Statistics for Everyone Workshop Summer 2011

1. Part 1 Statistics as a Tool in Scientific Research: Review of Summarizing and Graphically Representing Data; Introduction to SPSS Statistics for Everyone Workshop Summer 2011 Workshop presented by Linda Henkel and Laura McSweeney of Fairfield University Funded by the Core Integration Initiative and the Center for Academic Excellence at Fairfield University and NSF CCLI Grant

2. Types of Research Questions Descriptive (What does X look like?) Correlational (Is there an association between X and Y? As X increases, what does Y do?) Experimental (Do changes in X cause changes in Y?) Different statistical procedures allow us to answer the different kinds of research questions Statistics as a Tool in Scientific Research

3.  Start with the science and use statistics as a tool to answer the research question Get your students to formulate a research question first: How often does this happen? Did all plants/people/chemicals act the same? What happens when I add more sunlight, give more praise, pour in more water? Statistics as a Tool in Scientific Research

4. Can collect data in class Can use already collected data (yours or database) Helping students to formulate research question: Ask them to think about what would be interesting to know. What do they want to find out? What do they expect? Statistics as a Tool in Scientific Research

5. Categorical: male/female blood type (A, B, AB, O) Stage 1, Stage 2, Stage 3 melanoma Numerical: weight # of white blood cells mpg Types of Data: Measurement Scales

6. Nominal: numbers are arbitrary; 1= male, 2 = female Ordinal: numbers have order (i.e., more or less) but you do not know how much more or less; 1st place runner was faster but you do not know how much faster than 2nd place runner Interval: numbers have order and equal intervals so you know how much more or less; A temperature of 102 is 2 points higher than one of 100 Ratio: same as interval but because there is an absolute zero you can talk meaningfully about twice as much and half as much; Weighing 200 pounds is twice as heavy as 100 pounds Types of Data: Measurement Scales Categorical Numerical

7. Entering Data into SPSS You will need to specify the following for each variable: • Name of the variable • Type of data: Numerical or String • Type of measure: Nominal, Ordinal, Scale (Interval or Ratio) • Labels or units

8. Descriptive: Organize and summarize data Inferential: Draw inferences about the relations between variables; use samples to generalize to population Types of Statistical Procedures

9. The first step is ALWAYS getting to know your data  Summarize and visualize your data It is a big mistake to just throw numbers into the computer and look at the output of a statistical test without any idea what those numbers are trying to tell you or without checking if the assumptions for the test are met. Descriptive Statistics

10. For each group/treatment: Numerical Summaries: Measures of central tendency Measures of variability Representing numerical summaries in tables Graphical Summaries: Histograms Bar Graph of Means/Mean Plots Boxplots Descriptive Statistics for Numerical Data

11. One numerical variable (e.g., Height): Histogram, Boxplot One numerical variable and one categorical variable (e.g., Height vs. Gender): Side-by-side Histograms, Boxplots, Bar Graph of Means, Mean Plots Choosing the Appropriate Type of Graph

12. Side by Side Histograms

13. Box Plots

14. Bar Graph of Means and Error Bars

15. Mean Plot with Error Bars

16. Normal (approximately symmetric) Skewed Unimodal/Bimodal/Uniform/Other Outliers Shapes of Distributions

17. The Normal Curve “Bell-shaped” Most scores in center, tapering off symmetrically in both tails Amount of peakedness (kurtosis) can vary

18. Skew = Asymmetrical distribution Positive/right skew = greater frequency of low scores than high scores (longer tail on high end/right) Negative/left skew = greater frequency of high scores than low scores (longer tail on low end/left) Variations to Normal Distribution

19. Histogram Showing Positive (Right) Skew

20. Bimodal distribution:two peaks Rectangular/Uniform:all scores occur with equal frequency Potential Outlier: An observation that is well above or below the overall bulk of the data Variations to Normal Distribution

21. Variations to Normal Distribution

22. Measuring Skewness Skewness measures the extent to which a distribution deviates from symmetry around the mean; SPSS computes value A value of 0 represents a symmetric or evenly balanced distribution (i.e., a normal distribution). A positive skewness indicates a greater number of smaller values (peak is to the left). A negative skewness indicates a greater number of larger values (peak is to the right).

23. Kurtosis Kurtosis is a measure of the “peakedness” or “flatness” of a distribution; SPSS computes value A kurtosis value near 0 indicates a distribution shape close to normal. A negative kurtosis indicates a shape flatter than normal. A positive value indicates more peaked than normal. An extreme kurtosis (e.g., |k| > 5.0) indicates a distribution where more of the values are in the tails of the distribution than around the mean.

24. Caution!! It is important to determine normality so you can • Choose appropriate measures of central tendency and variability • Use hypothesis tests appropriately

25. Assessing Normality Method 1: Make a histogram of numerical data and compare with normal curve, Check if the histogram is unimodal and symmetric, bell-shaped Method 2: Kurtosis and skewness values are between +1 is considered excellent, but a value between +2 is acceptable in many analyses in the life sciences.

26. Assessing Normality Method 3: Conduct a hypothesis test for normality • Shapiro-Wilk (n<2000) • Kolmogorov-Smirnov (n>2000) Ho: Data come from a population with a normal distribution Ha: Data do not come from a population with a normal distribution So if p-value < .05, conclude the distribution is not normal

27. Once you know what type of measurement scale the data were measured on, you can choose the most appropriate statistics to summarize them: Measures of central tendency: Most representative score Measures of dispersion: How far spread out scores are Descriptive Statistics

28. Central tendency = Typical or representative value of a group of scores Mean: Average score Median: middlemost score; score at 50th percentile; half the scores are at or above, half are at or below Mode: Most frequently occurring score(s) Measures of Central Tendency

29. Measures of Central Tendency

30. Three Different Distributions That Have the Same Mean

31. Knowing what the center of a set of scores is is useful but…. How far spread out are all the scores? Were all scores the same or did they have some variability? Range, Standard deviation, Interquartile range Measures of Variability

32. Variability = extent to which scores in a distribution differ from each other; are spread out

33. Difference between lowest score in the set and highest score Ages ranged from 27 to 56 years of age There was a 29-year age range The number of calories ranged from 256 to 781 The Range as a Measure of Variability

34. Sample Standard Deviation • Standard deviation = How far on “average” do the scores deviate around the mean? • s = SD = • In a normal distribution, 68% of the scores fall within 1 standard deviation of the mean • (M  SD) • The bigger the SD is, the more spread out the scores are around the mean

35. Variations of the Normal Curve(larger SD = wider spread)

36. Quartile 1: 25th percentile Quartile 2: 50th percentile (median) Quartile 3: 75th percentile Quartile 4: 100th percentile Interquartile range = IQR = Score at 75th percentile – Score at 25th percentile So this is the midmost 50% of the scores Interquartile Range

37. Interquartile Range on Positively (Right) Skewed Distribution IQR is often used for ordinal data or for interval or ratio data that are skewed (does not consider ALL the scores)

38. Relationship between Quartiles and the Boxplot Box is formed by Q1, Median and Q3 Whiskers extend to the smallest and largest observations that are not outliers Extreme outliers lie outside the interval Q1 – 3*IQR and Q3+ 3*IQR (denoted by *) Mild outliers lie outside of the interval Q1 – 1.5*IQR and Q3 + 1.5*IQR (denoted by o)

39. Measures of Variability

40. The number of fruit flies observed each day ranged from 0 to 57 (M = 25.32, SD = 5.08). Plants exposed to moderate amounts of sunlight were taller (M = 6.75 cm, SD = 1.32) than plants exposed to minimal sunlight (M = 3.45 cm, SD = 0.95). The response time to a patient’s call ranged from 0 to 8 minutes (M = 2.1, SD = 0.8) Sentences should always be grammatical and sensible. Do not just list a bunch of numbers. Use the statistical information to supplement what you are saying Presenting Measures of Central Tendency and Variability in Text

41. Presenting Measures of Central Tendency and Variability in Tables (Symmetric Data) Be sure to include the units of measurement! You can include an additional column to put the sample size (N)

42. Presenting Measures of Central Tendency and Variability in Tables (Skewed Data) Be sure to include the units of measurement! You can include an additional column to put the sample size (N)

43. Sometimes instead of standard deviation, people report the standard error of the mean (SE or SEM) in text, tables, and figures Standard deviation (SD) = “Average” deviation of individual scores around mean of scores Used to describe the spread of your (one) sample Standard error (SE = SD/N) = How much on average sample means would vary if you sampled more than once from the same population (we do not expect the particular mean we got to be an exact reflection of the population mean) Used to describe the spread of all possible sample means and used to make inferences about the population mean  Use the one that makes sense for your research question. Are you describing one data set only (SD) or generalizing to the population (SE)? What’s the Difference Between SD and SE?

44. Dangers of low N: Be sure to emphasize to students that with a small sample size, data may not be representative of the population at large and they should take care in drawing conclusions Dangers of Outliers: Be sure your students look for outliers (extreme values) in their data and discuss appropriate strategies for dealing with them (e.g., eliminating data because the researcher assumes it is a mistake instead of part of the natural variability in the population = subjective science) Teaching Tips

45. Working with SPSS Getting descriptive statistics Making appropriate graphs Looking at shapes of distributions Teaching tips: Hands-on practice is important for your students Sometimes working with a partner helps Time to Practice