2.1 Density Curves and the Normal Distribution

 Differentiate between a density curve and a histogram  Understand where mean and median lie on curves that are symmetric, skewed right, and skewed left.  Use a normal distribution to calculate the area under a curve

 1-Always plot your data: make a graph, usually a histogram or a stem plot.  2-Look for the overall pattern (shape, center, spread) and for striking deviations such as outliers.  3-Calculate a numerical summary to briefly describe the center and spread.  median-5 # summary mean-μ and σ We now add one more step!  4-Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

 it is an idealized description. It gives a compact picture of the overall pattern of the data but ignores minor irregularities as well as many outliers.

 Histogram Actual count of observations that fall within an interval  Density Curve The proportion of values that fall within an area under the curve.

 A Density Curve is a curve that: - Is always on or above the horizontal axis - Has an area of exactly 1 underneath it.  A density curve describes the overall pattern of a distribution. The area under the curve and any above range of values is the proportion of all observations that fall in that range.  A normal curve is one that is symmetrically skewed.

The following density curve is skewed to the right.  What does the shaded area mean?  The proportion of observations taking values between 9 and 10.

 The median is point where half the observations are on either side.  The quartiles divide the area under the curve into quarters.  The median of a symmetric density curve is at the center.

What do we know about the mean and median of the following 3 curves? Draw lines to represent the mean and median on each curve. symmetrically skewed skewed to the right skewed to the left

Ex: pg a-c 2.3a-d

1- all normal dist. have the same overall shape (symmetric, single-peaked, bell shaped)  The exact density curve for a particular normal distribution is described by giving its: 1- mean (μ)  and 2- standard deviation (σ) μ=mu σ=sigma

 Draw a normal curve with μ=10 and σ=2  Draw a normal curve with μ=10 and σ= 5 What do you notice? σ controls the spread. The larger σ, the more spread out the curve.

 In a normal distribution with mean (µ) and standard deviation (σ):  -68% of observations fall within 1σ of μ.  -95% of observations fall within 2σ of μ.  -99.7% of observations fall within 3σ of μ.

 Draw a curve. 1-What height of women do the middle 68% fall? 2-What height is the 84 th percentile? 3-What height is the highest 2.5% of women?