Stat 226 - Section 1.3 - Exploratory Data Analysis From...

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Section 1.3 The Normal Distribution Section 1.3 1 Exploratory Data Analysis From Sections 1.1 and 1.2, we have a basic framework for exploring the distribution of a single quantitative variable. 1. Always plot the data. We have seen histograms, boxplots, and stemplots. 2. Look for the overall pattern (shape, center, spread) and for striking deviations (outliers). 3. Calculate a numerical summary to describe the distribution completely (center and spread). Section 1.3 2 Describing the Overall Pattern Sometimes the overall pattern can be described by a smooth curve. Think about approximating the histogram with a curve. We will investigate an important group of these “smooth curves.” The smooth curve can serve as a mathematical model , or idealized description, of the distribution. Often the form of the mathematical model can be given in terms of an actual equation (more on this later). Section 1.3 3 Example - Unemployment Unemployment Density Curve Unemployment Rate Density 0.0 0.1 0.2 0.3 2 4 6 8 Section 1.3 4 Density Curve Unemployment Density Curve Unemployment Rate Density 0.0 0.1 0.2 0.3 2 4 6 8 The smooth curve drawn for the unemployment data is a type of density curve . A density curve has two important characteristics: 1. It is always on or above the horizontal axis (never below). 2. It has an area of exactly 1 underneath it. The histogram bars have been scaled so their area adds to 1. Section 1.3 5 Density Curve The density curve is yet another way to describe the overall pattern of a distribution. Why might we like to use the density curve instead of the histogram? Section 1.3 6
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Density Curve Other useful characteristics of a density curve The area under the curve and above any range of values is the proportion of all observations that fall in that range. Density curves, like distributions, come in many shapes. The mathematical equation that often describes a density curve is known as a probability density function . Section 1.3 7 Center of a Density Curve The measures of center and spread used for distributions of observations apply to density curves as well. The median of a density curve is the point that divides the area under the curve in half (0.5 to the left and right). The mean of a density curve is its center of mass, or the point at which the curve would balance if it were made of solid material. Section 1.3 8 Symmetric Density Curve 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Symmetric Density X Density Section 1.3 9 Right Skewed Density Curve 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Right Skewed Density X Density Section 1.3 10 Left Skewed Density Curve 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Left Skewed Density X Density Section 1.3 11 Central Terminology Here we will make a brief excursion from Section 1.3 to address some terminology that is central to Statistics.
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