Calculate a percentile rank only when measurements use the same scale. For example, if one teacher gave a 50-point test and another gave a 100-point test, you could not use the raw data to determine a percentile ranking. Someone who earned 49 points on a 50-point test has scored very well, while someone who earned 49 points on a 100-point test has done very poorly. Using a percentile ranking in this case would obscure that fact.
Count the total number of scores in the group being considered. For example, if you are trying to determine your percentile ranking on a test, count the total number of people taking the test. Suppose for the purposes of illustration that 50 people took the test.
Determine the number of people who scored lower than you did. Often, instructors will put this information on the board by listing the number of people who earned a particular score. Suppose that you scored an 89 and that according to your instructor's chart, 24 people had lower scores than you did.
To calculate your percentile, substitute into the following formula:
(number of scores lower than yours)/(total number of scores) X 100.
In this example, the equation becomes 24/30 X 100. Performing the calculations gives an answer of 80 percent. In other words, 80 percent of people taking the test scored lower than you did. Note that the percentile ranking is different from the percentage of questions you answered correctly.
Determine whether is z-score is the best calculation to use by asking if the sample size is larger than 30 and if the data are normally distributed. The z-score is good to use when making comparisons of data in which different scales have been used. If you are trying to compare intelligence and you are given I.Q. scores, grade-point averages and reaction times, you may want to use a z-score to tell you how specific people rank relative to others.
Determine the particular score of interest, the mean of all scores and the standard deviation of all scores.
Put that information into the following equation: z = (individual score -- mean of all scores) standard deviation. As an example, if your child has an I.Q. score of 140, and you want to determine how that compares to other children, you would use the information that the average I.Q. score in the U.S. is 100 and the standard deviation is 15. Substituting into the equation gives (140 - 100)/15 = 40/15 = 2 2/3. Your child's I.Q. is 2 2/3 standard deviations above the mean. Generally, anything within two standard deviations of the mean is considered rather ordinary, but a score more than two standard deviations above or below the mean is considered unusual.
Determine whether or not to use the t-score by asking if the sample size is less than 30, if the data are normally distributed, and if the population standard deviation is unknown. If all of these cases apply, use the t-score.
Find the proper t-score using a table or statistical calculator. Suppose you have a sample size of 12 with a sample mean of 10, and want to find the interval within which the population mean is 95 percent likely to fall. Remember that the sample mean and the population mean are usually different. In this case, the degrees of freedom, 11, will be one less than the sample size, which is 12. Because we want a 95 percent confidence interval, we use a two-tailed distribution of 0.5. For this example, t = 2.201.
Compute the margin of error, E, by multiplying the t-score by the population mean and dividing by the square root of the sample size. In this example, the equation becomes: 2.201(10)/sqrt(12). Performing this calculation gives 6.35.
To find the confidence interval, use the following formula:
sample mean -- error < population mean < sample mean + error.
10 - 6.35 < population mean < 10 + 6.35
3.65 < population mean < 16.35