When would you use the median rather than the mean as a measure of central tendency?

The mean

The arithmetic mean is what most people simply know as the average. To be exact, however, we have to note that the mean is just one type of average. Before getting lost in the intricacies of these terms, let’s move on to the definition of the mean

Inhaltsverzeichnis Show

When to Use the Mean
When to Use the Median
Additional Resources
When would you use median rather than mean?
Why would we choose to use the median as the measure of central tendency rather than the mean?
When would you use median as a measure of central tendency?
When should you use the median rather than the mean to measure the center of a distribution of quantitative data?

Given a vector $x$ with $n$ entries, the mean is defined as

\[\overline{x} = \frac{1}{n} \sum_{i = 1}^n x_i\]

Assume we have $x = (30, 25, 40, 41, 30, 41, 50, 33, 40, 1000)$, what would be the mean? We can compute it in the following ways:

x <- c(30, 25, 40, 41, 30, 41, 50, 33, 40, 1000)
# the way of the beginner (don't do this!):
x.mean <- 0
for (xi in x) {
    x.mean <- x.mean + xi
}
x.mean <- x.mean / length(x)
print(x.mean)

## [1] 133

# a better way:  
x.mean <- sum(x) / length(x)
print(x.mean)

## [1] 133

# the right way:
x.mean <- mean(x)
print(x.mean)

## [1] 133

As you can see, you can simply use the mean function rather than having to implement the mean by yourself.

The mean of a dataset represents the average value of the dataset. It is calculated as:

Mean = Σxi / n

where:

Σ: A symbol that means “sum”
xi: The ith observation in a dataset
n: The total number of observations in the dataset

The median represents the middle value of a dataset. It is calculated by arranging all of the observations in a dataset from smallest to largest and then identifying the middle value.

For example, suppose we have the following dataset with 11 observations:

Dataset: 3, 4, 4, 6, 7, 8, 12, 13, 15, 16, 17

The mean of the dataset is calculated as:

Mean = (3+4+4+6+7+8+12+13+15+16+17) / 11 = 9.54

The median of the dataset is the value directly in the middle, which turns out to be 8:

3, 4, 4, 6, 7, 8, 12, 13, 15, 16, 17

Both the mean and the median estimate where the center of a dataset is located. However, depending on the nature of the data, either the mean or the median may be more useful for describing the center of the dataset.

When to Use the Mean

It’s best to use the mean to describe the center of a dataset when the distribution is mostly symmetrical and there are no outliers.

For example, suppose we have the following distribution that shows the salaries of residents in a certain city:

Since this distribution is fairly symmetrical (if you split it down the middle, each half would look roughly equal) and there are no outliers, we can use the mean to describe the center of this dataset.

The mean turns out to be $63,000, which is located approximately in the center of the distribution:

When to Use the Median

It is best to use the median when the distribution is either skewed or there are outliers present.

Skewed Data:

When a distribution is skewed, the median does a better job of describing the center of the distribution than the mean.

For example, consider the following distribution of salaries for residents in a certain city:

The median does a better job of capturing the “typical” salary of a resident than the mean. This is because the large values on the tail end of the distribution tend to pull the mean away from the center and towards the long tail.

In this example, the mean tells us that the typical individual earns about $47,000 per year while the median tells us that the typical individual only earns about $32,000 per year, which is much more representative of the typical individual.

Outliers:

The median also does a better job of capturing the central location of a distribution when there are outliers present in the data. For example, consider the following chart that shows the square footage of houses on a certain street:

The mean is heavily influenced by a couple extremely large houses, while the median is not. Thus, the median does a better job of capturing the “typical” square footage of a house on this street compared to the mean.

Summary

In summary:

Both the mean and the median can be used to describe where the “center” of a dataset is located.
It’s best to use the mean when the distribution of the data values is symmetrical and there are no clear outliers.
It’s best to use the median when the the distribution of data values is skewed or when there are clear outliers.

Additional Resources

How Do Outliers Affect the Mean?
How to Estimate the Mean and Median of Any Histogram
How to Find the Mean & Median of Stem-and-Leaf Plots

When would you use median rather than mean?

The answer is simple. If your data contains outliers such as the 1000 in our example, then you would typically rather use the median because otherwise the value of the mean would be dominated by the outliers rather than the typical values. In conclusion, if you are considering the mean, check your data for outliers.

Why would we choose to use the median as the measure of central tendency rather than the mean?

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average. For data from skewed distributions, the median is better than the mean because it isn't influenced by extremely large values.

When would you use median as a measure of central tendency?

The median is less affected by outliers and skewed data than the mean, and is usually the preferred measure of central tendency when the distribution is not symmetrical.

When should you use the median rather than the mean to measure the center of a distribution of quantitative data?

The median is usually preferred to other measures of central tendency when your data set is skewed (i.e., forms a skewed distribution) or you are dealing with ordinal data. However, the mode can also be appropriate in these situations, but is not as commonly used as the median.

When would you use the median rather than the mean as a measure of central tendency?

The mean

When to Use the Mean

When to Use the Median

Summary

Additional Resources

When would you use median rather than mean?

Why would we choose to use the median as the measure of central tendency rather than the mean?

When would you use median as a measure of central tendency?

When should you use the median rather than the mean to measure the center of a distribution of quantitative data?

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