Research & Statistics Looking for Conclusions

Statistics Mathematics is used to organize, summarize, and interpret mathematical data 2 types of statistics Descriptive Inferential

Descriptive Statistics Organize and summarize Provide an overview Measures of central tendency Measures of variability Coefficient of correlation

3 measures of central tendency Mean (average) Median Mode (most frequent) (the one in the middle)

Range the difference between the highest and lowest scores in a distribution

AP quiz on 8/23 (14 possible points) Mean? Median? Mode?

Mean? 5 Median? 4 Mode? 4 AP quiz on 8/23 (14 possible points) Should this influence how the quiz is graded?

A Skewed Distribution: How much could you really expect to make? Mode Median Mean One FamilyIncome per family in thousands of dollars

Variability Plays a crucial role in deciding if the results of a study support the hypothesis

Variability How much do the scores vary? – From each other – From the mean Standard Deviation: a computed measure of how much the scores vary around the mean The higher the variability, the higher the deviation

The equation for standard deviation is: SD = (score - mean) 2 number of scores OR…..

You can memorize these percentages 1 st deviation = 68% 2 nd deviation = 95% 3 rd deviation = 99% Outliers remaining 1%

You can memorize these percentages Normal Distribution a bell-shaped curve, describing the spread of a characteristic throughout a population.

Curving the grade O 2,2,2 4,4,4,4,4,4,4,4 10, scores F D C B A 6,6,6 8,8,8

Correlation, Prediction, Causation Correlation does not equal causation!

When two variables are related to each other (i.e. cigarette smoking and cancer) A relationship between variables, in which changes in one variable are reflected in changes in the other variable A statistical index used to represent the strength of a relationship between two factors, how much and in what way those factors vary, and how well one factor can predict the other. Correlation

Using correlations does NOT provide you with cause and effect information; It will not tell you if one factor causes or is caused by the other. This was an important component in the court cases against the tobacco companies in the late 1990's. The studies conducted on the effects of smoking indicated a positive correlation between smoking and cancer. This means that the studies found that as the rate of smoking increased, so did the occurrence of cancer; BUT, this does not demonstrate that smoking causes cancer, only that there is a relationship between the two factors.

Correlation coefficient a numerical index of the degree of relationship between two variables Is the relationship positive? Or negative? How strongly are the two variables related?

Positive and Negative Correlations Scores with a positive correlation coefficient go up and down together (as smoking goes up, cancer rates go up) A negative correlation coefficient indicates that as one score increases, the other score decreases (as self-esteem increases, the rate of depression decreases).

a graphed cluster of dots, each of which represents the values of two variables the slope of the points suggests the direction of the relationship the amount of scatter suggests the strength of the correlation little scatter indicates high correlation Scatterplot

Scatterplots, showing patterns of correlations Perfect positive correlation (+1.00) No relationship (0.00)Perfect negative correlation (-1.00)

Positive, Negative, Random Correlations: ABC

As smoking goes up, cancer rates go up A B As smoking goes up, cancer rates go down

Positive and Negative Correlation Positive 2 variables are highly related They co-vary in the same direction High scores on variable x are associated with high scores on variable y Negative 2 variables are not really related They co-vary in the opposite direction High scores on variable X are associated with low scores on variable y

Students who do well in high school, tend to do well in college A B Students with a high absent rate tend to have low grades

a statistical measure of the extent to which two factors vary together and how well either factor predicts the other Correlation Coefficient

Strength of Correlation A positive or negative (+/-) indicates the direction of a correlation The size of the coefficient indicates strength to 0 (negative) 0 to (positive) The strength of the correlation depends on the size of the coefficient

Strength of Correlation between variables A coefficient at 0 means no relationship between variables A coefficient near 0 means a weak relationship A coefficient near or means a strong relationship A coefficient at or means a perfect relationship

Correlation coefficient Indicates direction of relationship (positive or negative ) Indicates strength of relationship (0.00 to 1.00) r = +.37

Correlation coefficient Indicates direction of relationship (positive or negative) Indicates strength of relationship (0.00 to 1.00) r = +.37 If people with high scores on one variable also have high scores on another variable, then the Correlation Coefficient is positive. If people with high scores on one variable have low scores on another variable, then the CC is negative.

Correlation coefficient Indicates direction of relationship (positive or negative) Indicates strength of relationship (0.00 to 1.00) +.37 If 28 out of 35 students score high on a calculus test AND score high on a science test, the CC would be +.80 If 30 out of 35 students score high on calculus but score low on an English test, the CC would be -.85 Which CC is stronger?

There is a strong correlation between economic vitality and the pace of life coefficient of +.74 There is a negligible correlation between population size and pace of life coefficient of -.07 There is an average correlation between height and income coefficient of.29

Height and Temperament of 20 Men Subject Height in Inches Temperament Subject Height in Inches Temperament plot data on a scatterplot….

Scatterplot of Height and Temperament Temperament scores Height in inches

Correlation and Prediction As a correlation increases in strength, prediction increases The closer you get to or -1.00, the better able you are to make a prediction If you found a strong correlation between the number of hours one studies and how high their grades are, you could predict a student’s grades based on the number of hours of study

But, Correlation does not prove Causation Most prison inmates ate bread growing up This does not mean that bread caused them to commit crimes Correlation only means there is a relationship

There is a positive correlation between self- esteem and academic performance. What can we conclude based on the correlation? Do low grades cause low self esteem? Is there an association between self esteem and academic achievement? Does high ability causes high self esteem and high achievement? Do kids with low self esteem tend to have lower grades, and those with high self esteem tend to have higher grades?

Is it Positive or Negative? The correlation between age and visual acuity The correlation between years in school and income The correlation between shyness and number of friends

Is it Positive or Negative? The correlation between age and visual acuity  Negative (as age increases, acuity decreases) The correlation between years in school and income  Positive (as education increases, income increases) The correlation between shyness and number of friends  Negative (as shyness increases, number of friends decreases)

Inferential Statistics used to interpret data and draw conclusions Are the scores from two or more groups essentially the same or different? Helps to eliminate chance Makes results meaningful (or not) Inferential as in to infer, for example, the study habits (hours of study) of excellent students compared to those of failing students

SIGNIFICANT DIFFERENCE A SIGNIFICANT DIFFERENCE is one that meets a certain criterion. Statistical Significance A researcher must ask whether the statistics collected occurred simply because of chance or whether there is a “real” or “significant” difference.

How do you calculate the statistical significance? The amount of variability in the data is an important consideration Standard deviation measures variability

SIGNIFICANT DIFFERENCE A SIGNIFICANT DIFFERENCE is when p <.05 Psychologists accept a difference between the groups as “real” or “significant” when the probability might be due to chance of less than 5 in 100

Statistical Significance Statistical Significance exists if the observed findings are unlikely to be due to chance Defined as being less than 5 out of 100 or the.05 level of significance When the statistical calculation prove not to be by chance, they are considered statistically significant

Random Sequences Your chances of being dealt either of these hands is precisely the same: 1 in 2,598,960.