Any percentile can be converted into its corresponding z score by using the z score table. However, since this conversion from percentile to z score is important when doing z score and T score problems, a percentile table has been made.
Next to each percentile is its corresponding z score. Notice that all percentiles in the first two columns have a negative z score. This reinforces the fact that z scores that fall below the mean, or the 50th percentile, are negative. In the last two percentile columns, from the 51st percentile and up, are all positive z scores.
Finding Raw Data Values Using Percentiles and the z Score Equation
When a percentile is known, the corresponding raw data value can also be determined by finding the corresponding z score to the given percentile and then using the equation x = zSD + M.
Although percentiles are easy to understand, they can lead to some distortions. This is because percentiles tend to compress the tails of the distribution, making extreme data values closer together than they actually are.
For example, a male adult height distribution has a mean of 69 inches and a standard deviation of 3.85. Comparing two heights, one at the 50th percentile and the other at the 54th percentile, a difference of 4 percentage points, the two corresponding z scores are found to be 0 and 0.10, respectively. Then the corresponding data values are calculated.
x = zSD + M = (0)(3.85) + 69 = 69
x = zSD + M = (0.10)(3.85) + 69 = 69.39
The difference between the two data values is 0.39 inches. Therefore, between the 50th and 54th percentile (percentiles in the center of the distribution), the difference in height is only 0.39 inches.
Now two new data values are compared. One at the 95th percentile and the other at the 99th percentile, which also differ by 4 percentage points, however both percentiles are at the end of the distribution (in this case, the high end) instead of at the center of the distribution.
The z scores for the 95th percentile and the 99th percentile is 1.65 and 2.41, respectively. The data values for each percentile is then calculated.
x = zSD + M = (1.65)(3.85) + 69 = 75.35
x = zSD + M = (2.41)(3.85) + 69 = 78.28
The difference between the two percentiles is approximately 3 inches, unlike the difference of 0.39 inches, despite both percentile pairs having the same difference of percentage points of 4.
Therefore, the difference between two percentiles in the center of the distribution is smaller than the difference between two percentiles at any end of the distribution.
Finding the Standard Deviation Using the z Score Equation
The z score equation can also be rewritten to find the standard deviation.
Therefore, if a raw data value, the mean, and the corresponding z score are known, then the standard deviation can be determined. The standard deviation can also be determined if a percentile is given instead of the z score.
Note that, although the z score and the raw data value can be negative, the standard deviation can never be negative.
The Range
Since the standard deviation can be found using the z score equation, the range can be approximated, because for normal distributions, the range is approximately six times the standard deviation.
Additionally, with the mean and the range known, the lowest value and the highest value of the distribution can be determined. Since the distribution is normal, one-half of the range falls above and below the mean.
To find the lowest value of the distribution, divide the range by two and subtract this quotient from the mean.
To find the highest value of the distribution, divide the range by two and add this quotient to the mean.
Finding the Mean Using the z Score Equation
The z score equation can also be rewritten so that the mean can be determined if the standard deviation, a raw data value, and the corresponding z score (or percentile) are known.
Since the mean, the median, and the mode all fall at the same point under the normal curve, the median and the mode are equal to the mean.
