AP Statistics Chapter 2 Notes. Measures of Relative Standing Percentiles The percent of data that lies at or below a particular value. e.g. standardized.

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Presentation transcript:

AP Statistics Chapter 2 Notes

Measures of Relative Standing Percentiles The percent of data that lies at or below a particular value. e.g. standardized test score reports baby weight/height/head size. Standardized value (z-score)

Z-score Z-score tells you how many standard deviations away from the mean a given observation is. Z-scores are more useful with symmetric distributions.

Density Curve An idealized mathematical model used to represent a distribution. Always on or above the horizontal axis. Has an area of exactly 1 underneath it. The area under the curve for any given interval is equal to the proportion of all observations that fall in that interval. median: equal areas point Mean: balance point

Notation used for density curves Observed dataIdealized Data (Sample) (population) Statistics Parameters MEAN μ standard deviation σ

Normal Distribution Always symmetric, but the exact shape depends upon μ and σ. Change in curvature (point of inflection) shows where 1 standard deviation from the mean is located.

Empirical Rule ( Rule)

Example IQ scores are Normally distributed with a mean of 100 and a std dev of 15. What % of people have IQ scores… Between 70 and 130? Less than 85? Greater than 145? Less than 115? Between 55 and 70?

Probability Calculations We define a Normal distribution by its mean and standard deviation. N(μ, σ) If we standardize the distribution by calculating z- scores, we create the distribution: N(0,1). The z-table provides the percentiles associated with various z-scores. When performing a calculation, be sure to draw a sketch of the region under the Normal curve that you are working with, and answer the question in context.

Examples IQ scores are Normally distributed with a mean of 100 and a std dev of 15. What percent of people have IQ scores less than 82? Less than 121? What percent of people have IQ scores greater than 107? What percent of people have IQ scores between 88 and 104? A person is considered a genius if they are in the top 2% in terms of IQ. What IQ score does a person need to be considered a genius?

Using the calculator To calculate the % of observations within a certain interval, use the z-table or the graphing calculator. 2 nd Vars (Dist), choose option 2. Normalcdf (min, max, μ, σ) To calculate raw data scores from percentiles: 2 nd Vars (Dist), choose option 3. invNorm(%, μ, σ)