What are the key differences between mean, median, and mode in skewed data?

The mean is a central tendency measure. As the name suggests it primarily describes what is going on in the central part of a data distribution. The most common mean is the arithmetic mean. This measure of central tendency is not robust to extreme values. In such a case if is preferable to rely on the median which is a robust central tendency indicator.

The mean is a descriptive measure related to the central tendency as the average of all it is values in a dataset, a skewed data tell us that the distribution is not skewed. When the mean and median are greater than the mode so the data is skewed to the right

The median is a central tendency measure. As the name suggests it primarily describes what is going on in the central part of a data distribution. It is the value of the ordered data distribution below which there are 50% of there data and above which there are 50% of the data. The median is a robust central tendency measured as opposed to the arithmetic mean. The median is used to construct box-Plots. It is also known as the second quartile.

The median is also a descriptive measure related to the central tendency as it is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. When the mean and median are less than the mode, the distribution is skewed to the left.

The mode is a central tendency measure. As the name suggests it primarily describes what is going on in the central part of a data distribution. The mode counts the most frequent values in the distribution. There can be one mode (unimodal distribution), two modes (bimodal distribution), more than two modes (multimodal distribution), or no mode at all whenever there are no repeats in the data distribution. The mode is particularly designed for discrete variables.

The mode is also used to describe the center of data and it tells you that the value that occurs most often in a data set is the mode. When the mean, median and mode are equal so we have a symmetric distribution

The focus solely on skewness impact on the mean, median, and mode ignores where it has the greatest impacts. This greatest impact is not at the middle of the distribution but in estimation of a distribution's tails. In normal theory, sample size is routinely considered "large" for n>=30. With skewness "large" is strongly related to the degree of skewness. Estimation of the asymptotic level of skewness in the population distribution is strongly biased low when skew is present and sample size is not "large". "Large" for estimation of skewness is a function of the population skewness and may require n to be in the hundreds or more to get close to the population skewness. It is possible to approximately correct for this underestimation bias.

Be sure to check the skewness impact on your data especially when dealing with the mean, as the mean is greatly influenced by outliers while the median is robust against outliers and tends to provide a better representation of the central tendency.

Um histograma pode te apresentar facilmente uma distor??o que faz com que a média seja significativamente diferente da mediana. Essas assimetrias podem indicar que o seu processo funciona de maneira "viesada" ou que você tem fontes severas de varia??o no seu processo.

When analyzing skewed data, the median often serves as a more robust measure of central tendency than the mean, particularly in the presence of outliers. Descriptively, the median offers valuable insights, while the mode—the most frequent value—can aid in identifying common observations. In inferential analysis, while the mean remains relevant, adjustments or robust methods may be necessary to address skewness. Techniques like bootstrapping, non-parametric methods, or log transformations can help. Engaging an experienced statistician ensures proper handling of these complexities, maintaining the integrity and reliability of the analyses.

There are several types of means (arithmetic, geometric, harmonic, ...). It is important to look at statistical textbook to understand what they mean, how they are calculated and in which context to use them.