4 Assumptions About the Mean Value in Statistics

Speelman & McGann (2013) emphasize that over-reliance on means may contribute to misleading, and possibly erroneous, findings.

The mean value represents the average performance of a group, as it is representative of the group’s data. In other words, the mean value represents a central tendency of the group, and a central tendency makes summarizing results easier.

Using the mean without care, however, can cause an illusion of stability in behavioral data. This is because the mean value does not deal with the complexity and variability of human behavior and cognition.

Therefore, the mean may be a way to summarize a data set, allowing us to know whether two groups differ in some way or another (e.g. a treatment vs placebo group). As such, it can be used for between-group comparisons, but sometimes, such comparisons are misleading because they only rely on the mean value (Speelman & McGann, 2013).normal distribution

It is believed that biological systems produce a normal distribution of scores, the so-called Gaussian distribution, and the mean sits perfectly in the middle of that distribution. ––>

Some people believe that scores on either side of the mean represent errors of measurement (the normal law of error), but how do we define what a mean value is? How many people do we need to measure to find a universal mean value? Relying on the mean of a set of scores, to represent the set of data, carry four assumptions (Speelman & McGann, 2013):

1. The mean value helps us find the “true value”

In experiments, a group of people are exposed to the same conditions. Everyone is assumed to respond similarly to the same conditions because their cognitive mechanism are similar. Unfortunately, the data we collect from these people are not identical, and we assume that this is because our measurements are not perfect.

But how come we assume that everyone has the same cognitive mechanisms? Indeed, some biological systems in the human body operate in similar ways such as the heart, the cardiovascular system and the musculoskeletal system. The brain, however, changes with experience (see brain plasticity), and as a result, it operates in a more unique fashion.

2. The mean value helps eliminate noise (imperfections) in data

If this statistical “noise” did not exist, we could measure “true” values directly, and there would be no need for inferential statistics.

3. When the mean value is not reliable, it is because of methodological flaws

Our methodologies are inherently faulty in that they do not provide perfect measures of the variables of interest. Even measurements of physical properties such as temperature carry with them error values. Measuring brain activity is slightly more difficult, so imagine the challenges that are associated with it.

Often, psychological variables are not directly observable so we need to construct measures that are directly observable that reflect the variable of interest.

Psychological Test Theory states that each score on a particular test reflects the true value plus error. But instead of assuming that each score is the true value plus some error, is it not be plausible that a certain score reflects the true value at that moment in time?

4. The noise in data represents the effects of variables unrelated to the one being measured

There is recognition in psychology that humans are sensitive to a vast range of variables that operate independently of the variable that we like to measure (see confounding variables). But does this variance reflect variables that are random and independent of the variable we seek to measure? Is it not possible that the variance is less random than we tend to think it is?

Speelman & McGann (2013) sum up by saying that we should investigate data beyond the means, and one’s confidence that the mean reflects the overall data should be low.

In other words, one should report the range and variation in data as frequently, and as clearly, as the central tendencies.Very often, theories originate from the use of means (i.e., central tendencies), but this sometimes leads to misleading findings.

For example, The Power Law of Learning is a learning effect, but the effect is only found between groups, not at the individual level. Nevertheless, some people continue to use learning strategies that stem from this learning effect, but in fact this is comic, considering that the effect does not exist at all (at least at the individual level).

Photo: Ken Whytock