Chapter 8a: Descriptive Statistics
8a.1 Introduction
8a.2 Kinds of Measurements
8a.3 Measures of Central Tendency
8a.4 Measures of Dispersion
8a.5 Displaying Data

8a.1 Introduction

One of the most frequent questions posed by mathematics students is "When will I ever really need to use this [expletive deleted] junk?" The answers vary, as do their relative truths. However, if we are talking about the part of mathematics called "Descriptive Statistics", and if we are part of modern society, then the answer is "Today, tomorrow, and almost every day. You will use this information, these skills, in daily conversations, in reading newspapers, magazines, and advertisements, in viewing television programs, and in browsing through databases and the Internet." In all of these situations we hear or read about "average" values, popular opinions, portions of people in various groups, trends, comparisons to historical values, and promises of change and improvement. This is the language of statistics.

Descriptive Statistics allow us to use a few standard values to characterize a set of data. We use these standard values to give meaning to the larger set of data. To appreciate these standard values we generally need to be looking at large a set of data. However, to teach students how to compute these values, we often give examples that use only a small set of data. This is unfortunate. Such small examples make our computations easy, but the small size hides the power and beauty of the real work that we are doing. For example, we might ask you to look at the set of numbers

{4, 7, 2, 1, 4, 6}
and to do some calculations to find certain values from that data. To just about anyone, this must seem like a waste of time. We can look at {4, 7, 2, 1, 4, 6} and get a great "feel" for the data. We do not need to compute values to characterize this set of numbers. It is small enough for us to comprehend it as it is. However, it is much harder to do the same for the set of values

[Caution: the previous set of values changes every time that this web page is loaded. Any effort that you make to get a feel for the numbers above will be useless when you reload the page and a new set of numbers appears.]

In a case such as the long list of values given above, with many numbers in the list, we need some agreed upon ways of describing the data with just a few values. For example, we would say that we have the following measures of central tendency for that data:

  1. The most frequent
  2. If we sort the values, then the middle value is (this is called the median value)
  3. If we add up the values and divide by the number of values in the list, then the quotient is (this is called the mean value).
  4. In addition, we say that we have the following measures of dispersion:

  5. The lowest number is (these give the range of values)
  6. 25% of the numbers are at or below , 50% are at or below , and 75% are at or below (these are the quartile points of the values)
  7. The standard deviation from the mean for the numbers in the list is

Remember that you can re-load this page and a new set of values will be displayed. This will give rise to new values for the measure of central tendency and for the measures of dispersion. This is an exercise worth doing a number of times so that you can see the changes in these standard measures.

Knowing these standard values characterizes our data set. We get a "feel" for the numbers in the list by knowing these computed values. We can even use these values if we want to compare our list with some other list. The list above had all of items in it. That was clearly enough items so that we can not get a good "feel" just by looking at the list of values. Imagine the case where there are thousands of items, or more, in the list of values. But even with a huge number of items, we can produce the few standard values introduced here to characterize the data set. However, before we even look at these standard values, we should take some time to examine the different "Kinds of Measurements" that we might find in a set of data.


Chapter 8a: Descriptive Statistics
8a.1 Introduction
8a.2 Kinds of Measurements
8a.3 Measures of Central Tendency
8a.4 Measures of Dispersion
8a.5 Displaying Data

©Roger M. Palay
Saline, MI 48176
November, 2010