2. Descriptive Research: Quantitative Method Descriptive Analysis –Limits generalization to the particular group of individuals observed. –No conclusions are extended beyond this group –Any similarities to those outside the group can not be assumed –The data describe one group and that group only –Provides information about the nature of a particular group of individuals.

3. Descriptive Research: Quantitative Method Raw Data –Frequencies Measures of Central Tendency –Mean Used with Interval and Ratio Scaled data Is the most stable measure of central tendency Extreme scores may have undo influence over the results used when the distribution of scores has approximately the same number of extremely high and low scores –Median Preferred when an Ordinal Scale is used Used when data is anticipated to be missing and a reporting of mean would be misleading (slow learners example) –Mode Quickest method Best used only when Nominal Data are being described Mode can fluctuate wildly with a small change in a few scores

4. Descriptive Research: Statistical Method Measures of Variability –Range The difference between the values of the largest and smallest scores in a distribution Changes of only a few scores can effect it greatly: least stable measure Can not be used with Nominal Data Use should be restricted to Ordinal or Interval Data –Standard Deviation Most widely used measure of Variability –The larger the standard deviation the greater the degree of variability –Can never be less than 0 The absolute value of the standard deviation has little meaning. SD should be considered a relative measure (i.e., SD’s of 5 and 60 mean exactly the same thing when foot and inch scales are used.

5. Descriptive Research: Statistical Method Standard Scores –Z-Score Scale Used to convert measurements from any arbitrary unit to a common standard Used to allow direct comparison of individual scores obtained from scales of measurement with quantitatively or even qualitatively different units of measurement. The unit of measurement on the z-score scale is the standard deviation of the distribution of the original measurements The mean of the z-score scale is 0. –Other Standard Scores in Common Use T score: M=50, SD=10 GRE, SAT: M=500, SD=100 IQ: M=100, SD=15 ACT: M= 20, SD=5

6. Statistical Method: Foundations –Type I and Type II Error He is cheatingHe is not cheating You decide he is cheating You decide he is not cheating You are correct You are wrong (Type I error – alpha) You are correct You are wrong (Type II error – beta)

7. Statistical Method: Foundations Probability –Definition of Probability The probability of an event is the number of favorable events divided by the number of possible events –Additive Law of Probability When the events are related by the word or Probability of pulling an Ace or King P(ace)= 4/52 (.08) P(king)= 4/52 (.08): P(ace or king) = 8/52 (.16) –Multiplicative Law of Probability When the events are related by the word and Probability of pulling an Ace and a King P(ace or king)=8/52 (.16) P(what was not drawn 1st)= 4/51 (.08): P(ace and king) =.16 x.08 =.01

8. Statistical Method: Foundations Subject Selection –Random All elements in a population have an equal chance of being chosen to participate in the sample –Independent The probability of an event does not depend on previous events (the gambler’s fallacy)

9. Descriptive Research: Statistical Method Chi Square (χ 2 ) –Typically used with Nominal data to compare proportions –Used to ascertain a probability that proportions of representation within categories are similar or different across groups –Compares what was Observed to what was Expected χ 2 = Σ ((f o – f e ) 2 / f e ) –Assigns a probability that what was observed was not a function of random error

10. Descriptive Research: Statistical Method Chi Square (χ 2 ) SPED 6370SPED Grad Male 3 33 Female 13 180

11. 6370 SPED M333 F13180 36 193 229 16 213 Compute Expected (16 x 33) / 229 = 2.31 (16 x 193) / 229 = 13.49 (213 x 33) / 229 = 30.69 (213 x 193) / 229 = 179.52 6370 SPED M3 (2.31) 42 (30.69) F13 (13.49) 180 (179.52) Compute χ 2 (3 – 2.31) 2 / 2.31 =.21 (13– 13.49) 2 / 13.49 =.02 (42 – 30.69) 2 / 30.69 =.17 (180 – 179.52) 2 / 179.52 =.00 χ 2 =.21 +.02 +.17 +.00 =.40 (ns) Table Lookup (1 df) *.05 must be > 3.84 **.01 must be > 6.64 χ 2 = Σ ((f o – f e ) 2 / f e )

13. Descriptive Research: Statistical Method Correlation –A measure of the relationship between two or more paired variables or two or more sets of data –The degree of relationship may be measured and represented by the coefficient of correlation: -1 to +1 –Represented by either the letter r or the Greek letter rho р) – Positive correlation Negative correlation Intelligence Academic Achievement Academic Achievement TV time Productivity Value of Farm Total corn production $ per bushel Height Shoe size Practice time Errors Income Value of Home Age of automobile trade-in value

14. Variables in Correlational Research n Predictor Variables: – ones in which participants' scores enable prediction of their scores on some criterion variable. May be thought of as independent variables n Criterion Variables: – the object of the research. the variable about which researchers seek to discover more information. May be thought of as dependent variables.

15. Critical Issues in Correlational Research n Development of Hypothesis – should be grounded in a theoretical framework and previous research – caution needed for "shot-gun" research n Selection of homogeneous groups – possess variables under study – requires precise definition n Collection and analysis of data – reliability and validity of measures critical – numerous statistical procedures available - caution needed

16. Calculation of Correlation (Z x1 * Z y1 ) + (Z x2 * Z y2 ) + (Z x3 * Z y3 ) +..... N r = a correlation is a covariance ratio ∑ (Z x )(Z y ) N r = 1 +1.5 +1.2 + 1.8 2 -.75 -.9 +.68 3 +.2 +.7 +.14 4 - 1.0 -.75 +.75 5 +1.4 +1.2 + 1.68 6 -.10 -.30 +.03 5.08 r = 5.08 / 6 =.85 x y xy

17. Correlation positive: +.99 negative: -.95 positive: +.45 none: 0

18. Interpreting Correlation n Look at correlation (r) separate from probability (p) correlation tells you amount of relationship positive and negative define direction of relationship r defines amount of relationship probability (p) tells you the odds that you observed the relationship by chance Interpretation n Coefficient Interpretation n.00 to.20 Negligible.20 to.40 Low.40 to.60 Moderate.60 to.80 Substantial.80 to 1.00 High to Very High

19. Common Correlations Pearson Product Moment Correlation –both variables are continuous Spearman Rank-order Correlation –both variables are measured as rank data Biserial Correlation –one variable is continuous and one is an ‘artificial’ dichotomy with an underlying normal distribution Point-Biserial Correlation –one variable is continuous and one is a ‘true’ dichotomy Phi Coefficient –both variables are ‘true’ dichotomies

20. More Correlations Tetrachoric Correlation –both variables are ‘artificial’ dichotomies with underlying normal distributions Polychoric Correlation –both variables are ordinally measured with both having underlying normal distributions Polyserial Correlation (r ps, D ps ) –one variable is continuous and one is ordinal with an underlying normal distribution Kendall Tau-b –measures agreement between two rankings Kendall’s Coefficient of Concordance –measures of the extent to which members of a set of m distinct rank orderings of N things tend to be similar