Following the American Psychological Association’s Guidelines
What are the characteristics of a population for which a mean/median/mode would be appropriate? Inappropriate?
The mean, median and mode are collectively known in statistics as measures of central tendency. A measure of central tendency is a numerical or categorical value that is supposed to be the middle-ground or center or average description of the entire data set whether obtained from a sample or the data population. Measures of central tendency are also known as summary statistics or measures of central location precisely because they describe the population data on the average across a wider range of values.
The mean is the most popular measure and its value as arithmetic mean is generated by taking the sum of all the values in the data set. This is then divided by the number of values or observations in the same data set. The mean may also be a weighted sum of values, and this process would yield the weighted arithmetic mean. The median is typically the middle value in a range of numerical values in a population data. The median is generated by numerically sorting a data set by order of magnitude, from lowest to highest. The mid-range value becomes the median of the population data. Often in population data, a range of data is summed up and the average or mean is obtained to generate the middle value, which is the median. The mode, which is the third measure of central tendency, captures the value that is the most-commonly observed in the data set. Graphically illustrated, the mode is the area in a bar chart or histogram that has longest or highest bar, denoting frequent observations of that particular value or characteristic. Thus, the mode is the most popular observed data response or option, having the most frequent occurrence in the population data available.
The mean is ideal for use with continuous data but it can be used with discrete data as well. The mean is particularly useful when you want to minimize error in predicting any value in the data set. The mean includes every value in your data set, therefore it truly represents a picture of the entire population. When the deviation or difference from the mean of every observation is added up, the sum is always equal to zero. The downside to the mean is that it is not reliable as a central measure in the presence of outliers, those values that are placed at the extreme of the data set, which may be unusually large or small. These outliers tend to place the mean deceptively away from the central population value.
Unlike the mean, the median is less affected by values that lie uniquely on the extreme ends of the data range. The median does not alter significantly or move away much from the central or mid-range value in the presence of these outliers. The median is preferred over the mean or mode when the population data does not conform to the characteristics of a normal distribution curve (that is, the frequently distribution is skewed or the curve is leaning to the right or left). With a normal distribution, the three measures of central tendency take on identical values, with the average value representing the most commonly observed value and the mid-range value as well. When data is skewed, the mean is moved farther away from the typical normal value. The median, however, will tend to stay on that same central position. The mode is generally used for categorical data when we want to know the most frequent common category. However, often the mode is not unique and when we adopt the mode there often is a problem when two or more values appear to indicate the central position in a range of values. The mode is inappropriate for continuous data. The mode also becomes unreliable as a central measure if the most common value is very far from the rest in the data set.
References
Laerd Statistics. (2013). Measures of Central Tendency. Retrieved from https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php.
QuickMBA. (2010). Statistics: Central Tendency. Retrieved from http://www.quickmba.com/stats/centralten/
Stat Trek. (2013). The Mean and Median: Measures of Central Tendency. Retrieved from http://stattrek.com/descriptive-statistics/central-tendency.aspx.
