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Measures of Relative Standing in Statistics

graybrobert avatar
graybrobert
11 diapositives · 189 Views · Jan 07, 2020

Learn about percentiles, Z-scores, density curves, and calculations in statistics to understand distribution characteristics. Practice using Normal Distribution and Empirical Rule for probability calculations.

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Measures of Relative Standing in Statistics
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  1. AP StatisticsChapter 5 Notes

  2. 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)

  3. 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.

  4. 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

  5. Notation used for density curves • Observed data Idealized Data • (Sample) (population) • Statistics Parameters • MEAN μ • standard deviation σ

  6. 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. Normal Distribution

  7. Empirical Rule (68-95-99.7 Rule)

  8. 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?

  9. 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.

  10. 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 3% in terms of IQ. What IQ score does a person need to be considered a genius?

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