As we can see, a distribution is just the way of showing how many times each value occurs for a single variable. By ordering the values numerically, we get
Fig. 3-9. Truncated curve.
a sense of how larger and smaller values are more or less common. The
distributions above are sample distributions, because they show thefrequency
of each value that is actually found in our sample. If we were dealing with a population, particularly a population of unknown size or infinite in size, we wouldn’t know exactly how many subjects would have each value for the variable. This is where the notion of a random variable comes in. For a population, instead of the exact count of each value, we will need to know the
probability that each value will occur.
A theoretical distribution for a variable is a curve with all of the possible values of the variable along the x-axis and the relative probability of each value along the y-axis. The height of the curve of a theoretical distribution is re-scaled so that the total area under the curve is exactly equal to one. Together, these two features mean that the area under any part of the curve is exactly equal to the probability that the variable will have the values that appear within that part of the curve.
Theoretical distributions are built from mathematical equations that are designed to match the shapes of the real sample distributions of various kinds
of data. The most common formula is thenormaldistribution, which has the
true bell-curve shape. Figure 3-11 shows the standard normal curve, which has a central tendency of zero and whose variability is standardized to a value
of one using a unit calledsigma (symbolized by ‘‘’’).
The exact formula for the curve normal distribution is too complicated for us here, and we can manage just fine without it. It is important to understand that the normal curve is defined by its specific formula and that having an exact formula for the normal curve gives it specific features that are vital to understanding statistical inference, and therefore applies to all the
conclusions about consequences of business decisions drawn from statistics
(the subject of Part Four ofBusiness Statistics Demystified).
Because the normal curve has a specific shape, the proportion of the area under each part of the curve is always the same, no matter what the specific values of the population distribution are. In the standard normal curve, these
areas are marked out in units of sigma. If we know any two points on the
x-axis under the standard normal curve in terms of sigma, we know the probability that the variable will have a value between those two points.
By adjusting the standard normal curve to the central tendency and the variability of our data, we can use the shape of the normal curve to link values of a variable to the probability that those values will occur. We slide the curve to the right or left (as in Fig. 3-8) to match the central tendency of our data. Then we widen or narrow it (as in Fig. 3-9) to match the variability. Now we can link any value of the variable to the corresponding sigma value. Suppose we handle financial transactions for the auto parts industry. We know that the price per sale is normally distributed. Small sales and big sales are costly to handle, so we want to offer a special rate to new customers for all transactions between $200 and $2000. In order to set the special rate, we need to know what proportion of sales will fall between these two values. If we know the central tendency and the variability of the price per sale, we can use the shape of the standard normal curve to calculate the probability of a sale being covered by our special rate. Even without looking at sales records, we can set our special rate.
This is the genius of statistical inference. Using distribution curves, we can associate real values measured from the real world with probabilities, which help us answer questions about the likelihoods of events, including future events, important to us.
In fact, the general technique shown above is even more powerful than that. So long as we know the precise mathematical form of the curve of the distribution of the variable in the population, that distribution does not even have to follow the normal curve. There are dozens and dozens of mathematical formulas for other sorts of distributions. And, for each mathematical formula, there is a way of calculating a probability and associating it with every point along the number line.
The trick is that we can never know for sure what the population distribution is. All we can ever really know is the sample distribution. There
is a mathematical proof, called theLaw of Large Numbers, which assures us
that, so long as we take a big enough sample, the statistics for that sample will be close to the statistics for the population distribution. That means that we can look at the sample distribution and be reasonably confident that it is close to the population distribution. Then, if the sample distribution matches
the shape of the normal distribution, or some other mathematically understood distribution, we can use the technique above to make inferences. And the above approach is even more powerful than that. Even if the
sample distribution is so oddly shaped that it doesn’t matchanydistribution
we know of, there is a less powerful form of statistical inference, called
nonparametric statistical tests (discussed in Chapter 14 ‘‘Nonparametric Statistics’’) that works without the mathematical formula of the population distribution.
We can now see a little bit of how the logic of statistical inference relates to probability. The most important thing to know is that statistics works whether or not we understand statistical theory and probability theory. We need to understand the logic of statistical inference in order to do quality statistics. We only need to know statistical theory and probability theory if it is on the test.