Which Measure S of central location is are meaningful when the data are ordinal?

If the data being analyzed is qualitative, then the only measure of central tendency that can be reported is the mode.  However, if the data is quantitative in nature (ordinal or interval/ratio) then the mode, median, or mean can be used to describe the data.

With quantitative data, the shape of the distribution of scores (symmetrical, negatively or positively skewed) plays an important role in determining the appropriateness of the specific measure of central tendency to accurately describe the data.  If the distribution of scores is symmetrical or nearly so, the median and mean (as well as the mode) will be real close to each other in value.  In this case, the mean is the value of central tendency that is usually reported.  However, if the distribution of scores is positively or negatively skewed, the mean will tend to either overestimate (in positively skewed distributions) or underestimate (in negatively skewed distributions)  the true central tendency of the distribution.  In extreme cases of skewed data, the mean can lie at a considerable distance from most of the scores.  Therefore, in skewed distributions, the median will tend to be the more accurate measure to represent the data than the mean because the median can never have more than one half the scores above or below it.
To describe data solely by its measure of central tendency, however, can be quite misleading.  Two distributions of scores may have the same mean, median, and mode but differ in their variability or dispersion of scores.  That is, the scores in one distribution may tend to cluster more closely around the measure of central tendency than the scores in the other distribution.  To further describe distributions, another statistical measure in addition to a measure of central tendency, is needed to reflect the amount of spread or variability of the scores.  Statisticians have suggested several measures, called measures of dispersion (variability), that indicate for any distribution the spread or variability of the scores in the distribution.

    As with measures of central tendency, different measures of dispersion are appropriate for different problems.  The most common measures of dispersion are the range, variance, and standard deviation.  The appropriateness of each would depend, in part, on the type of data that you have and which measure of central tendency you are using.  If the data is qualitative, then there is no measure of variability to report.  For data that is quantitative (ordinal and interval/ratio) all three measures are possible.  However, the shape of the distribution of scores and the measure of central tendency reported will determine which measure of variability  to use.  If the distribution of scores is symmetrical in nature, then the measures of variability usually reported are the variance and standard deviation, although the standard deviation would be more interpretable.  However, if the data is skewed, then the measure of variability that would be appropriate for that data would be the range.

In summary, with qualitative data, the only additional measure to be concerned with to further describe that data would be the mode.  With quantitative data, the mean, variance, and standard deviation would be appropriate with symmetrical distributions while the median and range would be appropriate when the distribution is skewed (either positively or negatively).
 

Measures of central tendency are numbers that describe what is average or typical within a distribution of data. There are three main measures of central tendency: mean, median, and mode. While they are all measures of central tendency, each is calculated differently and measures something different from the others.

The Mean

The mean is the most common measure of central tendency used by researchers and people in all kinds of professions. It is the measure of central tendency that is also referred to as the average. A researcher can use the mean to describe the data distribution of variables measured as intervals or ratios. These are variables that include numerically corresponding categories or ranges (like race, class, gender, or level of education), as well as variables measured numerically from a scale that begins with zero (like household income or the number of children within a family).

A mean is very easy to calculate. One simply has to add all the data values or "scores" and then divide this sum by the total number of scores in the distribution of data. For example, if five families have 0, 2, 2, 3, and 5 children respectively, the mean number of children is (0 + 2 + 2 + 3 + 5)/5 = 12/5 = 2.4. This means that the five households have an average of 2.4 children.

The Median

The median is the value at the middle of a distribution of data when those data are organized from the lowest to the highest value. This measure of central tendency can be calculated for variables that are measured with ordinal, interval or ratio scales.

Calculating the median is also rather simple. Let’s suppose we have the following list of numbers: 5, 7, 10, 43, 2, 69, 31, 6, 22. First, we must arrange the numbers in order from lowest to highest. The result is this: 2, 5, 6, 7, 10, 22, 31, 43, 69. The median is 10 because it is the exact middle number. There are four numbers below 10 and four numbers above 10.

If your data distribution has an even number of cases which means that there is no exact middle, you simply adjust the data range slightly in order to calculate the median. For example, if we add the number 87 to the end of our list of numbers above, we have 10 total numbers in our distribution, so there is no single middle number. In this case, one takes the average of the scores for the two middle numbers. In our new list, the two middle numbers are 10 and 22. So, we take the average of those two numbers: (10 + 22) /2 = 16. Our median is now 16.

The Mode

The mode is the measure of central tendency that identifies the category or score that occurs the most frequently within the distribution of data. In other words, it is the most common score or the score that appears the highest number of times in a distribution. The mode can be calculated for any type of data, including those measured as nominal variables, or by name.

For example, let’s say we are looking at pets owned by 100 families and the distribution looks like this:

Animal   Number of families that own it

• Dog: 60
• Cat: 35
• Fish: 17
• Hamster: 13
• Snake: 3

The mode here is "dog" since more families own a dog than any other animal. Note that the mode is always expressed as the category or score, not the frequency of that score. For instance, in the above example, the mode is "dog," not 60, which is the number of times dog appears.

Some distributions do not have a mode at all. This happens when each category has the same frequency. Other distributions might have more than one mode. For example, when a distribution has two scores or categories with the same highest frequency, it is often referred to as "bimodal."

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Crossman, Ashley. "The Difference Between the Mean, Median, and Mode." ThoughtCo. https://www.thoughtco.com/measures-of-central-tendency-3026706 (accessed December 1, 2022).

Which type of central tendency measure is used ordinal scale?

Which measure of location is most suitable for an ordinal variable?

Which measure of central location is meaningful when the data are qualitative?

What is the measure of central location for the data?