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.2 Kinds of Measurements

Our raw material in this chapter is the measurements that are taken to create our set of data values. We might measure the height of people, the weight of cars, the temperature on a roof, the age of pets, the number of cars in different aisles in the parking lot, the number of vehicles from each manufacturer in each aisle in the parking lot, or the opinions of students to questions on a course evaluation survey. All of these are measures, but they are not all the same kind of measure.

There are four kinds of measurements: nominal, ordinal, interval, and ratio. The differences in these kinds of measures are important.

Nominal Measurements assign numbers merely as names. We do this all the time. For example, we could open a bag of M&M's® and inspect each piece of candy in it. For each piece we could assign a number depending upon the color of the piece. We could assign 1 for a red, 2 for a green, 3 for a yellow, 4 for blue, 5 for a light brown, 6 for a dark brown, and 7 for orange. By doing this we would get a data set that might look like
{1, 1, 2, 1, 6, 6, 6, 7, 4, 1, 4, 3, 2, 3, 6, 2, 1, 6, 6, 6, 6, 7, 6, 6, 7, 2, 1, 6, 6, 7, 6, 4, 4, 2, 1, 6, 2, 4, 7, 6, 7, 3, 6, 6, 6, 6, 3, 6, 3, 7, 1, 6, 6, 2, 6}
In fact, that is exactly the data set derived by examining the pieces taken from a real 47.9g bag of M&M's® purchased on August 27, 1999. The nominal measurements (i.e., the number names) for the pieces were recorded as pieces were poured out of the bag, one at a time. Note that the numerals are being used simply as names for the different colors of the candy pieces. There is no particular importance attached to the numeric names. A red candy, given the name 1, is not better or worse than is a green candy, givne the name 2. The numeric names are not measuring anything. They are just numeric names, assigned to represent the different colors of candy.

Looking at the manufacturer of vehicles in an aisle of the parking lot would give us another example of a nominal measurement. We could use 1 for any GM vehicle, 2 for any Ford, 3 for any Chrysler, 4 for any Toyota, 5 for any Honda, and so on. We would need to have a list of manufacturers and we would need to assign a numeric name to each item on the list. Then we could move down an aisle of the parking lot and simply record the appropriate numeric name for any particular vehicle that we find. As before, having a numeric value higher or lower than the numeric value of a different vehicle says nothing about the vehicles other than that they are from different manufacturers.

We fill out school applications, registration forms, credit applications, and other questionnaires that ask us to indicate our gender, male or female. It is common to code our responses using a number, perhaps 1 for female and 2 for male. There is no inherent meaning to the numbers that we use. The makeres of the survey or form could just as easily have assigned 75 to female and 28 to male, or 1 for male and 2 for female. There is no implied order to a nominal measurement, no implied value.

As a final example of a nominal measurement, consider your social security number. It is recorded on any number of different computer systems. It is used as a means to identify different people. It is a government assigned name for those people.

Ordinal Measurement uses the order of our numbers to reflect the natural order of the items we are measuring. Opinion scales are among the most common ordinal measurements. Surveys meaure our opinions by giving us statements and then asking us to rate our answer as

  • Strongly Disagree
  • Disagree
  • Neutral or no opinion
  • Agree
  • Strongly Agree
After we have filled out the survey, our responses may be coded as
  1. Strongly Disagree
  2. Disagree
  3. Neutral or no opinion
  4. Agree
  5. Strongly Agree
In doing this we have created an ordinal measurement. The higher numbers correspond to higher levels of agreement with the original statement.

As another example of an ordinal measurement, we could ask people to rate the quality of motion pictures on a scale from 1 to 10 where 1 is the worst and 10 is the best. The order of various rankings would reflect the order of opinions on the quality of the movies. We understand that a ranking of 4 indicates a higher quality than does a ranking of 3. Note, however, that there is no implication that the change from 3 to 4 is the same as is the change from 4 to 5. All we know is that for the system we are using, a movie rated as 4 is believed to be of a lesser quality than is a movie that is rated as 5. Furthermore, there is no "yardstick" that is being used to determine these rankings. A movie rated as an 8 by one person does not necessarily mean the same as does an 8 from another person. All that we do know is that if both people rate one move lower than another then the people agree on the relative order of the quality of the movies.

We need to emphasize that point. Let us consider an example where George rates one movie as an 8 and another as a 7, while Betty rates the same movies as 8 and 3, respectively. Assuming that these ratings are all that we know about George and Betty, then all that we can really say is that George and Betty agree on the relative order of the qualities of those two movies. The fact that they both rate the first movie as an 8 tells us nothing. George's 8 does not indicate in any way an agreement with Betty's 8. The fact that that George rates the two movies close together (8 and 7) while Betty rates them further apart (8 and 3) tells us nothing. The spread in George's ratings (from 8 to 7) does not in any way indicate that George really considers the two movies to be of a more similar quality than does Betty, even though Betty has a wider range in her ratings. These are two different people and we have no reason to believe that they are using the same internal measurement system to determine their opinion on the quality of the two movies. Nonetheless, we do know that they agree that they view the first movie as being of higher quality than is the second movie.

It is tempting to read more into such ordinal measures than we should. It is tempting to say that George and Betty agree on the quality of the first movie and that they disagree on the quality of the second. It is tempting to say that George thinks that the movies are close in quality while Betty sees the two movies as being far different in quality. Those temptations are not justified. Unfortunately, in far too many real life cases, the temptation wins out and people jump to such unfounded conclusions.

Interval and Ratio Measurements are similar to each other. They both refer to a measurement for which there is a consistent "yardstick", where two people can derive the same measurement of a item by using that "yardstick". Temperature measurement in Fahrenheit is good example of an interval measurement. Not only is 86°F warmer than is 43°F, it is also the case that 86°F is always the same temperature, no matter who reads the thermometer (assuming that they read it correctly). The thermometer acts as the standard "yardstick" for this measurement.

Weight in pounds is an example of a ratio measurement. Something that weighs 8 pounds is heavier than is something that weighs 4 pounds. 8 pounds of sugar may be sweeter than is 8 pounds of salt, but 8 pounds of sugar is no more or less heavy than is 8 pounds of salt, or 8 pounds of iron, or 8 pounds of anything. A scale provides the standard "yardstick" for measuring weight.

We have seen an example of an interval measurement (temperature in Fahrenheit) and of a ratio measurement (weight in pounds). The two examples seem quite similar. What then is the difference between an interval and a ratio measurement? Note that it makes sense to say that something that weighs 8 pounds is twice as heavy as is something that weighs 4 pounds. In fact, 8 pounds of sugar weighs just as much as does two 4 pound packages of sand. However, we would not make the same kind of statement about temperature. 86°F is not twice as warm as is 43°F. Two 43°F days are not the same as one 86°F day. Furthermore, a weight of 0 pounds means no weight, that something is weightless, whereas 0°F does not mean no temperature, or no heat energy (which is what temperature measures). Ratio measurements have a true zero value, interval measurements do not.

0°F is not true zero. It does not signify the absence of heat energy. Temperature measured in Celcius is also an interval measurement. 0°C is not true zero, and 20°C is not half as warm as is 40°C. It is important to note that there is a ratio measurement of temperature, namely the Kelvin scale. 0°K represents absolute zero, the total lack of heat energy. And, something that is at 100°K has half the heat energy as does something at 200°K.

Interval and ratio measurements have a degree of accuracy that needs to be examined. Your body temperature is probably quite close to 98.6°F. However, it is most certainly not exactly 98.6°F. Even if a digital thermometer were to give a 98.6°F reading, we need to recognize that every true temperature value from 98.55°F through 98.64999°F will show up as 98.6°F on a thermometer that is only accurate to the nearest tenth of a degree. Height is a ratio measurement. How tall are you? Whatever answer you gave, it is certainly wrong. Your answer may well be true to the nearest inch, or even to the nearest quarter inch, but your answer is certainly not true to the nearest thousandth of an inch. Interval and ratio measurements tend to be taken on things that have a continuous nature. Generally, we can always get more accurate interval and ratio measures if we just use a more accurate "yardstick". However, in a practical sense, we always settle on some approximation to the true value, and we round off our answers to some degree of accuracy. It is interesting to note that this does not happen with nominal and ordinal measurements.

The mechanical processes of computing descriptive statistics can be done with any set of data. However, certain processes are more or less appropriate for different kinds of measurements. It is important to be able to identify kinds of measurements, nominal, ordinal, interval, and ratio, so that you can determine the most appropriate descriptive statistics to use for the kind of measurement that you have. The first descriptive statistics that we want to examine are the ones that tell us about the "center" of the data, namely, the Measures of Central Tendency.

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