Math Portal
Introductory Statistics
Section 6.1 - The Standard Normal Distribution
The standard normal distribution, denoted Z, is the special normal distribution that has mean μ = 0 and standard deviation σ = 1. Due to its unique parameters, a point z along the horizontal axis of a standard normal distribution expresses the position of the value relative to the mean, with negative values lying to the left of the mean and positive values lying to the right of the mean. Furthermore, the z value is in terms of standard deviations. For instance, the z value -1 is one standard deviation below the mean, and the z value 1 is one standard deviation above the mean. Similarly, the z value -1.58 is 1.58 standard deviations below the mean, while the z value 2.45 is 2.45 standard deviations above the mean.
PROBLEM: Describe the location of the point z = -2.34 along the horizontal axis of a standard normal distribution in terms of standard deviations from the mean.
SOLUTION: The point z = -2.34 is 2.34 standard deviations below the mean.
PROBLEM: Describe the location of the point z = 0 along the horizontal axis of a standard normal distribution in terms of standard deviations from the mean.
SOLUTION: The point z = 0 is zero standard deviations from the mean; that is, its location is the mean.
PROBLEM: For the standard normal distribution, find the probability that z assumes a value between -3 and 3.
SOLUTION: Since the standard normal distribution has mean μ = 0 and standard deviation σ = 1,
P(-3 < Z < 3) = 0.997. or 99.7%
PROBLEM: For the standard normal distribution, find the probability that z assumes a value between -2 and 0.
SOLUTION: Since the standard normal distribution has mean μ = 0 and standard deviation σ = 1,
P(-2 < Z < 0) = 0.135 + 0.34 = 0.475 = 47.5%
The standard normal distribution, denoted Z, is the special normal distribution that has mean μ = 0 and standard deviation σ = 1. Due to its unique parameters, a point z along the horizontal axis of a standard normal distribution expresses the position of the value relative to the mean, with negative values lying to the left of the mean and positive values lying to the right of the mean. Furthermore, the z value is in terms of standard deviations. For instance, the z value -1 is one standard deviation below the mean, and the z value 1 is one standard deviation above the mean. Similarly, the z value -1.58 is 1.58 standard deviations below the mean, while the z value 2.45 is 2.45 standard deviations above the mean.
PROBLEM: Describe the location of the point z = -2.34 along the horizontal axis of a standard normal distribution in terms of standard deviations from the mean.
SOLUTION: The point z = -2.34 is 2.34 standard deviations below the mean.
PROBLEM: Describe the location of the point z = 0 along the horizontal axis of a standard normal distribution in terms of standard deviations from the mean.
SOLUTION: The point z = 0 is zero standard deviations from the mean; that is, its location is the mean.
PROBLEM: For the standard normal distribution, find the probability that z assumes a value between -3 and 3.
SOLUTION: Since the standard normal distribution has mean μ = 0 and standard deviation σ = 1,
P(-3 < Z < 3) = 0.997. or 99.7%
PROBLEM: For the standard normal distribution, find the probability that z assumes a value between -2 and 0.
SOLUTION: Since the standard normal distribution has mean μ = 0 and standard deviation σ = 1,
P(-2 < Z < 0) = 0.135 + 0.34 = 0.475 = 47.5%