Statistics...

1. Statistics... • A set of mathematical procedures for describing, synthesizing, analyzing, and interpreting quantitative data …the selection of an appropriate statistical technique is determined by the research design, hypothesis, and the data collected

2. Preparing data for analysis... Data must be accurately scored and systematically organized to facilitate data analysis: scoring: assigning a total to each participant’s instrument tabulating: organizing the data in a systematic manner coding: assigning numerals (e.g., ID) to data

3. Types of Statistics • Descriptive Statistics • Describe data • e.g.,frequencies, percentiles, mean, median, mode, ranges, inter-quartile ranges, sds, Zs • Inferential Statistics • Allow for inferences about population to be drawn from sample data • e.g., t, ANOVA (F), correlations (r), regression weights (ß); variance explained (R2)

4. Variable • property of an object or event that can take on different values.  For example, college major is a variable that takes on values like mathematics, computer science, English, psychology, etc. • Discrete Variable - a variable with a limited number of values (e.g., gender (male/female), college class (freshman/sophomore/junior/senior). • Continuous Variable - a variable that can take on many different values, in theory, any value between the lowest and highest points on the measurement scale.

5. Independent Variable - a variable that is manipulated, measured, or selected by the researcher as an antecedent condition to an observed behavior.  In a hypothesized cause-and-effect relationship, the independent variable is the cause and the dependent variable is the outcome or effect. Dependent Variable - a variable that is not under the experimenter's control -- the data.  It is the variable that is observed and measured in response to the independent variable . Qualitative Variable - a variable based on categorical data. Quantitative Variable - a variable based on quantitative data.

6. Types of Variables • Categorical • Nominal; Ordinal • Can compute frequencies & mode for nominal • For ordinal variables, carefully interpret descriptive statistics • Continuous • Interval; Ratio • Can compute descriptive statistics

7. Representation of data • Tabular • Tables • Frequency distribution • Graphical • Bar diagram • Pie diagram • Histogram • Frequency polygon

8. Data summarization • Measures of central tendency • It describes the center of the data • Measures of dispersion • It describes how data is scattered around the center

9. Descriptive Statistics • Frequencies, percentiles • Central Tendency • Mean • Sum of all observations divided by total number of observations • Median • After arraying all observations in ascending/descending order, the obs that divides the sample into two • For even number of observations take avg of 2 central obs • Mode • Most frequently occurring observation • When would one use means vs. medians? (Economist article)

10. Descriptive Statistics • Variability • Range • Difference between the two most extreme observations • Inter-quartile range • Divide observations into quarters & use the middle half • Standard Deviation • Take each observation’s difference from the mean, square it, add all such squared differences, and divide the result by number of observations • Variance • Square of standard deviation

11. Skewness Describes visually the distribution of the data. Examples: Symmetrical distribution: two halves of data are mirror images. Skewed distribution: Data is pulled by extreme scores to the left or right.

12. Symmetrical Distribution

13. Skewed DistributionsHint: The tail points to the skew. Negatively Skewed Median right of mean More high scores Positively Skewed Median left of mean More low scores

14. Descriptive Statistics • Variability (cont’d) • Confidence intervals • The range of values in which the mean occurs 95% of the time • Typically includes scores that are two standard errors above or below statistic • Standard error: Type of standard deviation (for more see p. 287 Sekaran) • Standard scores (Zs) • Deviation from the mean divided by standard deviation • Mean of all Zs =0, sd=1 • Useful for computing interaction scores in regression analyses

15. Measures of relationship… Correlation Regression

16. Correlation Finding the relationship between two quantitative variables without being able to infer causal relationships Correlation is a statistical technique used to determine the degree to which two variables are related

17. Scatter diagram • Rectangular coordinate • Two quantitative variables • One variable is called independent (X) and the second is called dependent (Y) • Points are not joined • No frequency table

18. Example

19. Scatter diagram of weight and systolic blood pressure

20. Scatter diagram of weight and systolic blood pressure

21. Scatter plots The pattern of data is indicative of the type of relationship between your two variables: • positive relationship • negative relationship • no relationship

23. Positive relationship

25. Negative relationship Reliability Age of Car

26. No relation

27. Correlation Coefficient Statistic showing the degree of relation between two variables

28. Simple Correlation coefficient (r) • It is also called Pearson's correlation or product moment correlationcoefficient. • It measures the nature and strength between two variables ofthe quantitative type.

29. The sign of r denotes the nature of association • while the value of r denotes the strength of association.

30. If the sign is +ve this means the relation is direct (an increase in one variable is associated with an increase in theother variable and a decrease in one variable is associated with adecrease in the other variable). • While if the sign is -ve this means an inverse or indirect relationship (which means an increase in one variable is associated with a decrease in the other).

31. The value of r ranges between ( -1) and ( +1) • The value of r denotes the strength of the association as illustratedby the following diagram. strong intermediate weak weak intermediate strong -1 -0.75 -0.25 0 0.25 0.75 1 Direct indirect perfect correlation perfect correlation no relation

32. If r = Zero this means no association or correlation between the two variables. • If 0 < r < 0.25 = weak correlation. • If 0.25 ≤ r < 0.75 = intermediate correlation. • If 0.75 ≤ r < 1 = strong correlation. • If r = l = perfect correlation.

33. How to compute the simple correlation coefficient (r)

34. A sample of 6 children was selected, data about their age in years and weight in kilograms was recorded as shown in the following table . It is required to find the correlation between age and weight. Example:

35. These 2 variables are of the quantitative type, one variable (Age) is called the independent and denoted as (X) variable and the other (weight)is called the dependent and denoted as (Y) variables to find the relation between age and weight compute the simple correlation coefficient using the following formula:

37. r = 0.759 strong direct correlation

38. EXAMPLE: Relationship between Anxiety and Test Scores

39. Calculating Correlation Coefficient r = - 0.94 Indirect strong correlation

40. Spearman Rank Correlation Coefficient (rs) • It is a non-parametric measure of correlation. • This procedure makes use of the two sets of ranks that may be assigned to the sample values of x and Y. • Spearman Rank correlation coefficient could be computed in the following cases: • Both variables are quantitative. • Both variables are qualitative ordinal. • One variable is quantitative and the other is qualitative ordinal.

41. Procedure: • Rank the values of X from 1 to n where n is the numbers of pairs of values of X and Y in the sample. • Rank the values of Y from 1 to n. • Compute the value of di for each pair of observation by subtracting the rank of Yi from the rank of Xi • Square each di and compute ∑di2 which is the sum of the squared values.

42. Apply the following formula • The value of rs denotes the magnitude and nature of association giving the same interpretation as simple r.

43. In a study of the relationship between level education and income the following data was obtained. Find the relationship between them and comment. Example

44. Answer: ∑ di2=64

45. Comment: There is an indirect weak correlation between level of education and income.

46. Regression Analyses • Regression: technique concerned with predicting some variables by knowing others • The process of predicting variable Y using variable X

47. Regression • Uses a variable (x) to predict some outcome variable (y) • Tells you how values in y change as a function of changes in values of x

48. Correlation and Regression • Correlation describes the strength of a linear relationship between two variables • Linear means “straight line” • Regression tells us how to draw the straight line described by the correlation

49. Regression • Calculates the “best-fit” line for a certain set of data The regression line makes the sum of the squares of the residuals smaller than for any other line Regression minimizes residuals

50. By using the least squares method (a procedure that minimizes the vertical deviations of plotted points surrounding a straight line) we areable to construct a best fitting straight line to the scatter diagram points and then formulate a regression equation in the form of: b