Chapter 10 http://members.aol.com/johnp71/javastat.html
Goal • Not only to be able to analyze your own data but to understand the literature that you read.
Data Analysis • Statistics • Parameter
Reporting your Results • With words…. • With numbers…. • With Charts/Graphs…
Data • Categorical • Quantitative
Quantitative • In this chapter: • Correlation • Frequency • distributions • Measures of Central Tendency • Mean • Variability • Standard deviation
Distributions • Skewed Distributions • Positive – scores trailing to the right with a majority at the lower end • Negative – scores trailing to the left
Distributions • Normal • Large majority of scores in the middle • Symmetrical • Bell-shaped • Mean, median, and mode are identical
Types of Curves... The Normal Curve:
Measures of Central Tendency • Mode • Median • Point at which 50% of scores fall above and below • Not necessarily one of the actual scores in the distribution • Most appropriate if you have skewed data
Measures of Central Tendency • Mean • Uses all scores in a distribution • Influenced by extreme scores • Mean = sum of scores divided by the number of scores
Variability • Range • Low to High • Quick and dirty estimate of variability • Standard Deviation
Standard Deviation • 1. Calculate the mean • 2. Subtract the mean from each score • 3. Square each of the scores • 4. Add up all the squares • 5. Divide by the total number of scores = variance • 6. Take the square root of the variance.
Standard Deviation • The more spread out the scores the larger the standard deviation. • If the distribution is normal then the mean + two standard deviations will encompass about 95% of the scores. (+ three SD = 99% of scores)
% of Scores in 1 SD 1 SD = 68% of sample
2 Standard deviations? 2 SD = 95% of sample
Group A 30 subjects Mean = 25 SD = 5 Median = 23 Mode = 24 Group B 30 subjects Mean = 25 SD = 10 Median = 18 Mode= 13 What can you tell me about these groups?
Use your text (p. 207-208) Check your scores with this link. Scores: 12, 10, 6, 15, 17, 20, 16, 11, 10, 16, 22, 17, 15, 8 Mean = ?? SD = ?? Calculate the Standard Deviation and Average
Excel • Now go to the following web page and click on “class data”: assignments • Calculate mean, median, mode, SD for the ACT and Writing column data.
Standard Scores • A method in which to compare scores • Z scores – expressed as deviation scores • Example: • Test 1= 80 • Test 2 = 75
Example • Test 1: mean = 85, SD = 5 • Test 2: mean = 65, SD = 10
Probability • We can think of the percentages associated with a normal curve as probabilities. • Stated in a decimal form. • If something occurs 80% of the time it has a probability of .80.
Example • We said that 34% of the scores (in a normal distribution) lie between the mean and 1SD. • Since 50% of the scores fall above the mean then about 16% of the scores lie above 1SD
Example • The probability of randomly selecting an individual who has a score at least 1SD above the mean? • P=.16 • Chances are 16 out of 100.
Example • Probability of selecting a person that is between the mean and –2SD?
Z-Scores • For any z score we know the probability • Appendix B
Z-Scores • Can also be calculated for non-normal distributions. • However, cannot get probabilities values if non-normal. • If have chosen a sample randomly many distributions do approximate a normal curve.
Determining Relationships Between Scores Correlation
Relationships • We can’t assign blame or cause & effect, rather how one variable influences another.
Correlation • Helpful to use scatterplots
Plotting the relationship between two variables Age = 11 Broad Jump = 5.0 feet Y axis Age 11 5.0 X axis 5 Feet
Plot some more (Age & Broad Jump) y Do you see a relationship?? Age x Feet
Outliers • Differ by large amounts from the other scores
Correlation…. • Is a mathematical technique for quantifying the amount of relationship between two variables • Karl Pearson developed a formula known as “Pearson product-moment correlation”
Correlation • Show direction (of relationship) • Show strength (of relationship) • Range of values is 0 - 1.0 (strength) • 0 = no relationship • 1 = perfect relationship • Values may be + or - (direction)
Correlation r= 1.0 r = 0 r = -1.0
Correlation Strength • Very Strong .90 - 1.0 • Strong .80 - .89 • Moderate .50 - .79 • Weak < .50
Types of relationships Curvalinear Sigmoidal Linear
Test Your Skill Guess the Correlation
Quick Assignment • For the same excel spreadsheet that we opened earlier calculate a correlation coefficient for the ACT vs. Tricep. • Make a scatterplot of tricep vs. ACT. • Scatterplot and correlation for ACT vs. Writing
Coefficient of Determination • Determines the amount of variability in a measure that is influenced by another measure • I.e. how much does the broad jump vary due to varied ages? • Calculated as r2 (Corr. Squared)
Example: • Say that strength and 40yard sprint time have an r = .60 • How much does a variation in strength contribute to the variation in sprint speed?
Summarizing Data • Frequency Table • Bar Graphs/Pie Charts • Crossbreak Table • A graphic way to report a relationship between two or more categorical variables.
Assignment • Under assignments on my web page there is an excel spreadsheet published entitled “assignment 1”. • Download the spreadsheet by clicking here assignments
Assignment • 1. Calculate the mean, mode, and median for body density, ACT Score, and Reading Score on sheet 1 • 2. Calculate the mean and SD for TC, Trig, HDL, and LDL on sheet 2
Assignment • 3. Calculate a correlation coefficient for body density and age, ACT and Reading Scores, TC and LDL, and Trig and HDL • 4. Make a scatterplot for HDL and Trig as well as LDL and Total
Assignment • 5. Make a bar graph for the mean Total, Trig, LDL, HDL values.