788 CHAPTER 12 Statistics The mean is used when each piece of data is to be considered and “weighed” equally. It is the most commonly used average. It is the only average that can be affected by any change in the set of data; for this reason, it is the most sensitive of all the measures of central tendency (see Exercise 33). Occasionally, one or more pieces of data may be much greater or much smaller than the rest of the data. When this situation occurs, these “extreme” values have the effect of increasing or decreasing the mean significantly so that the mean will not be representative of the set of data. Under these circumstances, the median should be used instead of the mean. The median is often used in describing average family incomes because a relatively small number of families have extremely large incomes. These few incomes would inflate the mean income, making it nonrepresentative of the millions of families in the population. Consider a set of exam scores from a mathematics class: 0, 16, 19, 65, 65, 65 68, 69, 70, 72, 73, 73, 75, 78, 80, 85, 88, 92. Which average would best represent these grades? The mean is 64.06. The median is 71. Since only 3 of the 18 scores fall below the mean, the mean would not be considered a good representative score. The median of 71 probably would be the better average to use. The mode is the piece of data, if any, that occurs most frequently. Builders planning houses are interested in the most common family size. Retailers ordering shirts are interested in the most common shirt size. An individual purchasing a thermometer might choose one, from those on display, whose temperature reading is the most common reading among those on display. These examples illustrate how the mode may be used. The midrange is sometimes used as the average when the item being studied is constantly fluctuating. Average daily temperature, used to compare temperatures in different areas, is calculated by adding the lowest and highest temperatures for the day and dividing the sum by 2. The midrange is actually the mean of the high value and the low value of a set of data. Occasionally, the midrange is used to estimate the mean, since it is much easier to calculate. Sometimes an average itself is of little value, and care must be taken in interpreting its meaning. For example, Jim is told that the average depth of Willow Pond is only 3 feet. He is not a good swimmer but decides that it is safe to go out a short distance in this shallow pond. After he is rescued, he exclaims, “I thought this pond was only 3 feet deep.” Jim didn’t realize that an average does not indicate extreme values or the spread of the values. The spread of data is discussed in Section 12.4. TECHNOLOGY TIP There’s an App for That! Many of the concepts that we discuss throughout this section and throughout this entire chapter can be further explored with the use of an app on your smartphone or tablet computer. Several of these apps allow you to calculate the mean, median, and mode. Although these can be helpful, be sure to read the directions carefully. Some apps will not support decimal numbers. As always, consult with your instructor before using these apps when completing work related to your course. Measures of Position Measures of position are used to describe the position of a piece of data in relation to the entire set of data. If you took the SAT before applying to college, your score was described as a measure of position rather than a measure of central tendency. Measures of position are often used to make comparisons, such as comparing the scores of individuals from different populations, and are generally used when the amount of data is large.
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