Stat350, Lecture#4 :Density curves and normal distribution Try to draw a smooth curve overlaying the histogram. The curve is a mathematical model for the distribution. It is an idealized description of the data capturing the overall pattern but ignores minor irregularities and outliers.

A density curve is always described by a non-negative function, so lies above the horizontal axis. The total area under the curve is 1. The area under the curve for any interval is the proportion of observations falling in that interval. No set of real data exactly follows a density curve.

Mean and median of a density curve Median is the point which divides the area under the curve into equal halves. Mean is a balance point, where the curve will balance if it was made of solid metal For symmetric distribution, mean and median are same. For skewed distributions, mean is pulled farther towards the longer tail than the median.

Normal Distribution Normal distributions : an idealized curve, symmetric, unimodal and bell-shaped The density function is described by two parameters, the mean of this idealized distribution, µ and the standard deviation ơ. Remember that xbar and s are always calculated from sample data, the mean and standard distribution of the hypothetical distributions are denoted by the greek symbols.

Locating µ and ơ by eye µ is the point of symmetry ơ is the point where the graph changes curvature, the slope gets flatter. These are special properties of normal distribution, not true for mean and s.d. of any distribution normal distribution describes many real data There are many other distributions which can be approximated by a normal distribution for large enough sample size.

The rule For a normal distribution with mean µ and s.d. , 68% of the observations fall within  of the mean µ, 95% fall within 2  of the mean µ, 99.7% fall within 3  of the mean µ.  Remembering this rule helps us to roughly guess about probabilities regarding normal distributions rather than making constant calculations.

Example: Using the rule Height of adult men ~ N(69,2.5) What percent of men are taller than 74 inches? The 95% rule says the middle 95% have height between 69-5 to 69+5 or between 64 to 74 inches. So the top 2.5% are taller than 74 inches.

Now suppose the height distribution is N(64,2.5). What percent of men are shorter than 66.5 inches? The 68% of the rule says the middle 68% have height between to or between 61.5 to 66.5, because of symmetry, the top 16% are taller than 66.5 and the lower 16% are shorter than 61.5 inches. So 84% men are shorter than 66.5 inches.

The standard normal distribution X~N( ,  ) The standardized value of x is: z= (x-  )/  This is called a z-score.  A z-score tells us how many standard deviations the actual observation is from the mean and which direction.  observations greater than the mean have positive z-score, below the mean have negative z-score.

Example The amount of rainfall in a month in Indiana follows Normal distribution with mean 4 inches and standard deviation 2.1 inches A rainfall of 3.3 inches is standardized as: (3.3-4)/2.1 =-.33 A rainfall of 3.3 inches is.33 standard deviations lower (because of the negative sign) than the mean.

Standard Normal Distribution If X~N(( ,  ) then Z=(X-  )/  ~ N(0,1) If you standardize a variable which has a normal distribution, you get a variable with normal distribution with mean 0 and standard deviation 1. N(0,1) distribution is called the standard normal distribution. Table A at the back of the book tabulates area under the standard normal curves.

Using the normal probability table State the problem in terms of the actual observed variable. Standardize the X variable to Z Use the standard normal probability table to find proportion or probability under the standard normal curve. We will do many normal calculations now on our good old blackboard….