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The Basic Practice of Statistics4th EditionDavid Moore 150 solutions Introduction: Connecting Your LearningIn real-world applications, you can use tables and graphs of various kinds to show information and to extract information from data that can lead to analyses and predictions. Graphs allow you to communicate a message from data.Measures of central tendency are a key way to discuss and communicate with graphs. The term central tendency refers to the middle, or typical, value of a set of data, which is most commonly measured by using the three m's: mean, median, and mode. The mean, median, and mode are known as the measures of central tendency. In this lesson, you will explore these three concepts. Focusing Your LearningLesson ObjectivesBy the end of this lesson, you should be able to:
Key TermsPresentationThe Mean, Median, and ModeMean, median, and mode are three basic ways to look at the value of a set of numbers. You will start by learning about the mean. The mean, often called the average, of a numerical set of data, is simply the sum of the data values divided by the number of values. This is also referred to as the arithmetic mean. The mean is the balance point of a distribution.
For instance, take a look at the following example. Use the formula to calculate the mean number of hours that Stephen worked each month based on the example below.
The calculations for the mean of a sample and the total population are done in the same way. However, the mean of a population is constant, while the mean of a sample varies from sample to sample.
Look at another approach. If you were to take a sample of 3 employees from the group of 8 and calculate the mean age for these 3 workers, would the results change? Use the ages 55, 29, and 46 for one sample of 3, and the ages 34, 41, and 59 for another sample of 3: The mean age of the first group of 3 employees is 43.33 years. The mean age of the second group of 3 employees is 44.66 years. The mean age for a sample of a population depends upon the values that are included in the sample. From this example, you can see that the mean of a population and that of a sample from the population are not necessarily the same. In addition to calculating the mean for a given set of data values, you can apply your understanding of the mean to determine other information that may be asked for in everyday problems.
All values for the means you have calculated so far have been for ungrouped, or listed, data. A mean can also be determined for data that is grouped, or placed in intervals. Unlike listed data, the individual values for grouped data are not available, and you are not able to calculate their sum. To calculate the mean of grouped data, the first step is to determine the midpoint of each interval or class. These midpoints must then be multiplied by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean. The following example will show how the mean value for grouped data can be calculated.
The mean is often used as a summary statistic. However, it is affected by extreme values (outliers): either an unusually high or low number. When you have extreme values at one end of a data set, the mean is not a very good summary statistic. Example: Outliers If you were employed by a company that paid all of its employees a salary between $60,000 and $70,000, you could probably estimate the mean salary to be about $65,000. However, if you had to add in the $150,000 salary of the CEO when calculating the mean, then the value of the mean would increase greatly. It would, in fact, be the mean of the employees' salaries, but it probably would not be a good measure of the central tendency of the salaries. In addition to calculating the mean for a given set of data values, you can also apply your understanding of the mean to determine other information that may be asked for in everyday problems. The MedianWhat is the Median? The median is the number that falls in the middle position once the data has been organized. Organized data means the numbers are arranged from smallest to largest or from largest to smallest. The median for an odd number of data values is the value that divides the data into two halves. If n represents the number of data values and n is an odd number, then the median will be found in the position.This measure of central tendency is typically used when the mean value is affected by an unusually low number or an unusually high number in the data set (outliers). Outliers distort the mean value to the extent that the mean value no longer accurately depicts the set of data. For example: If one of the houses in your neighborhood was broken down and maintained a low property value, then you would not want to include this property when determining the value of your own home. However, if you are purchasing a home in that neighborhood, you may want to include the outlier since it would drive down the price you would have to pay. Try a few examples to follow the steps needed to calculate the median.
Another way to look at the example is to narrow the data down to find the middle number.
Here is another example of how to calculate the median of a set of numbers.
The ModeWhat is the Mode? The mode of a set of data is simply the value that appears most frequently in the set. If two or more values appear with the same frequency, each is a mode. The downside to using the mode as a measure of central tendency is that a set of data may have no mode, or it may have more than one mode. However, the same set of data will have only one mean and only one median.
When determining the mode of a data set, calculations are not required, but keen observation is a must. The mode is a measure of central tendency that is simple to locate, but it is not used much in practical applications.
Remember that the mode can be determined for qualitative data as well as quantitative data, but the mean and the median can only be determined for quantitative data. Now that you have added to your knowledge by reviewing the lesson and the examples, it is time to watch the following Khan Academy videos. These videos will provide additional explanations and working examples of how to determine the mean, median, and mode to help you gain a better understanding of this new concept. In this lesson, you have learned how to calculate the mean, median, and mode of a set of data values. In addition, you have been introduced to other key terms such as measures of central tendency, unimodal, bimodal, and outliers. You also learned that the mode is the only measure of central tendency used in both quantitative and qualitative data. As with every lesson and module, you are encouraged to research how these topics pertain to your particular area of study within the world of information technology. By now you are very aware that not every topic in mathematics will be directly implemented in your future career field. However, do not rule out the possibility that this topic might be an integral part of your future until you do some research. 1) Complete the Statistics: Finding Mean, Median, and Mode. “Chapter 5: Measures of Central Tendency” by Merry, B. © 2012 retrieved from http://www.ck12.org/flexbook/chapter/9079 and used under a Creative Commons Attribution http://creativecommons.org/licenses/by/3.0/. This is an adaption of the lesson titled, “Measures of Central Tendency: Mean, Median, and Mode” by the National Information Security and Geospatial Technologies Consortium (NISGTC) is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0 |