Question

Standardise some distribution to standard normal distribution

Answer 1

Suppose you have a normally distributed random variable and would like to calculate the probability of its value occurring in the interval of the mean plus or minus .5 standard deviations. How would you go about calculating the probability?

One method is to standardize the random variable. Standardization of a normally distributed random variable enables an analyst or researcher to determine with ease the probability associated with a range of values for that variable by using a standardized distribution table. A normally distributed random variable can be standardized using a formula. The letter Z represents the standardized random variable and the probabilities associated with ranges of values of Z can be found in a Z distribution table. Read on to learn more about how to calcultate (and apply) the standardized normally distributed random variable Z.

Standard Normal Distribution

The standardized value of a normally distributed random variable is called a Z score and is calculated using the following formula.

x = the value that is being standardized

m = the mean of the distribution

s = standard deviation of the distribution

As the formula shows, a random variable is standardized by subtracting the mean of the distribution from the value being standardized, and then dividing this difference by the standard deviation of the distribution. Once standardized, a normally distributed random variable has a mean of zero and a standard deviation of one. The standard normal distribution (Z distribution) is shown in the graph below. As you can see from the notation to the right of the curve, mz = 0 and sz = 1.

Learn How to Calculate the Mean of a Standard Normal Distribution

To see how a random variable is standardized, imagine you have a random variable, x, that is normally distributed with a mean of 20 and a standard deviation of 10. What would be the standardized value (Z score) of 40? To solve this problem, you would use the Z score formula. Using the information from the example, you know that

x = 40

m = 20

s = 10

Substitute these numbers into the formula and solve.