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AP Statistics

AP Statistics. 2.1 Density Curves and the Normal Distribution. Learning Objective. Differentiate between a density curve and a histogram Understand where mean and median lie on curves that are symmetric, skewed right, and skewed left .

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AP Statistics

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  1. AP Statistics 2.1 Density Curves and the Normal Distribution

  2. Learning Objective • Differentiate between a density curve and a histogram • Understand where mean and median lie on curves that are symmetric, skewed right, and skewed left. • Use a normal distribution to calculate the area under a curve

  3. How do we explore data with a single quantitative variable? • 1-Always plot your data: make a graph, usually a histogram or a stem plot. • 2-Look for the overall pattern (shape, center, spread) and for striking deviations such as outliers. • 3-Calculate a numerical summary to briefly describe the center and spread. • median-5 # summarymean-μ and σ We now add one more step! • 4-Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

  4. Mathematical Model • it is an idealized description. It gives a compact picture of the overall pattern of the data but ignores minor irregularities as well as many outliers.

  5. Density Curves versus Histograms • Density Curve The proportion of values that fall within an area under the curve. • Histogram Actual count of observations that fall within an interval

  6. A Density Curve is a curve that: - Is always on or above the horizontal axis - Has an area of exactly 1 underneath it. A density curve describes the overall pattern of a distribution. The area under the curve and any above range of values is the proportion of all observations that fall in that range. A normal curve is one that is symmetrically skewed.

  7. Density curves, like normal distributions come in many shapes. The following density curve is skewed to the right. • What does the shaded area mean? • The proportion of observations taking values between 9 and 10.

  8. The median is point where half the observations are on either side. The quartiles divide the area under the curve into quarters . The median of a symmetric density curve is at the center.

  9. What do we know about the mean and median of the following 3 curves? Draw lines to represent the mean and median on each curve. symmetrically skewedskewedto the right skewed to the left

  10. Ex: pg. 71 2.2 a-c 2.3a-d

  11. Normal Distributions 1- all normal dist. have the same overall shape (symmetric, single-peaked, bell shaped) • The exact density curve for a particular normal distribution is described by giving its: 1- mean (μ) • and 2- standard deviation (σ) μ=mu σ=sigma

  12. What happens to two normal curves with different standard deviations? • Draw a normal curve with μ=10 and σ=2 • Draw a normal curve with μ=10 and σ= 5 What do you notice? σ controls the spread. The larger σ, the more spread out the curve.

  13. The 68-95-99.7 rule • In a normal distribution with mean (µ) and standard deviation (σ): • -68% of observations fall within 1σ of μ. • -95% of observations fall within 2σof μ. • -99.7% of observations fall within 3σof μ.

  14. The average height of women is 64.5 inches with σ=2.5 • Draw a curve. 1-What height of women do the middle 68% fall? 2-What height is the 84th percentile? 3-What height is the highest 2.5% of women?

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