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AP Statistics 2.1 Density Curves Objectives: Explain what is mean by a mathematical model. Define a density curve. Explain where the mean and median of a density curve are to be found. Describe the relative position of the mean and median in a symmetric density curve and in a skewed density curve. Density Curve Our strategy for exploring data from a single quantitative variable: 1. Always plot your data: make a graph, usually a histogram or a stemplot. 2. Look for the overall pattern (SOCS!) 3. Calculate a numerical summary to briefly describe center and spread. ADD A STEP FOUR: 4. Sometimes the overall pattern of a large number of observations is so regular that we can draw a smooth curve. Mathematical Model: An idealized description of the overall pattern of the data ignoring minor irregularities and possible outliers. Density Curve: (a mathematical model of a histogram) see pg. 124 • Is always ON or ABOVE the horizontal axis, AND • Has area EXACTLY 1 underneath it. • The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval. A density curve describes the overall pattern of a distribution. In the graph to the right, a density curve is shown. It appears to have data values from 0 to 10 and is skewed right. The shaded area under this density curve between 2 and 4 is .3109. That is saying that 31.09% of the data lies between 2 and 4. The Median and Mean of a Density Curve • The measures of center still apply to Density Curves. • NOTE: Because density curves are “idealized patterns” a symmetric distribution is EXACTLY symmetrical. o The mean and median are identical. • A skewed distribution pulls x-­‐bar to the tail. The median: the “equal-­‐areas point”, divides the curve into two halves of equal areas. • The quartiles divide the area under the curve into quarters. • The rough location of the quartiles can be done by eye. The mean: the “balance point”, at which the curve would remain level if it were made of solid material. In a symmetric density curve the mean and the median are the same In a right skewed curve the mean is to the right of the median In a left skewed curve, the mean is to the left of the median •
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Since a density curve is NOT the actual data, we use: o µ (mu) ! mean of a density curve or a population o σ (sigma) ! standard deviation of a density curve or a population. Parameter vs. Statistics: STATISTICS: calculated from a set of data, a SAMPLE of a particular population. • Usually, but not always, English letters are used for statistics (x-­‐bar, s for stand. dev) PARAMETER: a value, usually estimated, and used to represent a particular POPULATION. • Greek letters are used for parameters (measures on idealized distributions: µ, σ) 2.12 (pg. 128) (a) Mean C, Median B (b) Mean A, Median A (c) Mean A, Median B 2.20 (pg. 132) a) area under curve is A + B + C ½ (.5)(.8) + ½ (.5)(.8) + 1(.6) = 1 b) The median is a .5, the quartiles are APPROXIMATELY Q1 = .3 Q3 = .7 THIS IS BY EYE ONLY. I tried to break it into four equal parts. c)
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The line is y = mx + b so y = 1.6x + .6. So the height at .3 is 1.6(.3) + .6 = 1.08 Total area = 1/2bh + lw 1/2(.48)(.3) + (.3)(.6) = .252 SO 25.2% of the observations lies below .3 Using symmetry of the density curve: 1 – 2(.252) = .496 so 49.6% of observations lie between .3 and .7 HW: pg. 126; 2.10, 2.23, 2.24