic) 41 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) 42

Areas under the normal curve 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 43 44 Non-normal distributions 45 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) 46 Non-normal distributions Histogram of weight 8 Histogram 6 4 Frequency 2 Std. Dev = 17.1 Mean = 69.6 N = 2. 4. 5. 6. 7. 8. 9. 1. 11. 47 WEIGHT 48

6 4 2 Very feminine Fairly feminine Androgynous 6 4 2 Very feminine Fairly masculine Very masculine Fairly feminine Androgynous Fairly masculine 5 4 3 2 1 Very masculine Fairly feminine Androgynous Fairly masculine Very masculine Histogram of daily calorie intake Histogram of fertility 49 5 6 5 Example normal distribution 1 6 Example normal distribution 2 This bimodal graph actually consists of two different, underlying normal distributions. 4 Frequency 3 Count 4 2 2 1 2 4 6 8 1 12 14 Mean =81.21 Std. Dev. =18.228 N =188 51 Very feminine Fairly feminine Androgynous Fairly masculine Femininity-Masculinity Very masculine 52 Distribution for females Count C ou nt Femininity-Masculinity Gender: female Gender: male Femininity-Masculinity Count Distribution for males Femininity-Masculinity 53 Non-normal distribution: Use non-parametric descriptive statistics Min. & Max. Range = Max.-Min. Percentiles Quartiles Q1 Mdn (Q2) Q3 IQR (Q3-Q1) 54

Effects of skew on measures of central tendency 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 55 56 Transformations Converts data using various formulae to achieve normality and allow more powerful tests Loses original metric Complicates interpretation Review questions 1.If a survey question produces a floor effect, where will the mean, median and mode lie in relation to one another? 57 58 Review questions 2.Would the mean # of cars owned in Australia to exceed the median? Review questions 3.Would you expect the mean score on an easy test to exceed the median performance? 59 6

Science is beautiful (Nature Video) Is Pivot a turning point for web exploration? (Gary Flake) (Youtube 5:3 mins) (TED talk - 6 min.) 63 64 Principles of graphing Graphs (Edward Tufte) Clear purpose Maximise clarity Minimise clutter Allow visual comparison 65 Visualise data Reveal data Describe Explore Tabulate Decorate Communicate complex ideas with clarity, precision, and efficiency 66

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 welllabelled (maximise clarity, minimise clarity; show the data; avoid distortion; reveal data at several levels/layers) 67 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 68 Cleveland s hierarchy 69 Univariate graphs } Bar graph Pie chart Histogram Stem & leaf plot Data plot / Error bar Box plot Non-parametric i.e., nominal, ordinal, or nonnormal interval or ratio } Parametric i.e., normally distributed interval or ratio 7 Count 13 12 12 11 11 1 1 9 9 Bar chart (Bar graph) 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) Count 1 2 1 1 1 9 8 7 6 5 4 3 2 1 Pie chart Use a bar chart instead Bio lo gy Hard to read Difficult to show Small values An thro p o lo gy Small differences Rotation of chart and position of slices influences perception So cio logy Information Technolo Bio lo gy So cio lo gy Information Technolo Bio lo gy Note P sy ch olo gy A nt h ro p o lo gy P sy ch o lo gy An thro p o lo gy truncated AREA AREA Y-axis 71 72 So cio lo gy P sy ch o lo gy Information T echnolo

3 2 1 1 2.5 2 2. 5 3 2.5 42.5 52.5 6 2.5 6 5 4 Std. D ev = 3 9.1 6 M e an = 24. N = 5 57 5. 2 1 Std. Dev = 9.1 6 M ea n = 24. N = 557 5. 1 8 6 4 2 9 1 3 17 21 25 29 33 3 7 4 1 45 49 53 57 61 6 5 Std. Dev = 9.16 Mean = 2 4 N = 5575. 569 5491 5328 1938 2224 332 63 34 4998 5564 2928 2151 2183 296 351 688 454 527 68 2241 2336 1488 438 3519 227 2718 267 2438 556 2265 5186 3653 2476 2923 432 2576 4279 1465 661 257 2321 2944 2743 641 278 2334 1493 2312 437 1425 292 438 1963 4167 3552 2814 2646 148 686 39 1747 5481 3998 1793 292 2319 2626 154 641 2822 2845 4141 2688 2262 3645 4438 412 4997 5315 3531 1955 2672 5274 417 4482 32 2614 2829 2179 3442 638 51 62 649 4998 2359 5536 425 2243 2527 3995 56 551 2962 3673 33 2187 5538 399 415 2965 39 1969 2335 211 269 436 3634 328 688 149 1421 4317 2293 3564 2317 335 678 2452 334 2138 5558 5316 1985 218 2515 3593 2743 5186 4228 3165 4284 645 2492 296 248 2291 266 647 2476 4193 438 68 17 2699 334 3556 345 4349 2552 3466 4162 2752 153 8 73 724 19 4351 4159 2596 122 3137 324 34 428 2659 188 2861 2114 727 2313 62 1978 36 517 233 4186 1823 75 3532 244 144 284 2596 1495 578 2821 4162 5591 4495 473 36 23 2898 2451 1983 4 5475 2847 3588 1419 32 3444 452 69 2116 5427 2834 2559 318 724 4268 295 571 521 3137 689 394 727 196 435 3635 342 442 2229 231 396 34 3383 3562 5545 5525 122 44 2358 733 417 2385 5563 3351 5524 Histogram For continuous data (Likert?, Ratio) X-axis needs a happy medium for # of categories Y-axis matters (can exaggerate) Histogram of male & female heights Participant Age Participant Age Participant Age 73 Wild & Seber (2) 74 Stem & leaf plot Use for ordinal, interval and ratio data (if rounded) May look confusing to unfamiliar reader 75 Stem & leaf plot Contains actual data Collapses tails Underused alternative to histogram Frequency Stem & Leaf 7. 1. & 192. 1. 22223333333 541. 1. 444444444444444455555555555555 61. 1. 6666666666666677777777777777777777 849. 1. 88888888888888888888888888899999999999999999999 614. 2. 111111111111111111 62. 2. 222222222222222233333333333333333 447. 2. 4444444444444455555555555 291. 2. 66666666677777777 24. 2. 88888889999999 167. 3. 1111 146. 3. 22223333 153. 3. 44445555 118. 3. 666777 99. 3. 888999 16. 4. 111 54. 4. 222 339. Extremes (>=43) 76 Box plot (Box & whisker) Useful for interval and ratio data Represents min., max, median, quartiles, & outliers Box plot (Box & whisker) Alternative to histogram Useful for screening Useful for comparing variables Can get messy - too much info Confusing to unfamiliar reader 1 8 6 4 2 T ime Managem ent-t 1 77 Missing Male Participant Gender Female Self-Confidence-T 1 78

Data plot & error bar Data plot Error bar Line graph Alternative to histogram Implies continuity e.g., time Can show multiple lines 8. 7.5 7. Mean 6.5 6. 5.5 79 5. OVERALL SCALES-T OVERALL SCALES-T 2 OVERALL SCALES-T1 OVERALL SCALES-T 3 8 Graphical integrity (part of academic integrity) "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." 81 Michael Friendly Gallery of Data Visualisation 82 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 83 Review exercise: Fill in the cells in this table Level Properties Examples Descriptive Statistics Nominal /Categorical Ordinal / Rank Interval Ratio Answers: http://goo.gl/ln9e1 Graphs 84

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. (26). 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. (21). Visualizing quantitative data. Cheshire, CT: Graphics Press. 6. Tukey J. (1977). Exploratory data analysis. Addison-Wesley. 7. Wild, C. J., & Seber, G. A. F. (2). Chance encounters: A first course in data analysis and inference. New York: Wiley. 85