Normal Distributions

Standardized Value

Often called z-score. Tells us how many standard deviations the original observation falls away from the mean, and in which direction.

Standard Deviation

The distance from µ to inflection point.The symbol is σ. It controls the spread of a normal curve.

Standard Normal Distribution

The normal distribution N(0,1) with mean 0 and standard deviation 1.

Percentile

The percent of the distribution that is at or to the left of the observation.

Mean

The point at which the curve would balance if made of solid material. The symbol is µ or it can be x bar.

Numbers you input in normalcdf

lower number, upper number, mean, standard deviation

The first quartile is the ____ percentile and the third quartile is the ____ percentile. The median is the ____ percentile.

25th 75th 50th

normalcdf

Calculator command that calculates the area under the normal curve between two given points.

invNorm

Calculator command that calculates the z-score for percentiles.

Normal Curves

Density curves that are symmetric, single- peaked, and bell- shaped. They describe normal distributions.

68-95-99.7 Rule

Descibe what percent of observations lie within one, two, and three standard deviations of the mean. 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

Assessing Normality

Evaluating whether a distribution is normal. There are two methods you can use. FIRST METHOD: Make a histogram or stemplot. See if graph is symmetric.Mark the points mean, mean ± s (standard deviation), and mean ± 2s. Then compare with the 68-95-99.7 rule. 2ND METHOD: Make a normal probability plot.

With your calculator find the area between -2.43<z<0.5 in a normal distribution.

Go to [2nd] [VARS], down to [2.normalcdf], Click [ENTER] normalcdf ( -2.43, 0.50, 0, 1) =0.6839

Table A

Has the areas under the standard normal curve. The table entry for each value z is the area under the curve to the left of z.

Normal Distribution

Have same overall shape, they are described by normal curves. The exact density curve is described by giving its mean and its standard deviation.Satisfy the 68-95-99.7 rule.

Table A only gives the areas to the _____ of z.

Left

Inflection Points

Points where the change of curvature take place. They are located at distance σ on either side of the mean µ.

First step in finding Normal Proportions

State the problem, draw a picture and shade the area of interest under the curve.

How many standard deviations are within the mean?

There are three standard deviations on each side of the mean.

The distribution of children's foot size is approximately normal with mean 48 inches and standard deviation 2.5 inches. What percent of children have a foot size less than 55 inches?

z=(x-μ)/σ z=(55-48)/2.5 z= 2.8=(.9974) 99.74% of children have a foot size less than 55 inches.