Chapter 6

The Normal Curve

A Density Curve is a curve that:

*is always on or above the horizontal axis

*has an area of exactly 1 underneath it

*describes the overall pattern of a distribution

Density curves are mathematical models. They are an “idealized description of a data set” not an actual set of data.

They represent continuous not discrete data.

The median is the 50th percentile. Exactly 50% of the data lies on each side of the median.

The population mean ( - lower case Greek mu) is the point at which the curve would balance if made of solid material.

The mean and the median are the same for a symmetric distribution. For non-symmetric dist. remember that a tail pulls the mean toward it.

The population standard deviation ( - lower case Greek sigma) controls the spread about .

Normal Curves are important density curves that are symmetric, single-peaked, and bell-shaped and they describe normal distributions.

On a normal curve, can be roughly located by eye – It is where the change in curvature (inflection points) takes place on either side of .

The Empirical Rule 68% of the observations fall within 1 of

. 95% of the observations fall within 2 of

. 99.7% of the observations fall within 3 of

.

Standardizing A Normal Distribution can be standardized so that

area at any value (not just the empirical rule values) can be found easily. The letter z is used for standardized observations on a normal curve, thus we call them z-scores.

z = x -

Using this method, we standardize any normal distribution so that we can use the areas from the standard normal distribution where the mean is 0 and the standard deviation is 1, or N(0,1).

The standardized score represents the number of standard deviations a given value is away from the mean of the distribution.

Normal distribution notation: N(,) Example:

for a normal distribution whose mean is 500 and standard deviation is 100, we would note that as N(500, 100)

Normal Distributions are common for 3 reasons:1) Some real data has a normal distribution.

2) They approximate many chance outcomes

in probability.

3) Many symmetric distributions that are large

can be approximated using normal dist.

However… Not all data follows a normal distribution, in fact many do not. Don’t assume data to be normal. Right now it must be given that it is Normal.

Percentiles: The “pth” percentile has p% that lies to the left and 100% - p% that lies to the right.

Finding Normal Proportions:1) State the problem (as an inequality)2) Standardize3) Draw a picture4) Use the table or calculator to find area.

Finding a value Given a Proportion:1) Draw a picture2) State the problem (in terms of z)3) Use the table or calculator to “unstandardize”..

***Please note that since this is an area model, single values, P(x = #), do not have any probability. Only probabilities associated with intervals can be found.

Is a Normal Distribution Appropriate? Assessing Normality – in other words, should I use a normal

distribution to approximate this data or not. Method 1: Plot the data, look for an approximately

symmetric, bell-shaped curve about the mean. Check that the mean and median are close together. You can check the empirical rule to see if numerical values are in line with the percentages expected. Small data sets rarely fit the empirical rule very well even if they actually come from a normal distribution.

Method 2: Use the calculator to produce a Normal Probability Plot. See page 119 in text for calculator keystrokes. If the data is close to normal then this plot will be close to a straight line, linear. Outliers will show up far away from the overall pattern of the plot.