Programmer's Reference

Tje z-score is often called the standardized value.  The standardized value or z-score can be interpreted as the number of standard deviations from the mean. 

So if x1 has a value of 4 and a z-score of 1.5, this would mean that 7 is 1.5 standard deviations more than the sample mean (goes right). 

Alternatively if x2 is equal to 1 and has a z-score of -1.3, this means that 1 is 1.3 standard deviations less than the sample mean (goes left).

Calculation:

z = (value in sample – sample mean) / standard deviation of sample

 the formula to compute a z-score is:

Z = (value in pop – pop mean) / standard deviation

the resulting value of a z-score should be looked up in a Z-Score chart (http://www.regentsprep.org/Regents/math/algtrig/ATS7/ZChart.htm)

The values are usually represented as 0.XXXX, where the X’s can be treated as a percentage.  That is to say, if your Z-Score is -0.10, then on the table it will have a value of .0398.  Expressed another way, there is a 3.98% chance that your x value or anything below it will be picked at random out of the entire population.  Also, this means that the x value that you are looking up is within 3.98% of the population you are examining.    

If the z score was positive 0.10, add .50 or 50% to the percentage above.  Expressed another way, the value you computed the z score for (or any value below it) has a 53.98% chance of being randomly selected from your population.  

The value for your z-score on your chart is referred to the area between the mean and z.  If the value is positive, we add .50 or 50% to note that the value is above the mean.  Otherwise if it is negative, the value is below the mean.

Rules for calculations between two z scores

To calculate the area between two z scores on opposite sides of the mean, add the z-scoe values together.

To calculate the area between two z scores on the same side of the curve, subtract the large z-valu (from the table) from the smaller z-value (from table).

Expressed as a ratio from 0 to 1.00.   Observed in a distribution of scores to the maximum variation that could exist in a particular distribution.

 0.00 indicates no variation.  1.00 represents maximum variation.

 The most common usage for this is for nominal variables and it can be used with any variable when scores have been grouped into a frequency distribution.

For example, given the following tables representing a city’s marital status in the hundreds, we may want to know which of the two cities is more diverse (or heterogeneous).

At a glance, it is difficult to discern which one of the two is more diverse.  So we need to compute the index of qualitative variation (IQV). To compute the (IQV) we apply the following formula. 

k(N² – sum(f²)) / N²(k – 1)

where:

k = number of categories

N = number of cases (total the frequency counts)

sum of f² = the sum of the squared frequencies (that is, square the frequencies first, then add them up)

The result should be something like this:

In the attached excel sheet, you will be able to audit the formula for IQV if you want excel to do your calculations.