15.6. The Normal Curve http://www.ck12.org
- Approximately 95% of the data will be within 2 standard deviations of the mean.
- Approximately 99.7% of the data will be within 3 standard deviations of the mean.
Some other important points about the normal distribution:
- The total area between the normal curve and thexaxis is 1 and this area represents all possible probabilities.
- If data is distributed normally, you can use the normal distribution to determine the percentage of the data
between any two values by calculating the area under the curve between those two values. When you take
calculus, you will learn how to calculate this area analytically, but for now you can use the normalcdf function
on your calculator. - Many histograms approximate a normal curve, but a true normal curve is infinitely smooth.
Example A
The amount of rain each year in Connecticut follows a normal distribution. What is the probability of getting one
standard deviation below the normal amount of rain?
Solution: You are looking for the area of the shaded portion of the normal distribution shown below. By the
empirical rule, you know that approximately 34% of the data is in between -1 and 0. Also, 50% of the data is above
- Therefore, approximately 84% of the data is unshaded. Therefore, 100%−84%=16% of the data is shaded. The
approximate probability is 16%.
To get the exact probability, use the normal cdf function on your calculator to calculate the exact area under the
curve. Go to [DISTR] (which is[ 2 nd][VARS]) and choose normalcdf. This is thenormal cumulative distribution
functionand calculates the area under the curve between twox-values. The syntax (how you will type it in) for
normal cdf is:
normalcdf(lower, upper, mean, standard deviation)
The lower bound for this shaded region is technically−∞, but the TI-84 cannot handle that so use -1E99. -1E99 is
− 1 × 1099 , an extremely small number, and will give identical results that are correct to many decimal places. The
upper bound is -1. For a standard normal distribution with a mean of zero and a standard deviation of 1 you don’t
need to type anything else in, but since you will be working with normal distributions with means and standard
deviations that are different, it will make sense to get used to using the whole syntax.
normalcdf(-1E99, -1, 0, 1)