Click here to purchase the entire book in PDF format. Psychometric function Let's do a taste test to see how detectable sugar is in a cup of coffee. First, you'll need to make a pot of coffee. Pour the coffee into 101 cups (hope you made a big pot...). In the first cup, don't add any sugar. Put 1 mg of sugar in the second cup, 2 mg of sugar in the third cup, and so on until you get 100 mg (about 21 teaspoons) of sugar in the 101st cup. Now let's design the test. We'll give each subject two cups of coffee, one without sugar, and the second with some amount of sugar between 1 and 100 mg. The question to the subject will be "which of these two cups of coffee is the one with sugar in it?" Then we'll give the subject another two cups of coffee (one without sugar and one with) and ask the question again. We'll do this again and again, making sure to repeat pairs of cups of coffee to check the answers. Once the taste test is over, we'll see that, if there is just a very small amount of sugar in the coffee, people will just be guessing as to which cup has the sugar in it. If there is a large amount of sugar in the coffee, everyone will get the right answer. If the amount of sugar in the coffee is somewhere in between, then some people will get it right, some will get it wrong. Or, another way to think of it is that a single person will have a reduced probability of getting the right answer when they're asked. They won't be guessing completely, but they won't get it right every time either... If we were to make a graph to show this behaviour, it would look like Figure 5.16. This graph shows the number of subjects that, on average, got the correct answer on our taste test plotted against the amount of sugar in the second cup of coffee. If there is no sugar in the coffee, everyone will be guessing. If there is a lot of sugar in the coffee 100% of the answers will be correct. The shape of this graph has a specific shape and is called the psychometric function. Figure 5.16: The psychometric function. This shows the probability of a difference in two stimuli being detected as a function of the amount of the physical difference in the stimuli. For example, the number of people that will notice a change in the sugar content of coffee (the Y-axis) for a given amount of sugar (the X-axis). If you test enough people, asking them to detect any change in a stimulus, you'll get this same shape of curve every time. This is because it's directly dependent on a standard normal distribution which describes everybody... This is shown in Figure 5.17. Figure 5.17: The relation between a standard normal distribution and the psychometric function. The top plot shows the number of people who require a given amount of change in the stimulus to detect that the change has happened. The bottom plot shows the area under the top plot to the left of a given value on the X-axis. Check the text for a more thorough explanation. Let's say, for example, that we're asking people to detect sugar in coffee. If you add 1 mg of sugar in the cup, almost nobody will be able to detect that sugar has been added. The top plot in Figure 5.17 shows this - for a small amount of sugar, you get almost no one being able to detect it. As you add more and more sugar, you get to a point where you're at the threshold of detection for the most people, and therefore you get a peak in the graph. There are some people with a very poor sense of taste, and they will require much more sugar in the coffee before they detect it. However, this type of person is rare, therefore, for large quantities of sugar, you get a small percentage of the population on the Y-axis. The result is a standard normal distribution - better known as a bell curve. This bell curve tells us the relationship between a given amount of sugar and the threshold of detection. So, for example, using that top plot in Figure 5.17, we can see that 40% of the population has a threshold of detection of 50 mg of sugar (remember that these are completely hypothetical values - I didn't really do this test...). Some people have a lower threshold, and some people have a higher threshold. This particular piece of information isn't really that useful to us. A more interesting question is ``If I put 50 mg of sugar in the coffee, how many people will detect it?'' This means that we're looking for the people whose threshold of detection is 50 mg or less - not just 50 mg. This number of people can be found by looking at the area under the bell curve up to and including the value for 50 mg. Since the bell curve is symmetrical, and since its peak is at 50 mg of sugar in our graph, then the area under it on the left of that value must be 50% of the total area under the graph. As we go higher and higher on the X-axis, we include a higher and higher percentage of the total area under the graph. The bottom plot in Figure 5.17 shows exactly this - it's the area under the top plot to the left of a given value on the X-axis expressed as a percentage of the total area under the graph. This is the psychometric function. Figure 5.18: The relation between the area under a standard normal distribution and the psychometric function.