Average

To describe a group of values, it is useful to have a typical value or an average value. An average is a summary, so the average value should be representative of a group of values. An average can be used primarily to describe a sample and can also be used to estimate a population value. For example, researchers might want to find an average value in their sample that lets them predict what the average value in the population might be.

There are various measures of central tendency, or average, of a data set, and all have different statistical properties, which makes them sometimes more and sometimes less useful descriptors. Specific problems associated with the distribution the group of values represents are the shape of the distribution (symmetrically distributed or skewed, for example) and the presence or absence of outliers in the data set. Either the average value can be computed, taking all or only some values in the group into account, or it can be a chosen value from the group, seen fit to represent the group.

Most commonly, people refer to the arithmetic mean as the average (and vice versa), although this language is ambiguous and should be avoided. The arithmetic mean is the sum of all the values divided by the number of values in the group. There are many variations of the mean, two of the most common ones being the geometric mean and the harmonic mean. A trimmed mean excludes a specific percentage of the upper and the lower end of a distribution, commonly 5% in either direction. A midmean is a special case of a trimmed mean in which the mean is calculated for the data between the 25th and 75th percentiles.

The mode, the most frequently occurring value, can also be a descriptor of a group of values. The median is also a frequently used term for expressing an average value. Especially in skewed data, the mean would be less informative than the median.

It is important to remember that an average value should be presented together with a measure of dispersion.

More about average:

• The mean as the most commonly used average measure is sensitive to extreme scores.
• The median is a suitable average for nonsymmetric distributions and is not affected by outliers.
• The mode is an average value actually represented in the group of values, whereas the mean, as a derivative, can take a value that is not actually represented in the group of values (e.g., the average number of children is 1.2).

Applying Ideas on Statistics and Measurement

The following abstract is adapted from See, W. Y., Wagner, T. H., Shuo, C., & Barnett, P. G. (2003). Average cost of VA rehabilitation, mental health, and long-term hospital stays. Medical Care Research and Review, 60(3 suppl), 40S–53S.

One of the most common methods used to better understand a collection of data points is through the calculation of an average (which includes such descriptive statistics as the mean, the mode, and the median). In this article, researchers Wei You See and his colleagues at [Page 66]Stanford University describe the development of a database for the cost of inpatient rehabilitation, mental health, and long-term care stays in the Department of Veterans Affairs from fiscal year 1998. As a unit of analysis, they used bedsection, which is similar to a hospital ward, and they classified inpatient services into nine categories, including rehabilitation, blind rehabilitation, spinal cord injury, psychiatry, substance abuse, intermediate medicine, domiciliary, psychosocial residential rehabilitation, and nursing home. For each of these nine categories, they estimated a national and a local average per diem cost. The next step was to calculate what they call encounter-level costs, which was done by multiplying the average per diem cost by the number of days of stay in the fiscal year. Their conclusion? The national cost estimates for hospitalization are more reliable than the local cost estimates for the same.

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