How do you standardize a measurement?

What is the normal distribution and standardizing?

The normal distribution is a symmetric, bell-shaped probability curve completely defined by its mean and standard deviation. The mean sets the center of the bell, while the standard deviation controls its spread. Because the curve is symmetric, exactly 50 percent of all values lie below the mean and 50 percent lie above it.

When working with data, standardizing a measurement transforms a raw measurement, x, into a standard score, z, using the formula z = (x - mean) / standard deviation. Standardizing converts any normal distribution into the standard normal distribution, which has a mean of 0 and a standard deviation of 1. A z-score has no units; it simply tells you how many standard deviations your value sits away from the center.

Step-by-step method and worked example

To standardize any raw value, follow two strict steps in order: first, subtract the mean from the raw score; second, divide that difference by the standard deviation.

Suppose exam scores follow a normal distribution with a mean of 70 points and a standard deviation of 10 points. To find the z-score for a student who scored 85, subtract 70 from 85 to get 15. Then divide 15 by 10. The z-score is 1.5, meaning the score is 1.5 standard deviations above the class average.

You can also reverse the process to find an original measurement from a known z-score: multiply the z-score by the standard deviation, then add the mean. For adult heights with a mean of 170 cm and standard deviation of 8 cm, a z-score of -2 corresponds to (-2 * 8) + 170 = 154 cm.

Reading tables and avoiding frequent errors

A standard normal table gives the cumulative probability of scoring below a positive z-score. To find the probability of a value falling above that score, subtract the table value from 1. To find the probability between two scores, subtract the lower cumulative probability from the higher one.

Watch out for two common calculation pitfalls. First, always divide by the standard deviation, never the variance. The variance is the square of the standard deviation; dividing by it ruins the scale. Second, use symmetry for negative values: the area below -z equals the area above +z, which equals 1 minus the area below +z.