Which of the following measures of central tendency involves dividing by how many values there are?
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Measures of central tendency are numbers that tend to cluster around the “middle” of a set of values. Three such middle numbers are the mean, the median, and the mode. Show
For example, suppose your earnings for the past week were the values shown in Table 1.
 Mean You could express your daily earnings from Table 1 in a number of ways. One way is to use the average, or mean, of the data set. The arithmetic mean is the sum of the measures in the set divided by the number of measures in the set. Totaling all the measures and dividing by the number of measures, you get $1,000 ÷ 5 = $200. Median Another measure of central tendency is the median, which is defined as the middle value when the numbers are arranged in increasing or decreasing order. When you order the daily earnings shown in Table 1, you get $50, $100, $150, $350, and $350. The middle value is $150; therefore, $150 is the median. If there is an even number of items in a set, the median is the average of the two middle values. For example, if we had four values—4, 10, 12, and 26—the median would be the average of the two middle values, 10 and 12; in this case, 11 is the median. The median may sometimes be a better indicator of central tendency than the mean, especially when there are outliers, or extreme values. Given the four annual salaries of a corporation shown in Table 2, determine the mean and the median.The mean of these four salaries is $275,000. The median is the average of the middle two salaries, or $40,000. In this instance, the median appears to be a better indicator of central tendency because the CEO's salary is an extreme outlier, causing the mean to lie far from the other three salaries. Another indicator of central tendency is the mode, or the value that occurs most often in a set of numbers. In the set of weekly earnings in Table 1, the mode would be $350 because it appears twice and the other values appear only once. The mean of a sample is typically denoted as
where n is the number of values. Occasionally, you may have data that consist not of actual values but rather of grouped measures. For example, you may know that, in a certain working population, 32 percent earn between $25,000 and $29,999; 40 percent earn between $30,000 and $34,999; 27 percent earn between $35,000 and $39,999; and the remaining 1 percent earn between $80,000 and $85,000. This type of information is similar to that presented in a frequency table. Although you do not have precise individual measures, you still can compute measures for grouped data, data presented in a frequency table.The formula for a sample mean for grouped data is
where x is the midpoint of the interval, f is the frequency for the interval, fx is the product of the midpoint times the frequency, and n is the number of values. For example, if 8 is the midpoint of a class interval and there are ten measurements in the interval, fx = 10(8) = 80, the sum of the ten measurements in the interval. Σ fx denotes the sum of all the products in all class intervals. Dividing that sum by the number of measurements yields the sample mean for grouped data. For example, consider the information shown in Table 3.
Substituting into the formula:
Therefore, the average price of items sold was about $15.19. The value may not be the exact mean for the data, because the actual values are not always known for grouped data. As with the mean, the median for grouped data may not necessarily be computed precisely because the actual values of the measurements may not be known. In that case, you can find the particular interval that contains the median and then approximate the median.Using Table 3, you can see that there is a total of 32 measures. The median is between the 16th and 17th measure; therefore, the median falls within the $11.00 to $15.99 interval. The formula for the best approximation of the median for grouped data is
where L is the lower class limit of the interval that contains the median, n is the total number of measurements, w is the class width, f medis the frequency of the class containing the median, and Σ f bis the sum of the frequencies for all classes before the median class. Consider the information in Table 4.
As we already know, the median is located in class interval $11.00 to $15.99. So L = 11, n = 32, w = 4.99, f med = 4, and Σ f b= 14. Substituting into the formula:
Figure 1.For a symmetric distribution, mean, median, and mode are equal.
Figure 2 shows a negatively skewed curve. Figure 2.A negatively skewed distribution, mean < median < mode.
Figure 3 shows a positively skewed curve. Figure 3.A positively skewed distribution, mode < median < mean.
Which measure of central tendency is divided?The median is the middle value in distribution when the values are arranged in ascending or descending order. The median divides the distribution in half (there are 50% of observations on either side of the median value).
What measure of central tendency is calculated by adding all the values and dividing the sum by the number of values?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.
What measure of central tendency that divides a set of scores in two equal number of scores?the median, symbolized Mdn, is the middle score. It cuts the distribution in half, so that there are the same number of scores above the median as there are below the median. Because it is the middle score, the median is the 50th percentile.
What measure of central tendency refers to the summation of all scores divided by the total number of cases?Mean. The arithmetic mean is the most common measure of central tendency. It is computed by summing all the scores (sigma or Σ) and dividing by the number of scores (N): Where X is the mean, ∑x is the addition or summation of all scores, and N is the number of cases.
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