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Density Curves and the Normal Distribution

Learn about density curves, an idealized graphical representation of data distributions, and the normal distribution. Explore concepts such as mean, median, and skewness in relation to these curves.

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Density Curves and the Normal Distribution

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  1. Chapter 2 DENSITY CURVES AND THE NORMAL DISTRIBUTION

  2. Density Curve • A graph that represents the relative frequency distribution for a set of data • It can be used to describe the overall pattern of a distribution. • Often an idealized version – “Mathematical Model” • ALWAYS above or on the horizontal axis • ALWAYS bounds AREA = 1 • AREA bounded is used to tell the proportion of observations within a range of values.

  3. 1 1/6 AREA = 1 1 1 2 3 4 5 6 1/2 AREA = 1 1 2 3 4 6 5 Examples

  4. FROM HISTOGRAM … TO … A DESITY CURVE TOTAL AREA OF ALL BARS = 1 A Mathematical Model – an idealized representation of reality

  5. ACTUAL % OF SCORES LESS THAN 6 .. IS THE AREA OF THE HISTOGRAM BARS AREA = 0.303

  6. AREA OF THE APPROXIMATED DENSITY CURVE IS NOT EXACT! AREA OF SHADED REGION = 0.293 … (i.e. 0.01 lower than the actual area)

  7. Total Area = 1.00 Area = .12 7 8

  8. Let’s ROLL A DISTRIBUTION • P.84 # 2.5 • Simulate the act of “Rolling a single die” • {1, 2, 3, 4, 5, 6} • Clear L1 • MATH >>> PRB 5:RandInt(1, 6, 100) • STO  L1 • WINDOW: X [1, 7]; Y [-5, 25] YScl = 5 • STAT PLOT: Histogram for L1 • Repeat … Are we all the same? Skip for now …

  9. Mean and Median • MEAN: If the distribution were to be made out of solid material … the MEAN would be the balancing point. • MEDAIN: The point where the area under the curve is divided into to equal halves. • Same ideas from last chapter … regarding the impact of skewing. • Skewed Data

  10. The “NORMAL” Distribution • Normal Distribution

  11. Points of Inflection …. ONE Standard Deviation from the MEAN

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