CHAPTER 3 CENTRAL TENDENCY ANALYSES The next concept in the sequential statistical steps approach is calculating measures of central tendency. Measures of central tendency represent some of the most simple forms of data analysis. Each of the measures explained in this chapter are calculated in an effort to identify a specific value that can be considered representative or typical of the entire distribution. In everyday language, most people refer to this value as the average. Students are often presented with information about national averages, class averages, and grade point averages. Statisticians refer to this statistic as the mean ( ). Two additional measures of central tendency are commonly employed by those working with statistics. These measures are the mode (Mo) and the median (Mdn). This chapter provides an explanation of each of these measures and how they can be calculated from a distribution of data. The mode (Mo) is the simplest measure of central tendency and is easy to derive. The mode is observed rather than computed. The mode (Mo) of a distribution of values is the value which occurs most often. Since all of the values in a distribution occur only once in a simple distribution, there is no mode for a simple distribution. For the frequency distribution in Figure 3:1, the mode (Mo) is 60 because this value occurs more times than any of the other values. 26
FREQUENCY DISTRIBUTION FIGURE 3:1 UNIMODAL POLYGON X f 90 1 80 2 70 3 60 (Mo) 4 50 3 40 2 30 1 The distribution shown in Figure 3:1 is said to be unimodal because it has only one mode. Some distributions may contain two or more modes. If two values in a distribution occur equally more often than the other values, this distribution is referred to as bimodal. The distribution in Figure 3:2 is an example of a bimodal distribution. FIGURE 3:2 BIMODAL DISTRIBUTION BIMODAL POLYGON X f 95 2 85 (Mo) 4 75 2 65 (Mo) 4 55 2 27
This distribution has two modes, 65 and 85, because each of these values occur four times. For a grouped frequency distribution, the mode is in the interval having the greatest frequency and is called the modal interval. An example of a mode for a grouped frequency distribution is shown in Figure 3:3. FIGURE 3:3 FINDING THE MODE FOR GROUPED DATA 1 X Midpoint f Cumulative Frequency (cf) 20-22 21 2 24 17-19 18 3 22 14-16 15 (Mo) 7 19 11-13 12 6 12 8-10 9 4 6 5-7 6 2 2 N= 24 The information provided in the frequency column indicates that the modal interval for this distribution is the 14-16 interval. The mode represents the midpoint of the modal interval (15). The mode of a distribution is not always at the middle or center of a distribution. It can be located at any point within the range of observations that make up the distribution examined. The reasons for this are explained later in the chapter. The mode can provide a quick reference point, but it does not always yield an accurate evaluation of the central tendency of a distribution of values. The other measures of central tendency are generally more useful in this regard. 1 W hen data are grouped, the determination of measures of central tendency will always be less accurate. 28
The second measure of central tendency is the median. In comparison to the mode, the median (Mdn) is a slightly more complex and useful statistic. The median represents the middle point of a distribution of data. It is the point at which exactly half of the observed values in the distribution are higher and half of the observed values are lower. When determining the median for a frequency distribution with an odd number of observations, the location of the median is 2 determined by multiplying the total number of observed values by (.5). Once the position is calculated, determine the value of the observation in that position within the distribution. If the median position is in between two positions in the distribution, the median is calculated by averaging those two numbers together. Figure 3:4 provides an example of this process. FIGURE 3:4 X f cf 70 2 25 60 3 23 50 4 20 40(Mdn) 7 16 30 4 9 20 3 5 10 2 2 N= 25 FINDING THE MEDIAN OF A DISTRIBUTION (Odd Number of Values) 2 Be very careful. Please note that this calculation only determines the position and NOT the actual median. 29
th The calculated value of 12.5 identifies the median is the 12.5 value. That position in the distribution falls among the 7 observations with values of 40 (40,40,40,40,40,40,40). These th th th th th th th seven observations represent the 10, 11, 12, 13, 14, 15, and 16 (cf) positions in the th th distribution. Since the values for the 12 and 13 positions are both 40, the median of this distribution is 40. Finding the median for grouped frequency distributions is a more complex procedure. The median for grouped data will not be explained at this point because it would divert our attention from measures of central tendency. The median for grouped data, which is equivalent to the 50th percentile, will be discussed in a later chapter of the text. The third and most widely used measure of central tendency is the mean( 3 ). The mean of a distribution of values is obtained by adding all of the values and dividing the sum by the number (N or n) of values. When obtaining the mean for a simple distribution of sample data, the formula for the mean is: The following are the explanations of the components of the formula for the mean. the mean. an individual value in the distribution. the Greek letter Sigma which means the "sum of". the sum of all the values of X. 3 The symbol for mean is a bar over a capital X for a sample and the Greek letter for a population 30
n = the number of values in the distribution. Calculating the mean for simple distributions is not a difficult task. Suppose a researcher obtained a simple distribution or set of scores on a civil service exam as follows: X = 70, 80, 50, 30, 90, 80, 75, 95, 100, 105. The mean would be determined by following two simple steps. First, sum all the values of X. Then divide by the number of values in the distribution to determine the value of the mean. Step 1: = 70 + 80 + 50 + 30 + 90 + 80 + 75 + 95 + 100 + 105 = 775 Step 2: 4 The mean civil service exam score is 77.5. The mean score is the typical performance level of all the candidates taking the exam. Obtaining the mean for a frequency distribution is only slightly more complicated. The formula for the mean of a frequency distribution is: The following are the explanations of the symbols for this formula: Mean the frequency of each value of X times that value The sum of all values of X multiplied by their frequency n = the total number of values. 4 Rounding in this text is always two places. For example, 77.5033 is 77.50, 77.5453 is 77.55. The decision is made at the third decimal place by rounding up if the digit is 5 or above and down if 4 or below. 31
A hypothetical distribution is shown in Figure 3:6. The use of the frequency (f) and cumulative frequency (cf) simplifies the calculations as well as the organization of the data. The mean for this distribution is not equivalent to the value which occurs most often (Mo) or the value that is in the middle of the distribution (Mdn). FIGURE 3:6 FINDING THE MEAN OF A FREQUENCY DISTRIBUTION X f fx cf 21 1 21 14 19 1 19 13 18 2 36 12 12 1 12 10 10 2 20 9 9 3 27 7 7 1 7 4 5 1 5 3 2 2 4 2 n=14 So far, the mean has been calculated for distributions of data which were not grouped. In actual research situations, the data may not always be ungrouped or organized in a simple frequency distribution. Suppose a researcher had data grouped into class intervals with a width of ten. In such cases, the mean is calculated in a slightly different manner. The process begins 32
with a determination of the midpoint for each class interval. this case, the mean must be obtained by determining the midpoint of each interval. The midpoint is calculated by adding the highest value included in the interval to the lowest value included in the interval and dividing the sum by two. The value for the midpoint is then used to calculate. Figure 3:7 provides an example of a grouped frequency distribution and the procedure for determining its mean. FIGURE 3:7 DETERMINING MEAN (Grouped Frequency Distribution) X Midpoint f cf fx 90-99 94.5 2 24 189 80-89 84.5 3 22 253.5 70-79 74.5 7 19 521.5 60-69 64.5 6 12 387 50-59 54.5 4 6 218 40-49 44.5 2 2 89 n=24 1658 In comparing the mean, mode, and median, one concludes that the mean is a more precise measure of central tendency for interval data. The mean is also used as a building block for many subsequent statistical calculations and in statistical inference. One of the most useful explanatory comparisons of the mode, median, and mean is related to the concept of symmetry or shape of a distribution. In a unimodal symmetrical distribution, the mode, median, and mean 33
are the same value. The shape of the distribution is a bell shaped curve with the mode, median, and mean occurring at the peak of the distribution. The comparative location of these measures of central tendency for a unimodal distribution is shown in Figure 3:8. If the distribution is bimodal and symmetrical the mean and median will be the same value, and the modes will be at the peaks of the distribution as shown in Figure 3:9. FIGURE 3:8 SYMMETRICAL DISTRIBUTION 34
FIGURE 3:9 BIMODAL DISTRIBUTION The mode, median, and mean for a skewed distribution are also important. Any asymmetrical frequency distribution with values spread out toward one direction more than the other is said to be skewed. The mode, median, and the mean are not the same value in a skewed distribution. The mean will always be located in the tail of a skewed distribution. In a positively skewed distribution, outlying values on the right side of the distribution have the effect of pulling the mean in that direction. The tail of the distribution will be on the right. The mean of a positively skewed distribution will have a higher value than the median. The mode will have the lowest value of the three measures These ideas are illustrated in Figure 3:10 for a positively skewed distribution. The relative positions of the measures of central tendency are shown. 35
FIGURE 3:10 POSITIVE SKEWED DISTRIBUTION If the tail of the distribution is skewed to the left, the skew is said to be negative. In a negatively skewed distribution, the mode is the largest value, the median in the center, and the mean is the smallest of the three measures of central tendency. A negatively skewed distribution with the relative positions of the mode, median, and mean is shown in Figure 3:11. FIGURE 3:11 NEGATIVE SKEWED DISTRIBUTION Since the measures of central tendency summarize distributions, these measures are an important first step in data analysis. Sometimes it is sufficient to report only the most useful of the three 36
measures of central tendency as a final interpretation of the data. For a hurried, rough estimate of central tendency, the mode may be sufficient. When a distribution is highly skewed, the median is the best measure of central tendency. Overall, the mean is generally the best measure of central tendency since it is often an important first step for subsequent analyses. The objectives of the research project and levels of measurement are probably the most important factors in determining which measure of central tendency is appropriate. The mode may be used for nominal, ordinal, and interval data. The median may be used with ordinal and interval. Interval data are required for calculating a mean. For a review of the measures of central tendency, the student should consult the sequential statistical steps at the end of this chapter. 38
Step 1 Organize Data SEQUENTIAL STATISTICAL STEPS CENTRAL TENDENCY ANALYSES What is the first step in calculating measures of central tendency? Create a frequency distribution. Step 2 Mode (Mo) What is the value (unimodal) or values (bimodal) which occur(s) most frequently in the distribution? Mode. Step 3 Median Position What is the position of value which divides the distribution in two equal halves? Identify the median position by multiplying the number of values in the distribution by.5 Step 4 Median Fifty percent of the values are above and below what value? Median Step 5 What is the sum of all the individual values? Add all of the values in a distribution including those that occur more than once. Step 6 What is the sum of the individual values divided by the total number of values? Mean Step 7 Symmetrical or Skewed What is the shape of the distribution? Skewed or Symmetrical. 39
EXERCISES - CHAPTER 3 (1) Find the mean, mode, and median for the following sets of data. Show all work and organize in a solution matrix format. (A) 10, 70, 40, 50, 10, 30, 30, 50, 60, 60, 50, 40, 30, 50, 40, 50, 40, 40, 40, 50, 50, 70, 80, 90, 60, 60, 60, 50, 10, 100, 60, 20, 20, 60 (B) 100, 90, 40, 50, 80, 70, 60, 50, 60, 70, 80, 70, 70, 90, 30, 30, 70, 40, 40, 40, 40, 70, 70, 20, 10, 15, 5, 70, 70 (C) 26, 32, 41, 58, 69, 73, 85, 97, 102, 114, 120, 130 (D) 10, 16, 18, 20, 21, 36, 64, 72, 75, 90, 101 (2) For 1A and 1B, construct a frequency polygon and indicate the skew. (3) Sixty-three students were asked to give their college GPA. Would calculating the median GPA be the most useful measure of central tendency? If not, why not? (4) For a unimodal distribution which has a severe positive skew, draw a frequency polygon showing the positions of the mode and median relative to the mean. (5) What is the mode of the following distribution? Find the mean and median. X = 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11 40
(6) For the following distribution find the mean, mode, and median. Draw a frequency polygon for this distribution. X f 100 1 90 2 80 3 70 5 60 3 50 2 40 1 30 1 (7) For the following classes find the mean, median, and mode number of minutes studied for a final exam. What is the skew for each class? On the average which class studied the most? Which studied the least? Freshmen Sophomores Juniors Seniors 30 20 10 105 20 30 40 50 60 25 30 80 40 40 40 40 50 60 60 120 80 30 25 40 5 30 40 50 10 20 40 50 20 60 60 50 30 30 50 90 41
(8) For the following data calculate the mean advertising dollars spent for both candidate A and B. ADVERTISING DOLLARS SPENT Selected Cities Candidate A Candidate B 1 101 59 2 210 205 3 162 79 4 345 340 5 57 24 6 750 720 7 33 41 8 205 200 9 38 35 10 300 200 On the average, which candidate spent the most? (9) Using a grouped frequency distribution with intervals of 101 and beginning with 100 calculate the mean and mode for the following distribution of values. 100, 100, 100, 100, 110, 110, 110, 110, 123, 145, 500, 300, 120, 420, 620, 540, 186, 310, 200, 690, 695, 220, 210, 200, 210, 210,210, 418, 423, 419, 475, 500, 520, 530, 560, 700 42