Normal Distribution

The concept of a normal distribution curve is used frequently in statistics. In essence, a normal distribution curve results when a large number of random variables are observed in nature, and their values are plotted. While this "distribution" of values may take a variety of shapes, it is interesting to note that a very large number of occurrences observed in nature possess a frequency distribution which is approximately bell-shaped, or in the form of a normal distribution, as indicated in Figure 1.

Figure 1 Graph of a Normal Probability Distribution

The significance of a normal distribution existing in a series of measurements is two fold. First, it explains why such measurements tend to possess a normal distribution; and second, it provides a valid basis for statistical inference. Many estimators and decision makers that are used to make inferences about large numbers of data, are really sums or averages of those measurements. When these measurements are taken, especially if a large number of them exist, confidence can be gained in the values, if these values form a bell-shaped curve when plotted on a distribution basis.

Probability

If E_l is the number of heads, and E_Z is the number of tails, E₁/(E₁ + E_Z)is an experimental determination of the probability of heads resulting when a coin is flipped.

P(E₁) = nlN

By definition, the probability of an event must be greater than or equal to 0, and less than or equal to 1. In addition, the sum of the probabilities of all outcomes over the entire "event" must add to equal 1. For example, the probability of heads in a flip of a coin is 50%, the probability of tails is 50%. If we assume these are the only two possible outcomes, 50% + 50%, the two outcomes, equals 100%, or 1.

The concept of probability is used in statistics when considering the reliability of the data or the measuring device, or in the correctness of a decision. To have confidence in the values measured or decisions made, one must have an assurance that the probability is high of the measurement being true, or the decision being correct.

To calculate the probability of an event, the number of successes (s), and failures (f), must be determined. Once this is determined, the probability of the success can be calculated by:

where

s + f = n = number of tries. Example:

Using a die, what is the probability of rolling a three on the first try?

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Higher Concepts of Mathematics STATISTICS

Solution:

First, determine the number of possible outcomes. In this case, there are 6 possible outcomes. From the stated problem, the roll is a success only if a 3 is rolled. There is only 1 success outcome and 5 failures. Therefore,

In calculating probability, the probability of a series of independent events equals the product of probability of the individual events.

Example:

Using a die, what is the probability of rolling two 3s in a row?

Solution:

From the previous example, there is a 1/6 chance of rolling a three on a single throw. Therefore, the chance of rolling two threes is:

1/6 x 1/6 = 1/36

one in 36 tries.

Example:

An elementary game is played by rolling a die and drawing a ball from a bag containing 3 white and 7 black balls. The player wins whenever he rolls a number less than 4 and draws a black ball. What is the probability of winning in the first attempt?

Solution:

There are 3 successful outcomes for rolling less than a 4, (i.e. 1,2,3). The probability of rolling a 3 or less is:

3/(3+3) = 3/6 = 1/2 or 50%.

The probability of drawing a black ball is:

7/(7+3) = 7/10.

Therefore, the probability of both events happening at the same time is:

7/10 x 1/2 = 7/20.

Summary

The important information in this chapter is summarized below.