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Part III Taking Chances for Fun and Profit
Chapter 8 Are Your Curves Normal? Probability and Why it Counts
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Why Probability? Basis for the normal curve
Provides basis for understanding probability of a possible outcome Basis for determining the degree of confidence that an outcome is “true” Example: Are changes in student scores due to a particular intervention that took place or by chance alone?
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The Normal Curve (a.k.a. the Bell-Shaped Curve)
Visual representation of a distribution of scores Three characteristics… Mean, median, and mode are equal to one another Perfectly symmetrical about the mean Tails are asymptotic (get closer to horizontal axis but never touch)
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The Normal Curve
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Hey, That’s Not Normal! In general, many events occur right in the middle of a distribution with few on each end.
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More Normal Curve 101
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More Normal Curve 101 For all normal distributions…
almost 100% of scores will fit between -3 and +3 standard deviations from the mean. So…distributions can be compared Between different points on the X-axis, a certain percentage of cases will occur.
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What’s Under the Curve?
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The z Score A standard score that is the result of dividing the amount that a raw score differs from the mean of the distribution by the standard deviation of the distribution. What do these symbols mean?
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The z Score Raw scores below the mean have negative z-scores Raw scores above the mean have positive z- scores A z score is the number of standard deviations from the mean z scores across different distributions (with different means and std devs) are comparable
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What z Scores Represent
The areas of the curve that are covered by different z scores also represent the probability of a certain score occurring. So try this one… In a distribution with a mean of 50 and a standard deviation of 10, what is the probability that a score will be 70 or above?
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What z Scores Really Represent
Knowing the probability that a z score will occur can help you determine how likely it is that a factor other than chance produced the observed outcome. Keep in mind… z scores are typically reserved for populations
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Hypothesis Testing & z Scores
Any event can have a probability associated with it. Probability values help determine how “likely” or “unlikely” the event might be Range from 0 to 1 The key --- if something has less than 5% chance of occurring, we have a significant result
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Using the Computer Calculating z Scores
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