AP Statistics Lesson 2.1 Density Curves and the Normal Distribution. Essential Question: How are Density Curves and Normal Curves used to analyze data? Objectives: To solve problems involving Density Curves. To use the Normal Curve to analyze data. Strategies for exploring data on a single quantitative variable. Always plot your data: make a graph, usually a histogram or a stemplot. Look for the overall pattern ( shape, center, spread) and for striking deviations such as outliers. Calculate a numerical summary to briefly describe the center and spread. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. Density Curves The smooth curve drawn through the tops of the histogram bars is a good description of the overall pattern of the data. The curve is a mathematical model for the distribution. The mathematical model is an idealized description. Density Curves A Density Curve is a curve that is always on or above the horizontal axis. and has area exactly 1 underneath it. A Density Curve describes: the overall pattern of a distribution. The area under the curve and above any range of values is the proportion of all observations that fall in that range. Density curves can be in any shape, skewed or symmetric. Median and Mean of a Density Curve The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean if a density curve is the balance point, at which the curve would balance if made of solid material. The median and mean are the same fro a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail. μ ( the Greek letter mu) is usually notation for the mean of an idealized distribution. σ ( the Greek letter sigma) is used for the standard deviation of a density curve. Normal Distribution Density curves are symmetric, single-peaked, and bell-shaped. They are called normal curves, and they describe normal distributions. All normal distributions have the same overall shape. The exact density curve for a particular normal distribution is described with μ as its mean and σ as its standard deviation. The 68 – 95 – 99.7 rule In the normal distribution with mean μ and standard deviation σ: 68% of the observations fall within σ of the mean μ. 95% of the observations fall within 2σ of μ. 99.7% of the observations fall within 3σ of μ.