IPS6eCh01_3bb - Looking at Data Distributions Density...

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Looking at Data - Distributions Density Curves and Normal Distributions IPS Chapter 1.3

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Objectives (IPS Chapter 1.3) Density curves and Normal distributions Density curves Measuring center and spread for density curves Normal distributions The 68-95-99.7 rule Standardizing observations Using the standard Normal Table Inverse Normal calculations Normal quantile plots
Density curves A density curve is a mathematical model of a distribution. The total area under the curve, by definition, is equal to 1, or 100%. The area under the curve for a range of values is the proportion of all observations for that range. Histogram of a sample with the smoothed, density curve describing theoretically the population.

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Density curves come in any imaginable shape. Some are well known mathematically and others aren’t.
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 of a density curve is the balance point, at which the curve would balance if it were made of solid material. The median and mean are the same for a symmetric density curve. The mean of a skewed curve is pulled in the direction of the long tail.

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Normal distributions e = 2.71828… The base of the natural logarithm π = pi = 3.14159… Normal – or Gaussian – distributions are a family of symmetrical, bell- shaped density curves defined by a mean μ ( mu ) and a standard deviation σ ( sigma ) : N( μ,σ ). 2 2 1 2 1 ) ( - - = σ μ π x e x f x x
exp(-5) = .0067 exp(-1) = .3679 exp(0) = 1 exp(5) = 148.4 For normal distribution probability higher when exponent Is close to 0 (can not be positive) Exponential distribution e x = exp(x) = inverse of natural log

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i) Plot the data – form histogram, stemplot, etc. ii) Look for patterns and deviations iii) Calculate summary statistics that describe center and spread iv) Sometimes the data follow a pattern, a common pattern is the bell shaped curve of a normal distribution. This is often the case for: - Test scores (for a very large group). - Measurements, such as height - Situations where measurement error exist (e.g., measuring the speed of light) Normal Distributions May be relevant for single quantitative variable :
The Normal Distribution is a mathematical model. Often real world data approximately fit this model - Normal distribution is described by a density curve - Density curves assume “continuous data,” and have a smooth shape - Real world data can be represented in a probability histogram, where relative frequency is on vertical axis - Area under a probability histogram, or a density curve = 1; it represents all data, or can be considered as representing all possible outcomes of an experiment.

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