5.1 Central Tendency
5.1: Central Tendency
5.1.1: Mean: The Average
The term central tendency relates to the way in which quantitative data tend to cluster around some value.
Learning Objectives
Define the average and distinguish between arithmetic, geometric, and harmonic means.
Key Takeaways
Key Points
- An average is a measure of the “middle” or “typical” value of a data set.
- The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.
- The arithmetic mean is the sum of a collection of numbers divided by the number of numbers in the collection.
- The geometric mean is a type of mean or average which indicates the central tendency, or typical value, of a set of numbers by using the product of their values. It is defined as the nth root (where n is the count of numbers) of the product of the numbers.
- The harmonic mean H of the positive real numbers X1, X2, … Xn is defined to be the reciprocal of the arithmetic mean of the reciprocals of X1, X2, … Xn. It is typically appropriate for situations when the average of rates is desired.
Key Terms
- average
- any measure of central tendency, especially any mean, the median, or the mode
- arithmetic mean
- the measure of central tendency of a set of values computed by dividing the sum of the values by their number; commonly called the mean or the average
- central tendency
- a term that relates the way in which quantitative data tend to cluster around some value
Example
The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that. One may find that . Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list for which we want an average, we get, for example, that the arithmetic mean of 2, 8, and 11 is found by solving for the value of A in the equation . One finds that A=7.
The term central tendency relates to the way in which quantitative data tend to cluster around some value. A measure of central tendency is any of a variety of ways of specifying this “central value”. Central tendency is contrasted with statistical dispersion (spread), and together these are the most used properties of distributions. Statistics that measure central tendency can be used in descriptive statistics as a summary statistic for a data set, or as estimators of location parameters of a statistical model.
In the simplest cases, the measure of central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. An average is a measure of the “middle” or “typical” value of a data set. In the most common case, the data set is a list of numbers. The average of a list of numbers is a single number intended to typify the numbers in the list. I