research & statistics looking for conclusions. statistics mathematics is used to organize,...

Research & StatisticsLooking 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)

121010108886664

44444442220

Mean?Median? Mode?

121010108886664

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Mean? 5Median? 4Mode? 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?

15 20 25 30 35 40 45 50 90 475 710

70

Mode Median Mean

One Family Income 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

1st deviation = 68%2nd deviation = 95%3rd deviation= 99%Outliers remaining 1%

You can memorize these percentages

Normal Distributiona bell-shaped curve, describing the spread of a characteristic throughout a population.

Curving the grade121010108886664

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O 2,2,2 4,4,4,4,4,4,4,4 10, 12

22 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 positivecorrelation (+1.00)

No relationship (0.00) Perfect negativecorrelation (-1.00)

Positive, Negative, Random Correlations:

A B C

As smoking goes up, cancer rates go up

A

B

As smoking goes up, cancer rates go down

Positive and Negative CorrelationPositive

• 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 - 1.00 to 0 (negative) 0 to +1.00 (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 -1.00 or +1.00 means a strong relationship

• A coefficient at -1.00 or +1.00 means a perfect relationship

Correlation coefficient

Indicates directionof relationship

(positive or negative)

Indicates strengthof relationship(0.00 to 1.00)

r = +.37

Correlation coefficient

Indicates directionof relationship

(positive or negative)

Indicates strengthof 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 directionof relationship

(positive or negative)

Indicates strengthof 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

123456789

10

11121314151617181920

80636179746962757760

64767166737063716870

75666090604242608139

48697257637530578439

SubjectHeight in

Inches Temperament SubjectHeight in

Inches Temperament

plot data on a scatterplot….

Scatterplot of Height and Temperament

55 60 65 70 75 80 85

959085807570656055504540353025

Temperamentscores

Height in inches

Correlation and Prediction

• As a correlation increases in strength, prediction increases

• The closer you get to +1.00 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

A SIGNIFICANT DIFFERENCE is one

that meets a certain criterion.

Statistical SignificanceA 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

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.