# Chapter 2: The Normal Distributions

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Chapter 2: The Normal Distributions
Density Curves Normal Distribution Standard Distribution

Density Curve Mathematical model for a distribution
Always on or above the horizontal axis Has an area of exactly 1 There are normal, skewed left, and skewed right curves To find a value, find the area under the curve Quartile 1=1/4 of the area under the curve Quartile 3=3/4 of the area under the curve

Median and Mean of Density Curves
Median—equal-areas point: divides the area under the curve in half Mean—the balance point: where the curve would balance if made solid Symbol: Mean and median are equal on a symmetric curve

Normal Curve

Skewed Right Curve

Skewed Left Curve

Normal Curve Must be: Symmetric mean=median Single Mound Bell Shaped

Normal Distribution Mean Standard Deviation
Can be written—N(mean, standard deviation) Changing the Standard Deviation changes the spread of the curve Changing the mean without changing the standard deviation moves the graph horizontally on the x-axis Larger Standard Deviation=curve more spread out Inflection Point-points where the curvature change takes place Located a from each side of the mean

The 68-95-99.7 Rule 68% of all observations fall within of
To find an observation’s percentile, use the Z-Score

Graph of Rule

Example The distribution of heights of adult American men is approximately normal with mean of 69 inches and a standard deviation of 2.5 inches. Between what heights do the middle 68% of men fall? 69+2.5=71.5 69-2.5=66.5 From 66.5 to 71.5

Z-Score To find what percentage a certain individual/value is
Tells us how many standard deviation we are from the mean Formula- Can use the table or the calculator Calculator normalcdf(lower bound, upper bound, mean, standard deviation)

Z-Score Table

Assessing Normality Method 1—Frequency Histogram or Stem Plot
Approximately bell-shaped Symmetric about the mean Check to see if rule works Method 2--Normal Probability Plot Interpret the graph—if the plotted points appear to be close to a line, then it is a normal distribution

The End

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