IM3.7.7 Standard Normal

Author:
00JonLind

Standardizing normal distributions with z-scores

When calculating probabilities associated with normal distributions, z-scores are used to standardize and compare scores from different distributions.

  • A z-score for a particular value measures the number of standard deviations away from the mean.
  • A positive z-score corresponds to a value that is above the mean, and a negative z-score corresponds to a value that is below the mean.
  • The letter z is used to represent a variable that has a standard normal distribution where the mean is 0 and standard deviation is 1.
  • The formula for calculating a z-score for any value x with standard deviation σ and mean μ is .
The U.S. Department of Agriculture (USDA), in its Official Food Plans (www.cnpp.usda.gov), states that the average cost of food for a 14- to 18-year-old male is $261.50 per month. Assume that the monthly food cost for a 14- to 18-year-old male is approximately normally distributed with a mean of $261.50 and a standard deviation of $16.25. The USDA also states that the average cost of food for a 14- to 18-year-old female is $215.20 per month. Assume that the monthly food cost for a 14- to 18-year-old female is approximately normally distributed with a mean of $215.20 and a standard deviation of $14.85. Use z-scores to compare Billy, who spends $270 a month on food, with Gina, who spends $250 a month on food.