Central Tendency Central Tendency In statistics, 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 number of ways of specifying this "central value". In practical statistical analyses, the terms are often used before one has chosen even a preliminary form of analysis: thus an initial objective might be to "choose an appropriate measure of central tendency". 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. However, the term is applied to multidimensional data as well as to univariate data and in situations where a transformation of the data values for some or all dimensions would usually be considered necessary: in the latter cases, the notion of a "central location" is retained in converting an "average" computed for the transformed data back to the original units.
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In addition, there are several different kinds of calculations for central tendency, where the kind of calculation depends on the type of data (level of measurement). Both "central tendency" and "measure of central tendency" apply to either statistical populations or to samples from a population. Several different measures of central tendency are defined below. The mode is the most frequently appearing value in the population or sample. Suppose we draw a sample of five women and measure their weights. They weigh 100 pounds, 100 pounds, 130 pounds, 140 pounds, and 150 pounds. Since more women weigh 100 pounds than any other weight, the mode would equal 100 pounds. To find the median, we arrange the observations in there is an odd number of observations, the median number of observations, the median is the average sample of five women, the median value would be middle weight.
order from smallest to largest value. If is the middle value. If there is an even of the two middle values. Thus, in the 130 pounds; since 130 pounds is the
The mean of a sample or a population is computed by adding all of the observations and dividing by the number of observations. Returning to the example of the five women, the mean weight would equal (100 + 100 + 130 + 140 + 150)/5 = 620/5 = 124 pounds. Proportions and Percentages When the focus is on the degree to which a population possesses a particular attribute, the measure of interest is a percentage or a proportion. A proportion refers to the fraction of the total that possesses a certain attribute. For example, we might ask what proportion of women in our sample weigh less than 135 pounds. Since 3 women weigh less than 135 pounds, the proportion would be 3/5 or 0.60.
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A percentage is another way of expressing a proportion. A percentage is equal to the proportion times 100. In our example of the five women, the percent of the total who weigh less than 135 pounds would be 100 * (3/5) or 60 percent. Notation Of the various measures, the mean and the proportion are most important. The notation used to describe these measures appears below: X: Refers to a population mean. x: Refers to a sample mean. P: The proportion of elements in the population that has a particular attribute. p: The proportion of elements in the sample that has a particular attribute. Q: The proportion of elements in the population that does not have a specified attribute. Note that Q = 1 - P. q: The proportion of elements in the sample that does not have a specified attribute. Note that q = 1 - p.
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