Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-2 Measures of Center1. The mean, median, mode, and midrange are measures of “center” in the sense that they each attempt to determine (by various criteria – i.e., by using different approaches) what might be designated as a typical or representative value. 2. The term “average” is not used in statistics because it is imprecisely used by the general public as a synonym for “typical” – as in, the average American has blue eyes. When referring to the result obtained by dividing a sum by the number of values contributing to that sum, the term “mean” should be used. 3. No. The price “exactly in between the highest and the lowest” would be the mean of the highest and lowest values – which is the midrange, and not the median. 4. No. Since the numbers are not measuring anything, their mean would be meaningless. In general, the mean is not meaningful or appropriate when there are no units (e.g., pounds, inches, etc.) associated with the values. NOTE: As it is common in mathematics and statistics to use symbols instead of words to represent quantities that are used often and/or that may appear in equations, this manual employs the following symbols for the various measures of center. mean = x mode = M median = x˜ midrange = m.r. This manual will generally report means, medians and midranges accurate to one more decimal place than found in the original data. In addition, these two conventions will be employed. (1)When there is an odd number of data, the median will be one of the original values. The manual follows example #2 in this section and reports the median as given in the original data. And so the median of 1,2,3,4,5 is reported as x˜ = 3. (2)When the mean falls exactlybetween two values accurate to one more decimal place than the original data, the round-off rule in this section gives no specific direction. This manual follows the commonly accepted convention of always rounding up. And so the mean of 1,2,3,3 is reported as x = 9/4 = 2.3 [i.e., 9/4 = 2.25, rounded up to 2.3]. In addition, the median is the middle score when the scores are arranged in order, and the midrange is halfway between the first and last score when the scores are arranged in order. It is usually helpful to arrange the scores in order. This will not affect the mean, and it may also help in identifying the mode. Finally, no measure of center can have a value lower than the smallest score or higher than the largest score. Remembering this helps to protect against gross errors, which most commonly occur when calculating the mean.