In our case it is the probability of a value being between 0 and 1 and it comes out very close to the 34% intuitive guess that we made earlier. The result is 0.8413 - 0.5 = 0.3413 This is a probability. We only want the area between the mean and our point so we will subtract 0.5 from out value (half the area is below the mean and not of interest to us in this case). But, this measures the probability from the far left-hand side of the probability distribution up to our point. In our case the z value is 1.00 so we will look in the row labeled 1.0 and the column labeled. The second decimal of the z value is shown across the top (yes I know this is strange but that is the way it is done). Thus you see 0.8, 0.9 and 1.0 in the fragment of the table shown above. How do I read the table ?The whole number and the first decimal place of a Z value is shown on the lefthand side of the table. Let's check our intuition by looking up the probability using Table A-2 for Positive Z Scores in the textbook. Consequently we would guess that there is a 68%/2 = 34% chance of our observing a value between 0 and 1.
Since the normal distribution is symmetric about the mean we can assume that half of the probability is above the mean and half is below.
HOW TO READ STANDARD NORMAL TABLE PLUS
We know from the empirical rule that there is a 68% chance that a value will fall within plus or minus one standard deviation of the mean. What is the likelihood that our guage will read 1 psi when it should read 0 psi? If we calculate the z Score we have z = (1 - 0)/1 = 1. We would like to be able to determine the probability of our being off by various amounts. So there will be times when our guage will read 0.5 psi even when pressures in the two areas are the same. Ours has a mean reading of 0 pounds per square inch when pressures are equal and a standard deviation of 1 pound per square inch (psi). Let us assume that we have a pressure difference guage that should always read zero when there is no difference in pressure between two containers. Standard Normal Distribution - For the standard normal distribution the mean is 0 (zero) and the standard deviation is 1. We will then use the properties of normal distributions to turn that value into a probability. In essesnce we are finding out how many standard deviations a point is from the mean. If we were to build normal distribution tables for evern conceivable value of the mean and standard deviation we would have a formidable tome! Fortunately there is a better way, and that is transform any normal distribution into the standard normal through use of the z Score. Even though our tables only went to n=15 they were sizeable. Why do we need to do this? If you remember our work with the Binomial Distribution you remember that there was a table for every value of n (sample size) with rows (x values) and columns (p values - probabilities) for a range of values of n, x and p. With the use of a z Score we will be able to transform any normal probability distribution into a "Standard Normal Distribution.". To determine a probability we will start with a z Score which is calculated as z = (X - Mean)/Standard Deviation. In our case one of the values will always be the mean of the probability distribution.
When dealing with with continuous distributions, and in our case it will usually be the normal distribution (pictured below) we will talk about the probability that x will be between certain values.įor continuous distributions the probability that the variable will fall between two values is the area under the curve between those values. Thus when we discussed the Empirical Rule we talked about 68% of the data being between the mean plus one standard deviation and the mean minus one standard deviation (mean +s and mean-s). In continuous probability distributions we can't point to specific values of x with spaces between the x values.
The Concept - When we were dealing with discrete probability distributions each value of x was related to a specific probability.