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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. If x equals the mean, then x has a z-score of 0 .

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 anormal 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 relative to their means .

The empirical rule

If X is a random variable and has a normal distribution with mean µ and standard deviation σ then the Empirical Rule says (See the figure below)
  • About 68.27% of the x values lie between -1 σ and +1 σ of the mean µ (within 1 standard deviation of the mean).
  • About 95.45% of the x values lie between -2 σ and +2 σ of the mean µ (within 2 standard deviations of the mean).
  • About 99.73% of the x values lie between -3 σ and +3 σ of the mean µ (within 3 standard deviations of the mean). Notice that almost all the x values lie within 3 standard deviations of the mean.
  • The z-scores for +1 σ and –1 σ are +1 and -1, respectively.
  • The z-scores for +2 σ and –2 σ are +2 and -2, respectively.
  • The z-scores for +3 σ and –3 σ are +3 and -3 respectively.
Empirical Rule
The Empirical Rule is also known as the 68-95-99.7 Rule.

Suppose X has a normal distribution with mean 50 and standard deviation 6.

  • About 68.27% of the x values lie between -1 σ = (-1)(6) = -6 and 1 σ = (1)(6) = 6 of the mean 50. The values 50 - 6 = 44 and 50 + 6 = 56 are within 1 standard deviation of the mean 50. The z-scores are -1 and +1 for 44 and 56, respectively.
  • About 95.45% of the x values lie between -2 σ = (-2)(6) = -12 and 2 σ = (2)(6) = 12 of the mean 50. The values 50 - 12 = 38 and 50 + 12 = 62 are within 2 standard deviations of the mean 50. The z-scores are -2 and 2 for 38 and 62, respectively.
  • About 99.73% of the x values lie between -3 σ = (-3)(6) = -18 and 3 σ = (3)(6) = 18 of the mean 50. The values 50 - 18 = 32 and 50 + 18 = 68 are within 3 standard deviations of the mean 50. The z-scores are -3 and +3 for 32 and 68, respectively.

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Source:  OpenStax, Principles of business statistics. OpenStax CNX. Aug 05, 2009 Download for free at http://cnx.org/content/col10874/1.5
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