QNT 275 Week 3 participation Essentials of Business Statistics, Ch. 6

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QNT 275 Week 3 participation Essentials of Business Statistics, Ch. 6
One of my Favorite Chapters! (Normal Distribution)

Team

This is one of my favorite aspects of statistics. It is called Normal Distribution and it is at the epicenter of statistical and hypothesis testing. I am going to explain Normal Distribution in more detail below and provide you with a lot of fun examples for you to walk through that illustrate how to find the Area of Curve which is directly associated to Normal Distribution:

What is Normal in the world of statistics?

You may be wondering this on a daily basis while in this class? Not really!

Under the normal curve, our distribution of scores is called the Empirical Rule and it is as follows:

The Robust Empirical Rule:

68% of all observations are within 1 standard deviation away from the mean

95% of all observations are within 1.96 standard deviations away from the mean

95.44% of all observations are within 2 standard deviations away from the mean

99% of all observations are within 2.57 standard deviations away from the mean

99.7% of all observations are within 3 standard deviations away from the mean

The normally distributed shaped bell curve is a purely theoretical concept in which the horizontal (X) axis represents all possible values of a variable (such as the Z-calculated). Examples include the Wisconsin Water table Z-score for the year 1960 (.11) that we covered in Week 2. You can see an this example under the Learning Activity called "The Z-score" . You can plot these points along the horizontal (X) axis of the bell curve and they will fall within a range. As you can see the 1960 Z-score of .11 is within 1 standard deviation of the mean (to the right).

The vertical (Y) axis of the scores tells us the probability of these values occurring. Look at the empirical rule, the Z-score for the year 1960 (.11) is within 68% of all observations being 1 standard deviation away from the mean. For the Z-calculated of -29, the scores are probably 99.99999999% of all observations. It is really hard to exactly determine the % of observations since we can go into infinity of Z-calculated and on the bell curve. Infinity of the Z-calculated will always be possible because our Bell Curve vertical (Y) axis NEVER touches our horizontal (X) axis. It just grows increasingly closer but the naked eye will never be able to distinguish a separation of the 2 axis. Hence, a Z-calculated of -29 is probably 99.9999999% of all scores away from the mean. It is impossible for me (or you) to determine this % since we do not have sophisticated software to tell us the exact % of all observations that this particular score falls under.

Normal distribution of the bell curve ties into our discussions about significance and non-significance for hypothesis testing which we will have next week. Essentially, we are determining if our observations are normal and if our observations are not normal. As you may recall, a relationship (of variables) is created by comparing our OBSERVED (behavior) calculation to our EXPECTED (behavior) critical.

Properties of the Normal Distribution are:

1.     The normal distribution curve is bell shaped.

2.     The mean (mode or median) is in the middle of the distribution of the bell curve (measurements of central tendency).

3.     Both the left side of the curve and the right side of the curve are equal and symmetric.

4.     The normal distribution curve has only one mode.

5.     The curve is continuous, the Y-axis (vertical) and the X-axis (horizontal) will never meet and your observed calculations can run into infinity. It will be possible to have observed scores be 99.9999999999% or greater of all observations being x amount of standard deviations away from the mean.

6.     For each value of X there is a value for Y.

7.     The total area under the curve (left side negative + right side positive) will always equal 1 or 100%

8.     The areas under the normally distributed bell curve follow the robust empirical rule

Now, let's determine the area under the bell curve that are particular Z-calculation or Z-score falls under. This means we want to know the % that our particular Z-score covers on the bell curve. I like to refer to it as the real estate the Z-score covers on the Bell curve.

As an example, say we calculated a Z-score of 2.34. Let us find the area between the mean (which is the middle of the curve and always denoted by a 0) Z = 0 and Z = 2.34. To determine the area under the bell curve go to your Normal Distribution Table located in your book (appendix). You need to go the Z-column and find 2.3 and then .04 in the top row, you should see .4904. This is where the column and the row meet in the table. The answer is .4904. Hence the area is .4904 or 49%.

This means that the area under the curve that the Z-score of 2.34 covers is 49%. Any questions?