The Normal Curve and Z-scores

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The Normal Curveand Z-scores

• Using the Normal Curve to Find Probabilities

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Outline

• 1. Properties of the normal curve.
• 2. Mean and standard deviation of the normal
  curve.
• 3. Area under the curve.
• 4. Calculating z-scores.
• 5. Probability.

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CarlGauss

• The normal curve is often called the Gaussian
  distribution, after Carl Friedrich Gauss, who
  discovered many of its properties.  Gauss,
  commonly viewed as one of the greatest
  mathematicians of all time (if not the greatest),
  is honoured by Germany on their 10 Deutschmark
  bill.
• From http//www.willamette.edu/mjaneba/help/norma
  lcurve.html

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The Histogram and the Normal Curve
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The Theoretical Normal Curve

• (from http//www.music.miami.edu/research/statisti
  cs/normalcurve/images/normalCurve1.gif

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Properties of the Normal Curve

• Theoretical construction
• Also called Bell Curve or Gaussian Curve
• Perfectly symmetrical normal distribution
• The mean of a distribution is the midpoint of the
  curve
• The tails of the curve are infinite
• Mean of the curve median mode
• The area under the curve is measured in
  standard deviations from the mean

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Properties (cont.)

• Has a mean 0 and standard deviation 1.
• General relationships 1 s about 68.26
• 2 s about 95.44
• 3 s about 99.72

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Z-Scores

• Are a way of determining the position of a single
  score under the normal curve.
• Measured in standard deviations relative to the
  mean of the curve.
• The Z-score can be used to determine an area
  under the curve known as a probability.
• Formula Z (Xi ) / S

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Using the Normal Curve Z Scores

• Procedure
• To find areas, first compute Z scores.
• Substitute score of interest for Xi
• Use sample mean for and sample standard
  deviation for S.
• The formula changes a raw score (Xi) to a
  standardized score (Z).

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Using Appendix A to Find Areas Below a Score

• Appendix A can be used to find the areas above
  and below a score.
• First compute the Z score, taking careful note of
  the sign of the score.
• Make a rough sketch of the normal curve and shade
  in the area in which you are interested.

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Using Appendix A (cont.)

• Appendix A has three columns (a), (b), and (c)
• (a) Z scores.
• (b) areas between the score and the mean

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Using Appendix A (cont.)

• ( c) areas beyond the Z score

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Using Appendix A

• Suppose you calculate a Z-score 1.67
• Find the Z-score in Column A.
• To find area below a positive score
• Add column b area to .50.
• To find area above a positive score
• Look in column c.

(a) (b) (c)
. . .
1.66 0.4515 0.0485
1.67 0.4525 0.0475
1.68 0.4535 0.0465
. . .
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Using Appendix A (cont.)

• The area below Z 1.67 is
• 0.4525 0.5000 0.9525.
• Areas can be expressed as percentages
• 0.9525 95.25

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Normal curve with z1.67
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Using Appendix A

• What if the Z score is negative (1.67)?
• To find area below a negative score
• Look in column c.
• To find area above a negative score
• Add column b .50

(a) (b) (c)
. . .
1.66 0.4515 0.0485
1.67 0.4525 0.0475
1.68 0.4535 0.0465
. . .
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Using Appendix A (cont.)

• The area below Z - 1.67 is .0475.
• Areas can be expressed as 4.75.
• Area .0475 z -1.67

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Finding Probabilities

• Areas under the curve can also be expressed as
  probabilities.
• Probabilities are proportions and range from 0.00
  to 1.00.
• The higher the value, the greater the probability
  (the more likely the event). For instance, a .95
  probability of rain is higher than a .05
  probability that it will rain!

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Finding Probabilities

• If a distribution has
• 13
• s 4
• What is the probability of randomly selecting a
  score of 19 or more?

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Finding Probabilities

1. Find the Z score.
2. For Xi 19, Z 1.50.
3. Find area above in column c.
4. Probability is 0.0668 or 0.07.

(a) (b) (c)
. . .
1.49 0.4319 0.0681
1.50 0.4332 0.0668
1.51 0.4345 0.0655
. . .
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In Class Example

• After an exam, you learn that the mean for the
  class is 60, with a standard deviation of 10.
  Suppose your exam score is 70. What is your
  Z-score?
• Where, relative to the mean, does your score lie?
• What is the probability associated with your
  score (use Z table Appendix A)?

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To solve

• Available information Xi 70
• 60
• S 10
• Formula Z (Xi ) / S
• (70 60) /10
• 1.0

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Your Z-score of 1.0 is exactly 1 s.d. above the
mean (an area of 34.13 50 ) You are at the
84.13 percentile.
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What if your score is 72?

• Calculate your Z-score.
• What percentage of students have a score below
  your score? Above?
• What percentile are you at?

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Answer

• Z 1.2
• The area beyond Z .1151
• (11.51 of marks are above yours)
• Area between mean and Z .3849 .50 .8849
• ( of marks below 88.49)
• Your mark is at the 88th percentile!

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What if your mark is 55?

• Calculate your Z-score.
• What percentage of students have a score below
  your score? Above?
• What percentile are you at?

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Answer

• Z -.5
• Area between the mean and Z .1915 .50 .6915
• ( of marks above 69.15)
• The area beyond Z .3085
• (30.85 of the marks are below yours)
• Your mark is only at the 31st percentile!

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Another Question

• What if you want to know how much better or worse
  you did than someone else? Suppose you have 72
  and your classmate has 55?
• How much better is your score?

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Answer

• Z for 72 1.2 or .3849 of area above mean
• Z for 55 -.5 or .1915 of area below mean
• Area between Z 1.2 and Z -.5 would be .3849
  .1915 .5764
• Your mark is 57.64 better than your classmates
  mark with respect to the rest of the class.

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Probability

• Lets say your classmate wont show you the
  mark.
• How can you make an informed guess about what
  your neighbours mark might be?
• What is the probability that your classmate has a
  mark between 60 (the mean) and 70 (1 s.d. above
  the mean)?

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Answer

• Calculate Z for 70......Z 1.0
• In looking at Appendix A, you see that the area
  between the mean and Z is .3413
• There is a .34 probability (or 34 chance) that
  your classmate has a mark between 60 and 70.

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The probability of your classmate having a mark
between 60 and 70 is .34
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Homework Questions

• Healey 8e and 1st Cdn 5.1, 5.3, 5.5
• Healey 2nd Cdn 4.1, 4.3, 4.5
• Answers can be found at back of text
• Read SPSS section at end of chapter