They are symmetric with scores more concentrated in the middle than in the tails Normal distributions are sometimes described as bell shaped Examples of
This distribution describes many human traits All Normal curves have symmetry, but not all symmetric distributions are Normal ? Normal distributions are
On the other hand, counting the number of heads/tails in a collection of coin tosses is not continuous (it is discrete) because the result can only be an
Behavioral data in a population tend to be normally distributed, meaning the data are symmetrically distributed around the mean, the median, and the mode, which
approximately normally distributed with a mean of 72 4 degrees (F) and a standard deviation of 2 6 degrees (F) Q1] Sketch the normal curve by hand here
Many populations have distributions that can be fit very The statement that X is normally distributed with captures upper-tail area 01
numbers used to describe what is a typical case value or how much variability is A curve like the one in Figure 4 1, which has a tail
center and less frequent scores fall into the tails Central tendency means most scores(68 ) in a normally distributed set of data tend to cluster in the
The normal probability distribution is the most commonly used probability There are many normal distributions, and each variable X which is nor-
255_6lesson5_normaldistn.pdf
5The Normal Distribution(Ch 4.3)
2The Normal DistributionThe normal distributionis probably the most important distribution in all of probability and statistics. Many populations have distributions that can be fit very closely by an appropriate normal (or Gaussian, bell) curve. Examples: height, weight, and other physical characteristics, scores on various tests, etc.
3The Normal DistributionDefinitionA continuous r.v. X is said to have a normal distribution with parameters µand σ> 0(or µand σ2), if the pdf of X isThe statement that X is normally distributed with parameters µand σ2 is often abbreviated X ~ N(µ, σ2). f(x;µ,!)=
1 ! 2"! e !(x!µ) 2 /2! 2 where"#
4The Normal DistributionFigure below presents graphs of f(x; µ,σ) for several different (µ,σ) pairs. Two different normal density curvesVisualizing µand σfor a normal distribution 5f(z;0,1)=
1 ! 2! e !z 2 /2 where"#The Standard Normal DistributionThe normal distribution with parameter values µ= 0 andσ= 1 is called the standard normal distribution.A r.v. with this distribution is called a standard normal random variableand is denoted by Z. Its pdf is: 6f(z;0,1)=
1 ! 2! e !z 2 /2 where"#The Standard Normal DistributionThe normal distribution with parameter values µ= 0 andσ= 1 is called the standard normal distribution.We use special notation to denote the cdf of the standard normal curve:!(z)= ! z !" f(y;0,1)dy 7The Standard Normal Distribution•The standard normal distribution rarelyoccurs naturally.•Instead, it is a referencedistribution from which information about other normal distributions can be obtained via a simple formula. •These probabilities can then be found A