Measures of Variability Variability: describes the spread or dispersion of scores for a set of data

Measures of Variation Range: difference between the highest and lowest scores in a distribution  Crude estimate of variation because a couple extreme scores create a deceptively large range

Measures of Variation Standard Deviation: computed measure of how much scores vary around the mean score  Tells us how different the scores are from each other Example: If your high school serves a community where most families have similar incomes, family income data will have a relatively small standard deviation compared with a school in a more diverse community population Table 2.4 – page 39

Measures of Variation Standard Deviation:  Lower standard deviations = more similar the scores  Higher standard deviation = less similar the scores If a set of test scores has a standard deviation of 5, then everyone who took the test scored similarly If the scores have standard deviation of 50, then the test-takers’ scores were not very similar at all!

More on Standard Deviation Standard deviation is kind of the “mean of the mean” and can often help you get the real story behind the data. It is how far, on average, scores deviate from the mean. The standard deviation is a statistic that tells you how tightly all the various examples are clustered around the mean in a set of data. When the examples are pretty tightly bunched together and the bell-shaped curve is steep, the standard deviation is small. When the examples are spread apart and the bell curve is relatively flat, that tells you that you have a relatively large standard deviation. See how standard deviation changes as you manipulate a graph!

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Correlations A correlation is a statistical technique that allows us to understand the degree to which two variables are related. The strength and direction of correlations can be illustrated graphically in a scatterplot, in which paired X and Y scores for each subject are plotted as single points on a graph.  The slope of a line that best fits the pattern of points suggests the degree and direction of the relationship between the two variables.

Scatterplots and Correlations Positive Correlation As one number increases, the other increases. Ex: Study time to GPA Negative Correlation As one number increases, the other decreases. Ex: Absences to GPA No Correlation Variables do not affect one another in a significant way Ex: Height to GPA

Illusory Correlation Illusory Correlation: We believe there is a relationship between two things when it actually doesn’t exist.

“Feline High-Rise Syndrome” The New York Times reported evidence for cats’ exceptional ability to survive falls from high-rise buildings. They cited evidence that from June – November, 1984, 132 such victims were admitted to the Animal Medical Center, only 8 of which died from their injuries.  What is the problem with this correlation?

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Inferential Statistics Whereas descriptive statistics simply describe a data set, inferential statistics attempt to make inferences about a larger population based on a data set.  For example, if you're interested in studying CHS student behavior, you would have a hard time collecting data from each and every student.  Instead, you collect data from a sample of students, representative of the entire CHS student body, and then make inferences about the student body based on the data collected from your sample.

Inferential Statistics (cont.) If this chance variability always exists in data collection, how can a researcher be confident that the inferences he or she makes about the larger population (the entire CHS student body) is accurate?  We use inferential statistics! Instead of making absolute conclusions about the population, researchers make statements about the population using the laws of probability and statistical significance.

Statistical Significance Statistical Significance: statistical statement of how likely it is that an obtained result occurred by chance

To Summarize Descriptive Statistics: Organize and summarize data Central Tendency: mean, median, mode Standard deviation: variation in data Range: distance from smallest to largest Inferential Statistics: Interpret data and draw conclusions Used to test validity of hypothesis

Biased or insufficient samples “Four out of five dentists surveyed recommended Brand X gum.”  The # of dentists surveyed is not clear  How were the dentists chosen? Was it a random sample, or were 5 dentists chosen because they hold stock in Brand X gum? Many mail-in surveys suffer from a selection bias – the people who send them in may differ in important ways from those who do not.

The Misleading Average Example: The principal of a small private school met the criticism that his faculty has no teaching experience by issuing the statement that the average experience of each member of the faculty was 5 years. This statement was technically true: there were five teachers in the school including the principal, but the principal neglected to mention that he had twenty-five years of experience while the remaining four members had none.

Exaggerated graphs Depending on the height and width of each of the axes, different pictures of the data will emerge. Also, the data will look different, depending on how the axes (particularly the y-axis) are labeled.