The ___________ are all valid measures of central tendency,

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.

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Which is the most reliable measure of central tendency?
What are the four types of central tendency?
Which measure of central tendency always exist?
Which of the following is not a valid measures of central tendency?

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.

Example 1

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.

Mode

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.

Notation and formulae

The mean of a sample is typically denoted as

(read as x bar). The mean of a population is typically denoted as μ (pronounced mew). The sum (or total) of measures is typically denoted with a Σ. The formula for a sample mean is

where n is the number of values.

Mean for grouped data

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.

Median 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 b is 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.

Which is the most reliable measure of central tendency?

Mean is generally considered the best measure of central tendency and the most frequently used one. However, there are some situations where the other measures of central tendency are preferred. There are few extreme scores in the distribution.

What are the four types of central tendency?

The four measures of central tendency are mean, median, mode and the midrange. Here, mid-range or mid-extreme of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set.

Which measure of central tendency always exist?

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average. For data from skewed distributions, the median is better than the mean because it isn't influenced by extremely large values.