Measures of Central Tendency
3.9 Experimental Research and the Mean: A Glimpse of Things to Come
In Chapter 1, several fundamental concepts of experimentation were presented.
At the most basic level, experimental designs compare the performances of dif- ferent groups of participants. The typical statistics used to determine whether the independent variable affected the dependent variable are the means of the various groups. Some examples of studies in which group means are used to reach conclusions are presented below.
► Example 3.1 An educational theorist is interested in comparing the effectiveness of two teaching techniques. Participants assigned to one group are exposed to educational material via an online teaching experience. Par- ticipants assigned to another group take part in a traditional classroom experience. The dependent variable is the amount of material learned.
The mean of the amount of material learned is computed for both groups.
Through the use of statistical analyses (presented in later chapters), the means of the groups are compared to decide if one teaching method is supe- rior to the other.◄
►Example 3.2 A social psychologist is interested in learning about the rela- tionship between different mood states and charitable giving. In one condition, it is arranged for participants to experience a pleasant interaction with the experimenter. In another condition, participants are treated in a cold, rude manner by the experimenter. Soon after leaving the laboratory, a person approaches the participant and asks for a donation to a homeless shelter.
The dependent variable is the amount of money donated. The means are computed for both groups and compared to see if mood states influence generosity.◄
3.9 Experimental Research and the Mean: A Glimpse of Things to Come 85
► Example 3.3 A child psychologist would like to evaluate two treatment techniques for helping children overcome their fear of the dark. Participants assigned to one group are taught to imagine themselves as a superhero on a mis- sion during the night. Participants assigned to a second group are told to repeat over and over,“I’m a big boy/girl.”The dependent variable is the amount of time the child is willing to stay in a dark room. Means are calculated for both groups and compared to judge whether one method is more effective than another in helping children tolerate the dark.◄
This chapter has presented several factors that should guide us in deciding which measure of central tendency to use when describing a distribution of scores. As we make our way through the text, we will discover that, when con- ducting an experiment, the mean is almost always the statistic that serves as the point of comparison between different conditions.
Box 3.2 presents a study that taught participants how to control their heart rate. Means were computed for groups of participants at two points in the study.
Statistical techniques discussed in later chapters will show us how to use the means to compare the two groups of participants and interpret the results.
Box 3.2 Learning to Control Our Heart Rate
For several decades, biofeedback was a popular treatment for many stress- related physical ailments. In the 1980s researchers started to investigate its effects. Biofeedback entails the provision of external feedback in the form of a visual display or varying auditory stimulus, which changes as some physiolog- ical response changes. Thousands of people have learned how to relax with bio- feedback training; there is little doubt that most people can achieve an impressive degree of control over their physiological responses, at least while they are attached to the biofeedback equipment. But therein lies the problem.
What good is it to learn how to relax if we can only experience that state when we’re hooked up to a machine? Posed as a research question, we might ask,
“When participants learn how to control one of their physiological responses, will they be able to transfer learning to control that response during their eve- ryday activities?”It was this question that led Gloria Balague-Dahlberg (1986) to conduct the following study.
Study Method
Eighteen participants who scored high on an anxiety questionnaire participated in the experiment. To assess the participants’heart rate throughout the day, they were asked to wear a Holter monitor (a device that continuously records heart rate). The participants were asked to try to keep their heart rate low while going about their usual daily routine.
Half of the participants were seen individually for five biofeedback sessions, during which time they tried to lower their heart rate as much as possible.
Although biofeedback is always conducted in a relaxed, comfortable atmos- phere, Balague-Dahlberg reasoned that the transfer of learning to the natural environment would be augmented if participants initially learned to control their heart rate in a setting filled with distractions. So with each successive ses- sion, participants attempted to lower their heart rate amid an increasing level of distractions. This was procedurally accomplished by having participants sit in a hard chair while performing a series of mental tasks. As the sessions progressed, a tape of distracting noises was played: people talking, phones ringing, machines running, and other“office noises.”
In addition to this experimental group, a control group was included: a group that received the same instructions but did not have experience with the bio- feedback equipment. After the training phase of the experiment, all partici- pants’heart rates were once again monitored for a 24-hour period.
Results
The data from this study are presented in the following tables. The baseline score (also called a pretest score) is the mean heart rate for the 24-hour period before training; the posttest score is the mean heart rate during the final 24-hour record- ing period. A graph (Figure 3.5) is presented so that we can easily see the differ- ence between the groups at each phase of the study. (Yes, truncation was used to highlight this difference. More will be said about this at the end of this box.)
Baseline (pretest) Posttest
Heart rate
75 76 77 78 79 80 81 82 83 84
Control
Biofeedback
Figure 3.5 Graphical presentation of the results of the Balague-Dahlberg study on heart rate control.
3.9 Experimental Research and the Mean: A Glimpse of Things to Come 87
Participant Baseline Posttest
1 92 92
2 64 70
3 93 86
4 70 71
5 67 69
6 93 74
7 63 62
8 86 93
9 84 79
Mpre= 79.11 Mpost= 77.33
Control group
Participant Baseline Posttest
10 90 95
11 92 99
12 79 82
13 85 86
14 75 73
15 82 84
16 78 73
17 80 83
18 61 63
Mpre= 80.22 Mpost= 82.00
The biofeedback and control groups have similar mean heart rates at the pretest baseline measure. This was to be expected because participants were randomly assigned to conditions and had not yet received the different treat- ments. It is evident from the 24-hour posttesting data that the biofeedback par- ticipants appear to have learned from the training and were able to keep their heart rate at a level lower than the control participants. However, one would not want to conclude anything by merely visually comparing the means. At this point, the experimenter would conduct the appropriate statistical test to deter- mine if these differences are unlikely to occur by chance. (We will learn about these tests later in the text.) If these posttest differences are unlikely to occur by chance, then tentative conclusions can be made about the superior effects of biofeedback training. (Furthermore, we can say that the graph does not mislead the viewer. If, however, the analysis suggests that chance factors can explain the posttest difference, then some could argue that the graphic appears to mislead unsuspecting viewers about the effects of biofeedback training.)
Summary
Descriptive statistics are statistical indices that summarize and communicate basic characteristics of a distribution. Values that communicate where scores center in the distribution are called measures of central tendency. Measures that communicate the degree to which scores are spread out around the center of a distribution are called measures of dispersion or variability. Statistical values that describe the distribution characteristics of a population are called para- meters; statistical values that describe the distribution characteristics of a sam- ple are called statistics.
The mean is the most important and most often used measure of central ten- dency. Not only can it be used as a descriptive index of central tendency, but the mean is frequently used in formulas designed to test experimental hypotheses.
The degree to which a score deviates from the mean isX–M. This deviation amount can be called a deviation score (or error score) and is symbolized asx.
Therefore,x=X–M. The sum of all the deviation scores equals 0. Therefore, Σ X−M =Σx= 0
The mean has several advantages. First, it takes into account not only all of the scores in a distribution but also their precise distance from the middle. Second, it is used in many statistical formulas. Third, as the size of the distribution increases, the mean becomes a very stable measure of central tendency.
A sample mean is usually a good estimation of the mean of a population. How- ever, since the mean is sensitive to extreme scores, it is oftentimes not seen as a good measure of centeredness when the distribution is skewed. Using the mean as a measure of central tendency can also present a problem when the distribu- tion is truncated, that is, when one or both ends of the distribution have been limited by the nature of the measuring instrument.
The median is the point in the distribution where 50% of the scores fall above and 50% fall below it. Since the median is not affected by the value of extreme scores, it should be used when the distribution is skewed, truncated, or has scale-limited upper or lower cutoff scores. The median is also the appropriate measure for cen- tral tendency when the values in the distribution come from an ordinal scale.
The mode is defined as the most typical or most frequent score. It is the least used measure of central tendency. The mode ignores all of the numbers in a distribution except the one value that occurs most often. On the other hand, the mode is the only measure of central tendency to use when evaluating scores measured on a nominal scale.
The particular shape of the distribution has implications for the relative posi- tion of the mean, median, and mode. If the distribution is symmetrical, then all three measures of central tendency will be identical. In a positively skewed
Summary 89
distribution, both the mean and median are pulled to the right, although the mean is pulled farther. In a negatively skewed distribution, both the mean and median are pulled to the left, although the mean is pulled farther.