module 13 wrap up probability distribution for normal continuous quantitative variables 19 of 19. let's summarize a continuous random variable is not limited to distinct values it is a measurement such as foot length we cannot display the probability distribution for a continuous random variable with a table or histogram we use the probability density curve to assign probabilities to intervals of x values we use the area under the curve to find these probabilities we use a normal probability density curve to model the probability distribution for many variables such as weight shoe sizes foot lengths and other human physical characteristics normal curves are mathematical models we use mu to represent the mean of a normal curve and sigma to represent the standard deviation of a normal curve we use greek letters to remind us that the normal curve is not a distribution of real data it is a mathematical model based on mathematical equations we use this mathematical model to represent the perfect bell shaped distribution for a normal curve the empirical rule tells us that there is a 68 chance that observations fall within one standard deviation of the mean 95 percent within two standard deviations of the mean and 99.7 percent within three standard deviations of the mean to compare x values from different distributions we standardize the values by finding a z-score a z-score measures how far x is from the mean in standard deviations in other words the z-score is the number of standard deviations x is from the mean of the distribution for example z is equal to 1 means that x value is one standard deviation above the mean if we convert the x values into z-scores the distribution of z-scores is also a normal probability density curve the curve is called the standard normal distribution we use an applet with the standard normal curve to find probabilities for any normal distribution we can also work backwards and find the x value for a given probability we use the different output to work backwards from probabilities to x values with this applet we found x values corresponding to quartiles and percentiles