General

DESCRIPTIVE STATISTICS

© James A. Geffert 2004

The purpose of this part of this week's lecture is to briefly review some things you have probably studied over and over but have forgotten. Well, maybe not forgotten, but have shoved into the back of your memory banks. These measures are generally termed "descriptive statistics", and are used to give more precise meaning to observations which might otherwise be stated as, "a whole bunch of 'em right over there."

Every collection of things that we measure exhibits two characteristics that interest us. First, things cluster together. Most things of interest are more alike than they are different. We call this characteristic "central tendency". The peaches I'm now picking from the tree in my back yard all look pretty much the same. It's easy to distinguish them from apples, say, or basketballs, or plums. They are all about "peach size", all have a color shading from a light green to a nice peachy color. They all weigh about the same amount, and have fuzz. It is the "alikeness" feature that we are measuring when we talk of "central tendency."

Second, even though peaches are more alike than they are different, each individual peach differs from each other in some way, however small. That is, they exhibit "variability." When we begin to measure the weight of individual peaches, or their firmness or their color, we find them different from one another. It is the differences we are talking about when we speak of "variability."

This brings us to the notion of "levels of measurement." Observations of the world can be classified by level of measurement. There are four levels of measurement which concern us in business data. These are: nominal, ordinal, interval and ratio.

Depending on what characteristic of my peaches we choose to measure, we will use different levels of measurement. The crudest measure is "nominal". That is, in my back yard there are a number of different things. If we list the "things" in my yard and start to count them, we come up with six trees, nine shrubs, a lot of ripe peaches, six pieces of furniture, five pieces of swimming pool equipment ..... Notice that what I've done is to count things classified by their name, hence a nominal level of measurement. The nominal level just names thing which are then counted.

Sometimes things which have the same name can be classified in a nominal way. For example, there are peaches still on the tree, peaches on the ground, peaches ripening on the table. Three nominal classifications.

The categories are mutually exclusive. No peach can be on the table and on the tree at the same time. Secondly, the categories are exhaustive. Every individual peach must appear in a category. There are no peaches in transit from the tree to the ground.

When we start to look at things with the same name we can begin to classify by an ordinal measure. Let's look at the peaches on the table. Further, let us rank them by

degree of ripeness. This level of measurement is ordinal. By comparing two, I can rank

them by which is riper, but it makes no sense to say, "Peach A is 3.5 times as ripe as

peach B." By the same kind of ordinal measure I can classify peaches by color - from dark green to dark orange. But again, it isn't sensible to say, "Peach A is two thirds as green as peach B."

The properties of an ordinal measure are those of the nominal measure with the addition of being ranked by some particular attribute. The ordinal measure is thus mutually exclusive, exhaustive, and ranked or ordered.

The third level of data, or level of measurement, is an interval level. The best example of

an interval level of measurement that I can think of is temperature. Consider the temperature in my back yard here in Las Vegas measured by degrees Fahrenheit. In the morning, at sunrise, it's about 80 degrees. By noon it's