統計学入門：測定の尺度

[0:05] Before we can conduct a statistical analysis, we need to measure our dependent variables. Exactly how the measurement is carried out depends on the type of variable involved in the analysis. Different types are measured differently. To measure the time taken to respond to a stimulus, you might use a stopwatch. Stopwatches are of no use, of course, when it comes to measuring someone's attitude towards a political candidate. A rating scale is more appropriate in this case, with labels like "very favorable", "somewhat favorable", etc.

[0:37] For a dependent variable such as favorite color, you can simply note the color word, like red, that the subject offers. Although procedures for measurement differ in many ways, they can be classified using a few fundamental categories. In a given category, all of the procedures share some properties that are important for you to know about. The categories are called scale types, or just scales, and are described in this section. There are four basic scales of measurement: nominal, ordinal, interval, and ratio.

[1:13] When measuring using a nominal scale, one simply names or categorizes responses. Gender, handedness, favorite color, and religion are examples of variables measured on a nominal scale. The essential point about nominal scales is that they do not imply any ordering among the responses. For example, when classifying people according to their favorite color, there is no sense in which green is placed ahead of blue. Responses are merely categorized. Nominal scales embody the lowest level of measurement.

[1:48] The next highest scale of measurement is ordinal. The ordinal scale is used when one wishes to name or categorize, and the order is meaningful. Let's look at an example. A researcher wishing to measure consumers' satisfaction with their microwave ovens might ask them to specify their feelings as either very dissatisfied, somewhat dissatisfied, somewhat satisfied, or very satisfied. The items in the scale are ordered, ranging from least to most satisfied. This is what distinguishes ordinal from nominal scales.

[2:20] Whereas nominal scales don't allow comparisons in the degree to which two subjects possess the dependent variable, just this kind of comparison is possible with ordinal scales. For example, our satisfaction ordering makes it meaningful to assert that one person is more satisfied than another with their microwave ovens. Other examples of variables measured at the ordinal level are military rank, private, lieutenant, captain, general, etc., and class ranking, first in class, second in class, and so on.

[2:55] scales fail to capture important information that will be present in the other scales we examine. In particular, the difference between two levels of an ordinal scale cannot be assumed to be the same as the difference between two other levels. In our satisfaction scale, for example, the difference between the responses very dissatisfied and somewhat dissatisfied is not necessarily the same as the difference between somewhat dissatisfied and somewhat satisfied. Nothing in our measurement procedure allows us to determine whether the two differences reflect the same difference in psychological satisfaction.

[3:31] Statisticians express this point by saying that the differences between adjacent scale values do not necessarily represent equal intervals on the underlying scale giving rise to the measurement. In our case, the underlying scale is the true feeling of satisfaction which we are trying to measure. What if the researcher had measured satisfaction by asking consumers to indicate their level of satisfaction by choosing a number from 1 to 4? Will the difference between the responses of 1 and 2 necessarily reflect the same difference in satisfaction as the difference between the responses 2 and 3?

[4:07] The answer is no. Changing the response format to numbers does not change the meaning of the scale. we are still in no position to assert that the mental step from 1 to 2, for example, is the same as the mental step from 3 to 4. The next highest scale of measurement is the interval scale. As you probably noticed, each successive scale builds upon the qualities of the lower scales, adding additional information. Interval scales are numerical scales in which intervals have the same interpretation throughout.

[4:42] So the interval scale names or categorizes, the order is meaningful, and the intervals are equal. As an example, consider the Fahrenheit scale of temperature. The difference between 30 degrees and 40 degrees represents the same temperature difference as the difference between 80 degrees and 90 degrees. This is because each 10 degree interval has the same physical meaning in terms of the kinetic energy of molecules. Interval scales are not perfect, however.

[5:14] In particular, they do not have a true zero point, even if one of the scaled values happens to carry the name zero. The Fahrenheit scale illustrates the issue. Zero degrees Fahrenheit does not represent the complete absence of temperature, the absence of any molecular kinetic energy. In reality, the label zero is applied to its temperature for quite accidental reasons connected to the history of temperature measurement. Since an interval scale has no true zero point, it does not make sense to compute ratios of temperatures.

[5:47] For example, there is no sense in which the ratio of 40 to 20 degrees Fahrenheit is the same as the ratio of 100 to 50 degrees. No interesting physical property is preserved across the two ratios. After all, if the zero label were applied at the temperature that Fahrenheit happens to label as 10 degrees, the two ratios would instead be 30 to 10 and 90 to 40, no longer the same. For this reason, it does not make sense to say that 80 degrees is twice as hot as 40 degrees.

[6:20] Such a claim would depend on an arbitrary decision about where to start the temperature scale. namely, what temperature to call zero. whereas the claim is intended to make a more fundamental assertion about the underlying physical reality. The ratio scale of measurement is the most informative scale. It is an interval scale with the additional property that its zero position indicates the absence of the quantity being measured. You can think of a ratio scale as the three earlier scales rolled up in one.

[6:53] Like a nominal scale, it provides a name or category for each object. The numbers serve as labels. like an ordinal scale, the objects are ordered in terms of the ordering of the numbers. Like an interval scale, the same difference at two places on the scale has the same meaning. And in addition, the same ratio at two places on the scale also carries the same meaning. An example of a ratio scale is the amount of money you have in your pocket right now. 25 cents, 55 cents, etc.

[7:25] Money is measured on a ratio scale because, in addition to having the properties of an interval scale, it has a true zero point. If you have zero money, this implies the absence of money. Since money has a true zero point, it makes sense to say that someone with 50 cents has twice as much money as someone with 25 cents. or that Bill Gates has a million times more money than you do. level of measurement is used for psychological variables.

[7:56] Rating scales are used frequently in psychological research. For example, experimental subjects may be asked to rate their level of pain, how much they like a consumer product, their attitudes about capital punishment, or their confidence in an answer to a test question. Typically, these ratings are made on a 5-point or 7-point scale. These scales are ordinal rather than interval scales, since there is no assurance that a given difference represents the same thing across the range of the scale. For example, there is no way to be sure that a treatment that reduces pain from a rated pain level of 3 to a rated pain level of 2 represents the same level of relief as a treatment that reduces pain from a rated pain level of 7 to a rated pain level of 6.

[8:44] In memory experiments, the dependent variable is often the number of items correctly recalled. What scale of measurement is this? you could reasonably argue that it is a ratio scale. First, there is a true zero point. Some subjects may get no items correct at all. Moreover, a difference of 1 represents a difference of one item recalled across the entire scale. It is certainly valid to say that someone who recalled 12 items recalled twice as many items as someone who recalled only 6 items.

[9:20] Number of items recalled is a more complicated case than it appears at first. Consider the following example in which subjects are asked to remember as many items as possible from a list of 10. Assume that there are five easy items and five difficult items. Half of the subjects are able to recall all the easy items and different numbers of difficult items. And the other half of the subjects are unable to recall any of the difficult items and remember different numbers of the easy items. Some sample data are shown here.

[9:53] Let's compare the difference between subject A's score of 2 and subject B's score of 3 with the difference between subject C's score of 7 and subject D's score of 8. The former difference is the difference of one easy item. The latter difference is a difference of one difficult item. Do these two differences signify the same difference in memory performance? We are inclined to respond "no" to this question, since only a little more memory may be needed to retain the additional easy item, whereas a lot more memory may be needed to retain the additional hard item.

[10:29] The general point is that it is often inappropriate to consider psychological measurement scales as either interval or ratio. Why are we so interested in the type of scale that measures a dependent variable? The crux of the matter is the relationship between the variable's level of measurement and the statistics that can be meaningfully computed with that variable. For example, consider a hypothetical study in which five children are asked to choose their favorite color from blue, red, yellow, green, and purple.

[11:03] the researcher codes and tabulates the results as shown in the displayed table. This means that if a child said her favorite color was red, then the choice was coded as 2. If the child said her favorite color was purple, then the response was coded as 5, and so forth. Consider the hypothetical data displayed in this table. Each code is a number, so nothing prevents us from computing the mean code assigned to the children. The mean happens to be 3, but you can see that it would be senseless to conclude that the average favorite color is yellow, the color with a code of 3.

[11:41] Such nonsense arises because favorite color is a nominal scale, and taking averages of its numerical labels is like counting the number of letters in the name of a snake to see how long the snake is. Does it make sense to compute the mean of numbers measured on an ordinal scale? This is a difficult question, one that statisticians have debated for decades. You will be able to explore this issue yourself in a simulation and reach your own conclusion. The prevailing, but by no means unanimous, opinion of statisticians is that for almost all practical situations, the mean of an ordinarily measured variable is a meaningful statistic.

[12:21] However, as you will see in the simulation, there are extreme situations