The Normal Curveand Z-scores

Using the Normal Curve to Find Probabilities

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.

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/normalcurve.html

The Histogram and the Normal Curve

The Theoretical Normal Curve

(from http://www.music.miami.edu/research/statistics/normalcurve/images/normalCurve1.gif

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

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%

-5 -4 -3 -2 -1 0 1 2 3 4 5

68.26%

95.44%

99.72%

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

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).

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.

Using Appendix A (cont.)

b b

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

Using Appendix A (cont.)

( c) = areas beyond the Z score

c c

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

. . .

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%

Normal curve with z=1.67

95.25%

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

. . .

Using Appendix A (cont.) The area below Z = - 1.67 is .0475. Areas can be expressed as %: 4.75%.

Area = .0475 z= -1.67Scores

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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!

Finding Probabilities

If a distribution has: = 13 s = 4

What is the probability of randomly selecting a score of 19 or more?

X

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

. . .

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)?

To solve:

Available information: Xi = 70

= 60

S = 10

Formula: Z = (Xi – ) / S

= (70 – 60) /10

= +1.0

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.

-5 -4 -3 -2 -1 0 1 2 3 4 5

< Mean = 60

Area 34.13%> <Area 34.13%

< Z = +1.0

68.26%

Area 50%-------> <-------Area 50%

95.44%

99.72%

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?

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!

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?

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!

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?

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 classmate’s mark with respect to the rest of the class.

Probability:

Let’s say your classmate won’t show you the mark….

How can you make an informed guess about what your neighbour’s 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)?

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%.

The probability of your classmate having a

mark between 60 and 70% is .34 :

-5 -4 -3 -2 -1 0 1 2 3 4 5

< Mean = 60

Area 34.13%> <Area 34.13%

< Z = +1.0 (70%)

68.26%

Area 50%-------> <------Area 50%

95.44%

99.72%

Homework Questions:

Healey 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