Measures of Central Tendency

Mode

The mode for a collection of data values is the data value that occurs most frequently (if there is one). Suppose the average number of colds in a family of six in a calendar year is as presented in Table 1.

Then, the mode is 1 because more family members (i.e., n = 2) caught one cold than any other number of colds. Thus, 1 is the most frequently occurring value. If two values occur the same number of times and more often than the others, then the data set is said to be bimodal. The data set is multimodal if there are more than two values that occur with the same greatest frequency. The mode is applicable to qualitative as well as quantitative data.

With continuous data, such as the time patients spend waiting at a particular doctor's office, which can be measured to many decimals, the frequency of each value is most commonly 1 because no two scores will be identical. Consequently, for continuous data, the mode typically is computed from a grouped frequency distribution. The grouped frequency distribution in Table 2 shows a grouped frequency distribution for the waiting times of 20 patients. Because the interval with the highest frequency is 30 – <40 minutes, the mode is the middle of that interval (i.e., 35 minutes).

Mean

Arithmetic Mean

The arithmetic mean, or average, is the most common measure of central tendency. Given a collection of data values, the mean of these data is simply the arithmetic average of these data values. That is, the mean is the sum of observations divided by the number of observations. If we use the following notation:

x is the variable for which we have data (e.g, test scores),

n is the number of sample observations (sample size),

x1 is the first sample observation (first test score),

x2 is the second sample observation (second test score),

xn is the nth (last) sample observation (last test score),

then the sample mean of a sample x1, x2,…, xn is denoted by

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