The Normal Probability Distribution and Z-scores

Using the Normal Curve to Find Probabilities

Carl Gauss

  The normal probability distribution or 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

       

Properties of the Normal Distribution:

Also called Bell Curve or Gaussian Curve Perfectly symmetrical normal distribution The mean ( µ) 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 ( σ) from the mean (also called

Z).

Total area under the curve is an area of 1.00

The Theoretical Normal Curve

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

Properties (cont.)

  Has a mean µ = 0 and standard deviation σ = 1.

General relationships: ±1 σ = about 68.26% ±2 σ = about 95.44% ±3 σ = about 99.72%* *Also, when z= ±1 then p=.68, when z=±2, p=.95, and when z=±3, p=.997

68.26% -5 -4 -3 -2 -1 95.44% 99.72% 0 1 2 3 4 5

Using Table A to Find the Area Beyond Z

   Table A (p. 668) can be used to find the area beyond the part of the curve cut off by the z-score.

This is the probability associated with the area of the curve beyond ±z.

The probability of falling within z standard deviations of the mean is found by subtracting the area beyond from .5000 for both the positive and negative sides of the curve

or

by doubling the area beyond z and subtracting it from a total area of 1.00.

Using Table A (cont.)

( p) = area beyond the Z score This area is the probability associated with your ±z-score p p

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 which is known as a probability.

Formula:

z



y

   

y i



y s

 The formula changes a “raw” score (y i ) to a standardized score (Z). Table A can then be used to determine the area (or the probability) beyond z, the area between the mean and z, or the area below z.

Using Table A (cont.)

 For instance, if you calculate

Z

area beyond

Z

to be +1.67, the = 1.67 would be .0475. The probability of a score lying beyond z = 1.67 would be p = .0475

 The area between the mean and

Z

would be .5000 - .0475 = .4525. Therefore the total probability associated with finding a score between the mean ( µ) and

Z

is p = .4575.

Probability of any score less than Z=1.67

This can be found by subtracting the area beyond Z = 1.67 from the total curve area of 1.00 (p=.9525) .9525

Using Table A (cont.)

 The area below Z = - 1.67 is .0475.

 Probabilities can be expressed as %: 4.75%.

Area = .0475

z= -1.67

Scores

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

 1.

2.

3.

4.

If a distribution has a mean of 13 and a standard deviation of 4, what is the probability of randomly selecting a score of 19 or more?

Find the Z score.

For y i = 19, Z = 1.50.

Find area above in Table A.

Probability is 0.0668 or 0.07.

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?

What percentile is your mark at?

Your Z-score of +1.0 is exactly 1 σ above the mean. The probability beyond z is .1587 and below z is .3413+.5000 = .8413.

This also means 16% of scores are above yours and 84% are below.

Your mark is at the 84 th percentile.

< Mean = 60 Area .3413>

< Z = +1.0

.

6826 Area .5000-------> <-------Area .5000

-5 -4 -3 -2 -1

.9972

0 1 2 3 4 5

Another Question…

 Suppose you have 72% and your classmate has 55%.  What is the probability of someone having a mark between your score and your classmate’s?

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

 The probability of having a mark between 72 and 55 for this distribution is p = .5764 or 58%

Probability:

 What is the probability of having a mark between 60% and 70%?

The probability of having a mark between 60 and 70% is .3413

:

-5

< Mean = 60% Area .3413>

< Z = +1.0 (70%) .6826

-4

Area .5000------> <------Area .5000

.9544

-3 -2 -1

.9972

0 1 2 3 4 5

Other points:

 Note that when:   z = 1.64, p = .05

z = 2.33, p = .01

z = 3.10, p = .001

 When the Normal Probability Distribution is used in inferential statistics, it is usually referred to as the Standard Normal Distribution. It has a mean µ = 0 and an s.d. σ = 1 

Try P. 112, #7(a-d) and #9(a-f).