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Understanding the Standard Normal Distribution: Areas and Z-scores

This chapter focuses on the standard normal distribution, characterized by a mean of 0 and a standard deviation of 1. Learn how to calculate the area under the standard normal curve, find Z-scores for given areas, and interpret these areas as probabilities. Explore methods to compute the area, including using statistical tables, technology like Excel, and software like StatCrunch. Three area calculations are discussed: to the left, to the right, and between Z-scores, providing a comprehensive understanding of standard normal distributions.

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Understanding the Standard Normal Distribution: Areas and Z-scores

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  1. Chapter 7Section 2 The Standard Normal Distribution

  2. 1 2 3 Chapter 7 – Section 2 • Learning objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

  3. 1 2 3 Chapter 7 – Section 2 • Learning objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

  4. Chapter 7 – Section 2 • The standard normal curve is the one with mean μ = 0 and standard deviation σ = 1 • We have related the general normal random variable to the standard normal random variable through the Z-score • In this section, we discuss how to compute with the standard normal random variable

  5. Chapter 7 – Section 2 • There are several ways to calculate the area under the standard normal curve • What does not work – some kind of a simple formula • We can use a table (such as Table IV on the inside back cover) • We can use technology (a calculator or software) • Using technology is preferred

  6. Chapter 7 – Section 2 • Three different area calculations • Find the area to the left of • Find the area to the right of • Find the area between • Three different area calculations • Find the area to the left of • Find the area to the right of • Find the area between • Three different methods shown here • From a table • Using Excel • Using StatCrunch

  7. Chapter 7 – Section 2 • "To the left of" – using a table • Calculate the area to the left of Z = 1.68 • "To the left of" – using a table • Calculate the area to the left of Z = 1.68 • Break up 1.68 as 1.6 + .08 • "To the left of" – using a table • Calculate the area to the left of Z = 1.68 • Break up 1.68 as 1.6 + .08 • Find the row 1.6 • "To the left of" – using a table • Calculate the area to the left of Z = 1.68 • Break up 1.68 as 1.6 + .08 • Find the row 1.6 • Find the column .08 • "To the left of" – using a table • Calculate the area to the left of Z = 1.68 • Break up 1.68 as 1.6 + .08 • Find the row 1.6 • Find the column .08 • The probability is 0.9535 Enter Enter Enter Enter Read

  8. Chapter 7 – Section 2 • "To the left of" – using Excel • The function in Excel for the standard normal is =NORMSDIST(Z-score) • NORM (normal) S (standard) DIST (distribution) Read Enter Enter Read

  9. Chapter 7 – Section 2 • To the left of 1.68 – using StatCrunch • The function is Stat – Calculators – Normal Read Enter Enter

  10. Chapter 7 – Section 2 • "To the right of" – using a table • The area to the left of Z = 1.68 is 0.9535 • "To the right of" – using a table • The area to the left of Z = 1.68 is 0.9535 • The right of … that’s the remaining amount • The two add up to 1, so the right of is 1 – 0.9535 = 0.0465 Enter Enter Read

  11. Chapter 7 – Section 2 • "To the right of" – using Excel • "To the right of" – using Excel • The right of … that’s the remaining amount of to the left of, subtract from 1 1 – 0.9535 = 0.0465 Read Enter Read

  12. "To the right of" – using StatCrunch • Change the <= to >= (the picture and number change) Read Enter Chapter 7 – Section 2 • "To the right of" – using StatCrunch • "To the right of" – using StatCrunch • Change the <= to >= Read Enter

  13. Chapter 7 – Section 2 • “Between” • Between Z = – 0.51 and Z = 1.87 • This is not a one step calculation

  14. Included too much Chapter 7 – Section 2 • The left hand picture … to the left of 1.87 … includes too much • It is too much by the right hand picture … to the left of -0.51

  15. We want We start out with, but it’s too much We correct by Chapter 7 – Section 2 • Between Z = – 0.51 and Z = 1.87

  16. Chapter 7 – Section 2 • We can use any of the three methods to compute the normal probabilities to get • The area between -0.51 and 1.87 • We can use any of the three methods to compute the normal probabilities to get • The area between -0.51 and 1.87 • The area to the left of 1.87, or 0.9693 … minus • We can use any of the three methods to compute the normal probabilities to get • The area between -0.51 and 1.87 • The area to the left of 1.87, or 0.9693 … minus • The area to the left of -0.51, or 0.3050 … which equals • We can use any of the three methods to compute the normal probabilities to get • The area between -0.51 and 1.87 • The area to the left of 1.87, or 0.9693 … minus • The area to the left of -0.51, or 0.3050 … which equals • The difference of 0.6643 • We can use any of the three methods to compute the normal probabilities to get • The area between -0.51 and 1.87 • The area to the left of 1.87, or 0.9693 … minus • The area to the left of -0.51, or 0.3050 … which equals • The difference of 0.6643 • Thus the area under the standard normal curve between -0.51 and 1.87 is 0.6643

  17. We want We delete the extra on the left We delete the extra on the right Chapter 7 – Section 2 • A different way for “between”

  18. Chapter 7 – Section 2 • Again, we can use any of the three methods to compute the normal probabilities to get • Again, we can use any of the three methods to compute the normal probabilities to get • The area between -0.51 and 1.87 • The area to the left of -0.51, or 0.3050 … plus • The area to the right of 1.87, or .0307 … which equals • The total area to get rid of which equals 0.3357 • Again, we can use any of the three methods to compute the normal probabilities to get • The area between -0.51 and 1.87 • The area to the left of -0.51, or 0.3050 … plus • The area to the right of 1.87, or .0307 … which equals • The total area to get rid of which equals 0.3357 • Thus the area under the standard normal curve between -0.51 and 1.87 is 1 – 0.3357 = 0.6643

  19. 1 2 3 Chapter 7 – Section 2 • Learning objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

  20. Chapter 7 – Section 2 • We did the problem: Z-Score  Area • Now we will do the reverse of that Area  Z-Score • We did the problem: Z-Score  Area • Now we will do the reverse of that Area  Z-Score • This is finding the Z-score (value) that corresponds to a specified area (percentile) • And … no surprise … we can do this with a table, with Excel, with StatCrunch, with …

  21. “To the left of” – using a table • Find the Z-score for which the area to the left of it is 0.32 • Look in the middle of the table … find 0.32 • “To the left of” – using a table • Find the Z-score for which the area to the left of it is 0.32 • Look in the middle of the table … find 0.32 • The nearest to 0.32 is 0.3192 … a Z-Score of -.47 Read Find Read Chapter 7 – Section 2 • “To the left of” – using a table • Find the Z-score for which the area to the left of it is 0.32

  22. Read Enter Read Chapter 7 – Section 2 • "To the left of" – using Excel • The function in Excel for the standard normal is =NORMSINV(probability) • NORM (normal) S (standard) INV (inverse)

  23. Enter Read Chapter 7 – Section 2 • "To the left of" – using StatCrunch • The function is Stat – Calculators – Normal

  24. Read Read Enter Chapter 7 – Section 2 • "To the right of" – using a table • Find the Z-score for which the area to the right of it is 0.4332 • Right of it is .4332 … left of it would be .5668 • A value of .17

  25. Read Enter Read Chapter 7 – Section 2 • "To the right of" – using Excel • To the right is .4332 … to the left would be .5668 (the same as for the table)

  26. Chapter 7 – Section 2 • "To the right of" – using StatCrunch • Change the <= to >= (watch the picture change) Enter Read

  27. Chapter 7 – Section 2 • We will often want to find a middle range, to find the middle 90% or the middle 95% or the middle 99%, of the standard normal • The middle 90% would be

  28. Chapter 7 – Section 2 • 90% in the middle is 10% outside the middle, i.e. 5% off each end • These problems can be solved in either of two equivalent ways • We could find • The number for which 5% is to the left, or • The number for which 5% is to the right

  29. Chapter 7 – Section 2 • The two possible ways • The number for which 5% is to the left, or • The number for which 5% is to the right 5% is to the right 5% is to the left

  30. Chapter 7 – Section 2 • The number zα is the Z-score such that the area to the right of zα is α • The number zα is the Z-score such that the area to the right of zα is α • Some useful values are • z.10 = 1.28, the area between -1.28 and 1.28 is 0.80 • z.05 = 1.64, the area between -1.64 and 1.64 is 0.90 • z.025 = 1.96, the area between -1.96 and 1.96 is 0.95 • z.01 = 2.33, the area between -2.33 and 2.33 is 0.98 • z.005 = 2.58, the area between -2.58 and 2.58 is 0.99

  31. 1 2 3 Chapter 7 – Section 2 • Learning objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

  32. Chapter 7 – Section 2 • The area under a normal curve can be interpreted as a probability • The standard normal curve can be interpreted as a probability density function • The area under a normal curve can be interpreted as a probability • The standard normal curve can be interpreted as a probability density function • We will use Z to represent a standard normal random variable, so it has probabilities such as • P(a < Z < b) • P(Z < a) • P(Z > a)

  33. Summary: Chapter 7 – Section 2 • Calculations for the standard normal curve can be done using tables or using technology • One can calculate the area under the standard normal curve, to the left of or to the right of each Z-score • One can calculate the Z-score so that the area to the left of it or to the right of it is a certain value • Areas and probabilities are two different representations of the same concept

  34. Example: Chapter 7 – Section 2 • Determine the area under the standard normal curve that lies • a. to the left of Z = –2.31. (0.0104) • b. to the right of Z = –1.47. (0.9292) • c. between Z = –2.31 and Z = 0. (0.4896) • d. between Z = –2.31 and Z = –1.47. (0.0603) • e. between Z = 1.47 and Z = 2.31. (0.0603) • f. between Z = –2.31 and Z = 1.47. (0.9188) • g. to the left of Z = –2.31 or to the right of Z = 1.47. (0.0812)

  35. Example: Chapter 7 – Section 2 • The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. The Department of Molecular Genetics at Ohio State University requires a GRE score no less than the 60th percentile. (Source: www.biosci.ohio-state.edu/~molgen/html/admission_criteria.html.) • a. Find the Z-score corresponding to the 60th percentile. In other words, find the Z-score such that the area under the standard normal curve to the left is 0.60. (0.25) • b. How many standard deviations above the mean is the 60th percentile? (0.25)

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