LIS 570

1. LIS 570 Session 6.1 Univariate Data Analysis

2. Objectives: Have answers to the following questions • Why is the normal distribution important for statistical analysis (the ones presented) to make sense? • What is the logic behind inferential statistics? (On what theories is it based?) • What is a Confidence Interval? • In what ways can we summarize quantitative data? • What are some visualization techniques to help us summarize and make sense of data?

3. Agenda • Exercise: understand “the problem” • Vocabulary • Functions of statistics • When to use what type • Descriptive statistics • Inferential statistics

4. Why and What • Why know statistics? • Informed consumer… • Informed user… • Informed professional… • … • What is a statistic? a descriptive summary (index) of a sample

5. Sample A set of observations, instances, individuals drawn from a population, usually intended to represent the population in a study Population (Universe) The totality of things we are interested in (e.g., the population of all students at the UW) Sample and Population Population New vocabulary Sample Average = 4.5 Average = 4.55 statistic parameter A statistic is a characteristic of a sample, while the same characteristic, if descriptive of a population, is called a population parameter.

6. 2 major functions of statistics • Help us describe characteristics of sample • Descriptive statistics • Procedures to summarize, organize, and simplify data • Help us describe characteristics of population • Inferential statistics • Techniques for studying samples, and then make generalizations about the population from which the samples were selected.* * Source: Gravetter, F. J. and Wallnau, L. B. (2002). Essentials of Statistics for the Behavioral Sciences. 4th edition. Pacific Grove, CA: Wadsworth, p. 5

7. Vocabulary • Variable—characteristic which has more than one value • e.g., Sex—male, female; hours of work/week—anything from 0 – 168 • Independent variable (X)—manipulated by the researcher or believed to be the cause of… • Dependent variable (Y)—variable observed to assess the effect of the manipulation, or changes depending on the independent variable • Data—observations (measurements) taken on the units of analysis

8. Choosing the Statistical Technique* Specific research question or hypothesis Determine # of variables in question Univariate analysis Bivariate analysis Multivariate analysis Determine level of measurement of variables Choose univariate method of analysis * Source: De Vaus, D.A. (1991) Surveys in Social Research. Third edition. North Sydney, Australia: Allen & Unwin Pty Ltd., p133 Choose relevantdescriptive statistics Choose relevantinferential statistics

9. What To Do with a Bunch of Numbers • Organize the observations • Interested primarily in normality and deviations from normality • Examine • Central tendency • Dispersion • Shape of distribution • Visualization aids • Frequency distribution (percentile) tables and charts • Histograms • Bar & pie charts (nominal data) • Frequency polygon • Cumulative percentage curve • Stem and leaf diagrams • Box plots

10. Frequency Distributions • Ungrouped frequency distribution • A list of each of the values of the variable • The number of times and/or the percent of times each value occurs • Grouped frequency distribution • A table or graph • Shows frequencies or percent for ranges of values

11. Frequency distributions Include in frequency distribution tables: • Table number and title • Labels for the categories of the variables • Column headings • Total number of cases (N) • The number of missing cases • Source of the data • Footnotes to explain anomalies and notes * Source: De Vaus, D.A. (1991) Surveys in Social Research. Third edition. North Sydney, Australia: Allen & Unwin Pty Ltd., p133

12. Grouped frequency distribution Table 1—Example of grouped frequency distribution Valid cases: 20 Missing cases: 0 Note 1: “Real limits” of a score extend from one-half of the smallest unit of measurement below the value of the score to one half unit above. Note 2: Percent (%) = (ƒ /N) * 100, Cumulative % = (Cf/N) * 100

13. The height of the bar corresponds to the frequency (ƒ) • The width of the bar extends to the real limits of the score • Used only on interval and ratio scales • No space between bars (that’s a bar chart)

14. What do graphs (histograms) show? • Normality (normal distributions) [Why are normal distributions important?] • Deviations from normality • Positive skewness • Negative skewness • Bimodality • And more…

15. Shapes of distribution Normal distribution:symmetrical Bell-shapedcurve symmetrical asymmetrical Negatively skewed:tail on the left, cluster towards high-end of the variable Positively skewed:tail on the right, cluster towards low end of the variable Bimodality: A double peak

16. Central Tendency • Central tendency is a single summary figure that ideally, is the most representative value of all values in the distribution. • Used to describe “typical” or representative value Mean (arithmetic mean), m • Sum all the observations; divide by N: use for interval variables when appropriate • Median: Value that divides the distribution so that an equal number of values are above the median and an equal number below • Mode: Value with the greatest frequency (uni-modal, bi-modal, etc.)

17. Variability, dispersion, spread • Why do we care about anything besides central tendency? • Variability refers to spread or dispersion • The extent to which a set of scores scatter about or cluster together • Measures of variability • Range • Interquartile range • Sum-of-squares • Variance • Standard deviation • Kurtosis Equal means, unequal variability

18. Kurtosis Two distributions: the same mean & variance Karl Pearson suggested names • Longer tailed: leptokurtic • Shorter tailed: platykurtic http://members.aol.com/jeff570/k.html

19. Mode (Mo): most common value • Best for nominal level data • Cautions: • most common may not measure typicality • not sensitive to outliers (good and bad) • may be more than one mode • unstable from sample to sample • Dispersion • variation ratio (v) • % of people not in the modal category

20. Median (Mdn): Even split of sample • For interval or ratio data, good for skewed distributions (mean would not be a good measure of central tendency) • Minimal calculation (need to know frequencies) • Reasonably insensitive to outliers (as long as there are only a few) • Reasonably stable from sample to sample • Example of ordinal variables • people are ranked from low to high (e.g., height) • median is the middle case • the median category is the one to which the middle person belongs

21. Median– simple examples • 1 2 3 4 5 6 7 • Mdn = 4 • 1 2 3 5 6 7 9 13 • Mdn = 5.5by interpolation between 5 & 6 (5+6)/2 = 11/2 = 5.5

22. Dispersion • The nth percentile of a set of numbers is a value such that n percent of the numbers fall below it and the rest fall above. • The median is the 50th percentile • The lower quartile is the 25th percentile • The upper quartile is the 75th percentile • Summary of sample using 5 numbers: median, quartiles, extremes

23. Dispersion Interquartile range Bottom 25% Top 25% Lower quartile Upper quartile Median

24. Boxplot Interquartile range (IQR) Variable 1 Variable 2 Variable 3 4 6 8 10 12 14 16

25. Mean • Uses the actual numerical values of the observations • Most stable from sample to sample • Most common measure of center • Makes sense only for interval or ratio data • Frequently computed for ordinal variables as well • Not a good representation of central tendency for skewed samples

26. Mean--Dispersion • The standard deviation and variance measure spread about the mean as center. • Deviation: distance and direction from the mean • Doesn’t work as a measure of variability because adds up to zero (see next slide). • Variance • mean of the squared deviation scores (of the deviations of observations from the mean). • Standard deviation • Conceptually: the typical distance of scores from the mean • Technically: the square root of the variance

27. Example Data (6,7,5,3,4) x = 6+7+5+3+4 = 25 = 5 5 5 • Variance (S2) • Calculate the mean for the variable • Take each observation and subtract the mean from it • Square the result from the above • Add (sum) all the individual results • Divide by n

28. Variance (s2) Variance = sum of the sq deviations = 10 = 2 number of observations 5

29. Standard deviation (s) • Square root of the variance 2 = 1.4 • An average deviation of the observations from their mean • Influenced by outliers • Best used with symmetrical distributions

30. Summary • Descriptive statistics – univariate analysis(central tendency, frequency distribution, dispersion) • Determine if variable is nominal, ordinal or interval • Nominal: frequency tables, mode • Ordinal • Frequency tables (grouped frequency tables) • histogram • Median or five number summary • Mode

31. Summary Interval Determine whether the distribution is skewed or symmetrical Compare median and mean Use the mean and the standard deviation if the distribution is not markedly skewed; otherwise use five number summary (median, extremes, quartiles) Use the mode in addition if it adds anything

32. 20-30 second synopsis; intent: to elicit interest Who you are and what you are doing With whom Where/How Why: What you hope to find, why the results may be important 100-300 words; elicit interest and summarize What type of study How approached When, where Why: what you hope to find, why the results may be important Abstract and Elevator Speech

33. Selecting analysis and statistical techniques* Specific research question or hypothesis Determine # of variables in question Univariate analysis Bivariate analysis Multivariate analysis Determine level of measurement of variables Choose univariate method of analysis * Source: De Vaus, D.A. (1991) Surveys in Social Research. Third edition. North Sydney, Australia: Allen & Unwin Pty Ltd., p133 Choose relevantdescriptive statistics Choose relevantinferential statistics

34. Why is normality important? 68% 95% 100% • Use proportions of the normal distribution to determine probabilities associated with any specific sample. • Sampling Error • Standard Error (SE)—a way for defining and measuring sampling error (specifically: how much error, on average, should exist between a sample mean and the unknown population mean, simply due to chance)

35. Standard Error of the Mean Standard error of the mean (Sm) Sm = N • Standard error is inversely related to square root of sample size • To reduce standard error, increase sample size • Standard error is directly related to standard deviation • When N = 1, standard error is equal to standard deviation S Standard deviation S Total number in the sample

36. Inferential statistics - univariate analysis Interval estimates and interval variables • Estimation of sample mean accuracy—based on random sampling and probability theory Standardize the sample mean to estimate population mean: t = sample mean – population mean estimated SE Population mean = sample mean + t * (estimated SE)

37. Exercise—sampling distribution • Coin tossing • Probability of head or tails—50% • Each of you is a “sample” for this activity. • Flip the coin 9 times, count the # of times you get a “head”. Live demo: http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

38. Confidence Interval Utilizes probability theory, assumes normal distribution • 95% of the samples will fallwithin 1 to 2 standarddeviations from the population mean • By the same token, for 95% of samples, the population mean will be within + or - 2 standard error units from the sample mean • E.g., for C.I. 80%, first find the lower and upper t-values that bind 80% area of the distribution. • Can state: with 80% confidence interval, the population mean is: sample mean + t (SE)

39. Standard Error(for nominal & ordinal data) Variable must have only two categories (may have to combine categories to achieve this) SB = PQ N P = the % in one category of the variable Q = the % in the other category of the variable Total number in the sample Standard error for binominal distribution