2.2 Density Curves and Normal Distribution

DENSITY CURVE

• a density curve is an idealized representation of a distribution and illustrates the distribution’s overall pattern

• easier to work with that histograms

• always on or above the horizontal axis and has an area exactly one underneath it

NORMAL DISTRIBUTION

• the most common continuous probability distribution is the normal distribution

• the graph of a normal distribution is called a normal curve

• normal distribution is an idealized version of a real-word distribution; analyzing the population

• greek letter “mu” will represent the mean and “sigma will represent standard deviation

  • these are called parameters

  • x and s are called statistics since they help to summarize the data we are studying

• if x is normally distributed with mean and standard deviation, we write x-N (mean, sd)

• the standard normal distribution is the normal distribution with mean=0 and sd=1

• a standard normal distribution is abbreviated z~N(0,1), due to standardized values will have a mean of zero and a standard deviation of 1

• since the normal distribution is continuous

  • P(x< -1)=P(z<=-1)

• assume normality or normality is satisfied

THE 68-95-99.7 RULE

• the 68-95-99.7 rules (the empirical rules) states that if the data set can be well approximated by a normal curve, then

  • 68% of observations will be within one standard deviation of the mean

  • 95% of observations will be within two standard deviations of the mean

  • 99.7% of observations will be within three standard deviations of the mean

    • normal probability plot or normal quartile plot