Central Tendency, Measures of

One of the most common statistical analyses used in descriptive statistics is a process to determine where the average of a set of values falls. There are multiple ways to determine the middle of a group of numbers, and the method used to find the average will determine what information is known and how that average should be interpreted. Depending on the data one has, some methods for finding the average may be more appropriate than others.

The average describes the typical or most common number in a group of numbers. It is the one value that best represents the entire group of values. Averages are used in most statistical analyses, and even in everyday life. If one wanted to find the typical house price, family size, or score on a test, some form of average would be computed each time. In fact, one would compute a different type of average for each of those three examples.

Researchers use different ways to calculate the average, based on the types of numbers they are examining. Some numbers, measured on the nominal level of measurement, are not appropriate to do some types of averaging. For example, if one were examining the variable of types of vegetables, and the labels of the levels were cucumbers, zucchini, carrots, and turnips, if one performed some types of average, one may find that the average vegetable was 3/4 carrot and 1/4 turnip, which makes no sense at all. Similarly, using some types of average on interval-level continuous variables may result in an average that is very imprecise and not very representative of the sample one is using. When the three main methods for examining the average are collectively discussed, they are referred to as measures of central tendency. The three primary measures of central tendency commonly used by researchers are the mean, the median, and the mode.

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Mean

The mean is the most commonly used (and misused) measure of central tendency. The mean is defined as the sum of all the scores in the sample, divided by the number of scores in the sample. This type of mean is also referred to as the arithmetic mean, to distinguish it from other types of means, such as the geometric mean or the harmonic mean. Several common symbols or statistical notations are used to represent the mean, including

, which is read as x-bar (the mean of the sample), and μ, which is read as mu (the mean of the population). Some research articles also use an italicized uppercase letter M to indicate the sample mean.

Much as different symbols are used to represent the mean, different formulas are used to calculate the mean. The difference between the two most common formulas is found only in the symbols used, as the formula for calculating the mean of a sample uses the symbols appropriate for a sample, and the other formula is used to calculate the mean of a population, and as such uses the symbols that refer to a population.

For calculating the mean of a sample,

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