Normal Distribution: Z-scores 1.5 2008/06/06 14:34:50 GMT-5 2008/07/21 03:07:26.246 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org elementary statistics If X is a normally distributed random variable and X~N(μ, σ), then the z-score is: z = x - μ σ The z-score tells you how many standard deviations that the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z-scores and values of x that are smaller than the mean have negative z-scores.Suppose X ~ N(5, 6). This says that X is a normally distributed random variable with mean μ = 5 and standard deviation σ = 6. Suppose x = 17. Then: z = x - μ σ = 17 - 5 6 = 2 This means that x = 17 is 2 standard deviations (2σ) above or to the right of the mean μ = 5. The standard deviation is σ = 6.Notice that: 5 + 2 ⋅ 6 = 17 (The pattern is μ + z σ = x . ) Now suppose x=1. Then: z = x - μ σ = 1 - 5 6 = - 0.67 (rounded to two decimal places) This means that x = 1 is 0.67 standard deviations (- 0.67σ) below or to the left of the mean μ = 5. Notice that: 5 + ( -0.67 ) ( 6 ) is approximately equal to 1 (This has the pattern μ + ( -0.67 ) σ = 1 ) Summarizing, when z is positive, x is above or to the right of μ and when z is negative, x is to the left of or below μ. Some doctors believe that a person can lose 5 pounds, on the average, in a month by reducing his/her fat intake and by exercising consistently. Suppose weight loss has a normal distribution. Let X = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of 2 pounds. X~N(5, 2). Fill in the blanks. Suppose a person lost 10 pounds in a month. The z-score when x = 10 pounds is z = 2.5 (verify). This z-score tells you that x = 10 is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?). This z-score tells you that x = 10 is 2.5 standard deviations to the right of the mean 5. Suppose a person gained 3 pounds (a negative weight loss). Then z = __________. This z-score tells you that x = -3 is ________ standard deviations to the __________ (right or left) of the mean. z = -4. This z-score tells you that x = -3 is 4 standard deviations to the left of the mean. Suppose the random variables X and Y have the following normal distributions: X ~ N(5, 6) and Y ~ N(2, 1). If x = 17, then z = 2. (This was previously shown.) If y = 4, what is z? z = y - μ σ = 4 - 2 1 = 2 where μ=2 and σ=1. The z-score for y = 4 is z = 2. This means that 4 is z = 2 standard deviations to the right of the mean. Therefore, x = 17 and y = 4 are both 2 (of their) standard deviations to the right of their respective means.The z-score allows us to compare data that are scaled differently. To understand the concept, suppose X ~ N(5, 6) represents weight gains for one group of people who are trying to gain weight in a 6 week period and Y ~ N(2, 1) measures the same weight gain for a second group of people. A negative weight gain would be a weight loss. Since x = 17 and y = 4 are each 2 standard deviations to the right of their means, they represent the same weight gain in relationship to their means.