Measures of central tendency: Focusing on the mean, median, and mode

Finding a dataset's middle or average is critical and involves measuring central tendency. The central tendency's most common measures include the mode, median, and mean. The mode is the most repeated measure in a data set. As the name suggests, the median is in the middle of an ordered data set, while the mean is determined by dividing the sum of all values by the number of observations. On top of the central tendency measures, the variability and distribution of a dataset should be assessed. To that end, one needs to look at how the data are distributed and which distributions are more relevant.

Defining distributions and central tendency measures

Suppose one has a dataset with n number of values. In that case, this data set is called a distribution. In statistics, two types of distribution exist: discrete and continuous distributions. Binomial and Poisson are examples of discrete distributions. In contrast, normal and exponential distributions are examples of continuous distributions. In discrete distributions, the random variable, x, can assume a finite number of values (all positive integers). Nonetheless, the x variable can take an infinite number of values in continuous distributions.

Normal distribution

Normal distribution, by far, is the most famous and used distribution. In a normal distribution, data are symmetric with no apparent skewness. In other words, one can see most values around the central region. However, some values may seem comparatively moving away from the center. The most distinct property of the normal distribution is that the arithmetic mean, median, and mode are the same. One of the most famous normal distributions is the distribution of IQ scores, meaning that if you know your IQ score, you can calculate how well you are doing compared to the rest of the population. In order to decide whether the data are normally distributed, the ideal option is to draw the histogram. Then, one can decide how well the data fit the bell-shaped curve (normal distribution). A histogram shows where the frequency of responses tends to concentrate or how it spreads out substantially.

How about skewed distributions?

When more values are on one side of the center relative to the other side, and the mean, median, and mode are not the same, the distribution is skewed. One can easily see that one side is profoundly more spread out and has a longer tail. It suggests that fewer scores are at one end compared to the other. The tail's direction dictates the side of the skew. Lower scores prevail when the distribution is positively skewed and a spread-out tail on the right is apparent. Contrarily, higher scores are predominant, and a spread-out tail is evident on the left if the distribution a negatively skewed.

The mode is the lowest, and the mean is the highest in the positively skewed distribution, whereas the mean is the lowest, and the mode is the highest in the negatively skewed distribution.

Should we use the mean, median, or mode?

Most researchers prefer using all these three measures of central tendency because they have specific strengths but also certain limitations. However, in particular cases, only one or two may be relevant and determinable. It all depends on what the measurement level is for the variable. Calculating mode is possible for all measurement levels, even if the variable is nominal. Nonetheless, the median is only possible for ordinal, interval, and ratio levels. The mean has the most limitation needing equal spacing; thus, it is only valid for interval or ratio levels. The mean is exceedingly sensitive to outliers. Therefore, using the median instead of the mean can be preferred if one has extreme values in a data set. For instance, the salary data in the US is generally given as median salaries because the salary is prone to extreme values.

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This article explains how to determine  central tendency in statistical studies. To give you an opportunity to practice proofreading, we have left a few spelling, punctuation, or grammatical errors in the text. See if you can spot them! If you spot the errors correctly, you will be entitled to a 10% discount.