Data Analysis with Microsoft Excel: Updated for Office 2007

Chapter 5 Probability Distributions 183

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p to now, you’ve used tools such as frequency tables, descriptive

statistics, and scatter plots to describe and summarize the proper-

ties of your data. Now you’ll learn about probability, which pro-

vides the foundation for understanding and interpreting these statistics.

You’ll also be introduced to statistical inference, which uses summary

statistics to help you reach conclusions about your data.

Probability

Much of science and mathematics is concerned with prediction. Some of

these predictions can be made with great precision. Drop an object, and the

laws of physics can predict how long the object will take to fall. Mix two

chemicals, and the laws of chemistry can predict the properties of the re-

sulting mixture. Other predictions can be made only in a general way. Flip

a coin, and you can predict that either a head or a tail will result, but you

cannot predict which one. That doesn’t mean that you can’t say anything. If

you fl ip the coin many times, you’ll notice that roughly half the fl ips result

in heads and half result in tails.

Flipping a coin is an example of a random phenomenon, in which

individual outcomes are uncertain but follow a general pattern of

occurrences.

When we study random phenomena, our goal is to quantify that general

pattern of occurrences in order to make general predictions. How do we do

this? One way is through theory. We imagine an ideal coin with two sides:

a head and a tail. Because this is an ideal coin, we assume that each side

is equally likely to occur during our coin fl ip. From this, we can defi ne the

theoretical probability for equally likely events:

Theoretical probability 5

Number of possible ways of obtaining the event

Total number of possible outcomes

In the coin-tossing example, there is one way to obtain a head and there

are two possible outcomes, so the theoretical probability of obtaining a head

is 1/2, or .5.

Another way of quantifying random phenomena is through observation.

For example to determine the probability of obtaining a head, we repeatedly

toss the coin. From our observations, we calculate the relative frequency of

tosses that result in heads, where

Relative frequency 5

Number of times an event occurs

Number of replications