06-22-12
          Random variables can either be discrete or continuous.  Discrete variables cannot assume all values between any two given values of the variables.  On the other hand, a continuous variable can assume  all values between any two given variables.
          Many continuous variables have distributions that are bell shape and are called approximately normally distributed variables.
       

Understand the Standard Normal distribution and its connection to all other Normal distributions

A value, x, from a normal distribution specified by a mean of m and a standard deviation of s can be converted to a corresponding value, z, in a standard normal distribution with the transformation z=(x-m)/s.  And, of course, in reverse, any value from a standard normal graph, say z, can be converted to a corresponding value on a normal distribution with a mean of m and a standard deviation of s by the formula x=m+z*s.  Remember that the standard normal distribution has a mean of 0 and a standard deviation of 1, i.e., m=0, s=1.

The ability to carry out this transformation is very important since we can do all our analysis with the standard normal distribution and then apply the results toevery other normal distribution, including the one of interest.  For example, to draw a normal curve with a mean of 10 and a standard deviation of 2 (m=10, s=2), draw the standard normal distribution and just re-label the axis.  The first figure below is the standard normal curve and the next figure is the curve with (m=10,s=2).

Each value along the x-axis represents that many standard deviations from the mean.  The 1 (or -1) x-value is one standard deviation from the mean.  Similarly, the 3 (or -3) represents three standard deviations from the mean.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one.

Standard Normal Distribution Table

A standard normal distribution table shows a cumulative probability associated with a particular z-score. Table rows show the whole number and tenths place of the z-score. Table columns show the hundredths place. The cumulative probability (often from minus infinity to the z-score) appears in the cell of the table.

For example, a section of the standard normal table is reproduced below. To find the cumulative probability of a z-score equal to -1.31, cross-reference the row of the table containing -1.3 with the column containing 0.01. The table shows that the probability that a standard normal random variable will be less than -1.31 is 0.0951; that is, P(Z < -1.31) = 0.0951.

z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
-3.0	0.0013	0.0013	0.0013	0.0012	0.0012	0.0011	0.0011	0.0011	0.0010	0.0010
...	...	...	...	...	...	...	...	...	...	...
-1.4	0.0808	0.0793	0.0778	0.0764	0.0749	0.0735	0.0722	0.0708	0.0694	0.0681
-1.3	0.0968	0.0951	0.0934	0.0918	0.0901	0.0885	0.0869	0.0853	0.0838	0.0823
-1.2	0.1151	0.1131	0.1112	0.1093	0.1075	0.1056	0.1038	0.1020	0.1003	0.0985
...	...	...	...	...	...	...	...	...	...	...
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990

There are a few ways to find the area under a normal distribution curve for two z-scores on opposite sides of the mean using a z-table. Once you know how to read the table, finding the area only takes seconds!

If you are looking for other variations (finding the area for a value between 0 and any z-score, or between two z-scores on the same side, see this normal distribution curve index).

normal distribution with z-scores on opposite side of mean

Step 1: Look in the z-table for the given z-scores (you should have two) by finding the intersections. For example, if you are asked to find the area from z= -0.46 to z= +0.16, look up both 0.46* and 0.16. The table below illustrates the result for 0.46  (0.4 in the left hand colum and 0.06 in the top row. the intersection is .6772).

z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.0000	0.0040	0.0080	0.0120	0.0160	0.0199	0.0239	0.0279	0.0319	0.0359
0.1	0.0398	0.0438	0.0478	0.0517	0.0557	0.0596	0.0636	0.0675	0.0714	0.0753
0.2	0.0793	0.0832	0.0871	0.0910	0.0948	0.0987	0.1026	0.1064	0.1103	0.1141
0.3	0.1179	0.1217	0.1255	0.1293	0.1331	0.1368	0.1406	0.1443	0.1480	0.1517
0.4	0.1554	0.1591	0.1628	0.1664	0.1700	0.1736	0.1772	0.1808	0.1844	0.1879
0.5	0.1915	0.1950	0.1985	0.2019	0.2054	0.2088	0.2123	0.2157	0.2190	0.2224

Step 2: Add both of the values you found in step 1 together.

*note. Because the graphs are symmetrical, you can ignore the negative z-values and just look up their positive counterparts. For example, if you are asked for the area of a tail on the left to -0.46, just look up 0.46.

 by: Jorey Mark A. Millamena