MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 1 STATISTICS MEASURES OF CENTRAL TENDENCY. In an experiment, these are applied to the dependent variable (DV)

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1 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 1 STATISTICS Descriptive statistics Inferential statistics MEASURES OF CENTRAL TENDENCY In an experiment, these are applied to the dependent variable (DV) E.g., MEASURES (DESCRIBERS) OF CENTRAL TENDENCY What is central tendency? Mean Formula: Easy Example: 4, 8, 8, 8

2 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 2 Second Example: 12, 15, 11, 20, 13, 10, 17 Sum of the deviations around the mean = 0 Simple Example Score X Deviation (X - M) Ex = E( X - M) = Balance Point

3 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 3 Median Conceptual Definition: Example 10, 17, 1, 5, 9, 10, 4 Put numbers on a number line: Mode Example 10, 17, 1, 5, 9, 10, 4 Mean versus Median The Median is Less influenced by extreme scores than the mean Case 1: 10, 17, 1, 5, 9, 10, 4 Case 2: 10, 69, 1, 5, 9, 10, 4 (69 is an extreme score)

4 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 4 Mean, Median, & Mode compared for Household Income

5 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 5 MEASURES OF VARIABILITY (DISPERSION) The Concept: What is variability (dispersion)? How spread out are scores around their center? Basketball Example: Two basketball players with same average shooting percentage Average shooting % is the same... but there is something different Effects of stress on performance Scientific Hypothesis: Stress will not affect the average performance of a group of people because it will increase the performance of some of them while decreasing the performance of others. BUT, stress will affect the variability of performance within the group. That is, people's performance will be less alike when they are stressed. DV is 1 to 7 rating of how good a person's performance is on a certain task Stress group: 6, 2, 1, 7 No-stress group: 4, 3, 5, 4 Mean of stress group vs. Mean of No-stress group

6 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 6 Range R = Hi - Lo Average deviation (around the mean) Formula: Stress Group Score X Deviation (X - M) Abs Value of Dev Ex = E( X - M) = Stress Group Score X Abs Value of Dev Ex =

7 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 7 Variance: Average squared deviation (around the mean) Definitional formula: S 2 = Sum of Squares SS = Stress Group Calculations: Stress Group Score X Deviation (X - M) Squared Deviation (X - M) Ex = E( X - M) = E( X - M) 2 = Non Stress Group Calculations:

8 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 8 Why this formula? deviations measure tendency away from the mean need to square the deviations because sum to zero magnifies larger deviations need to sum all the squared deviations need to divide by n because... Summary: Average Squared Deviation

9 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 9 Computational formula for variance S 2 = Stress Group Calculations Non Stress Group Calculations Computational formula for SS Computational SS = Calculating Variance using SS S 2 = Stress Group calculations

10 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 10 Formulas S = Standard Deviation Definitional Formula S = Computational Formula S = Stress and Non Stress Example Calculations Rationale: Variance squares the measurement operations of the DV Example: What is IQ 2? Standard Deviation (S) returns measurements to their original scale

11 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 11 Interpreting standard deviations Standard scores (z-scores) We ve defined z for the Normal Curve previously Now we define the same concept for a data set formula z = Tells how many standard deviations a score is away from... rationale: puts everything on the same scale of sd's gives the number of sd's a raw score is above or below the mean allows comparison of scores across populations: Example calculations: Psychology 1010 test Mean = S = A student scores 72, what is the z score of a raw score of 72?

12 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 12 Example: An English test and a math test A student gets a 36 in Math and a 72 in English On which test did he or she do better? Mean of Math test = Mean of English test = Student s z score on Math test S of Math test = z(36) = Student s z score on English test S of English test = z(72) = The z scores tell us a great deal about how well the student did on the two tests

13 MALLOY PSYCH 3000 MEAN & VARIANCE PAGE 13 Bell curve Bell curves such as the Normal are closely related to S Rules of Thumb for any Bell Curve Example: Psych 1010 test Mean = 65 and S = 5 Draw a rough bell curve showing from -3 to +3 S s around the mean If a student has a raw score of 72, find its z score and draw the score of 72 on your rough sketch of the Bell Curve