The Advantages and Limitations of the Arithmetic Mean: "When to Use and Not Use
The arithmetic mean, often simply called the "average", is one of the most widely used measures of central tendency in statistics. While it has many advantages in summarizing data with a single value, there are also important limitations to be aware of. Let us examine some of the key arguments on both sides:
In favour of the arithmetic mean being an "Advantage" statistic:
It incorporates every single data point into its calculation by summing them all up and dividing by the count. This gives an equal "democratic" weight to each observation when determining the central value.
For symmetric, bell-shaped distributions without outliers or skewness, the arithmetic mean provides an excellent representation of the "middle" or typical value of the data.
It follows the intuitive principle that positive and negative deviations from the mean should balance out and cancel each other.
The arithmetic mean has many useful mathematical properties, such as the sum of deviations from the mean equalling zero, that make it well-behaved analytically
Arguments that the arithmetic mean is a "Disadvantage":
It is heavily influenced by outliers or extreme values in the data.
Just a few unusually large or small values can dramatically skew the arithmetic mean away from the bulk of the data.
For skewed distributions (e.g. income), the arithmetic mean may not align with our intuitive sense of the "middle" value and can give a distorted representation.
By reducing an entire dataset to a single number, the arithmetic mean discards all other information about the shape, spread, outliers etc. in the distribution
For categorical/ordinal data, calculating the arithmetic mean makes little sense since values represent categories rather than true numerical quantities.
Other measures of central tendency like the median or trimmed mean may provide more robust and resistant alternatives for skewed or heavy-tailed data
In essence, the arithmetic mean succeeds at providing a balanced, intuitive summary for symmetric data distributions without outliers. Its equal weighting of all values provides a "democratic" central value.
However, it can indeed become a "disadvantage" when applied to skewed data with extreme values, ordinal data, or any situation where a single number cannot adequately capture the full complexity of the distribution.
Like any statistical tool, using the arithmetic mean judiciously, being aware of its limitations, and considering alternative robust measures for non-ideal data, is crucial to avoid being led astray. The arithmetic mean is powerful but not a universal panacea - understanding when it is a fair statistic versus an oversimplification is key.
In conclusion, while the arithmetic mean is one of the most widely used and well-known measures of central tendency, it does have some significant disadvantages and limitations that must be carefully considered.
A key drawback is its sensitivity to outliers and extreme values in the data. Just a few unusually large or small observations can dramatically skew and distort the arithmetic mean, causing it to no longer reflect the true central value of the bulk of the data distribution.
Additionally, for skewed or heavy-tailed distributions like income or wealth data, the arithmetic mean may counterintuitively fail to align with our sense of the "middle" or typical value. Its single summarizing number cannot adequately capture the asymmetry and nuances of the full data shape.
The arithmetic mean also discards all other information about spread, outliers, skewness and kurtosis when reducing an entire dataset down to one central value.
This oversimplification means important details about the distribution are lost.
Furthermore, calculating the arithmetic mean makes little sense for categorical, ordinal or ranked data where values represent qualitative groups or ranks rather than true numerical quantities.
In such non-ideal scenarios, more robust alternative measures of central tendency like the median or trimmed mean may provide better representative values that are resistant to outlier effects and skewness.
While simple and intuitive for symmetric, well-behaved distributions, the arithmetic mean has clear limitations. Using it judiciously, understanding its assumptions, and considering alternative robust statistics for skewed or outlier-prone data is crucial. The arithmetic mean is powerful but not a universal panacea - its appropriate use depends on closely matching the data characteristics.