Introduction

When a data set is skewed or has outliers, the mean is not the most reliable measure of center. We say that the mean is sensitive to skew and outliers.

To understand this, consider the following dataset:

0, \quad 0, \quad 1, \quad 1, \quad 2, \quad 2, \quad 134

Most of the numbers in this data set are between 0 and 2. However, because of the outlier 134, the mean is much larger than the majority of the data points:

\begin{align*} \text{mean} &= \dfrac{0+0+1+1+2+2+134}{7}\\[5pt] &= \dfrac{140}{7}\\[5pt] &= 20 \end{align*}

On the other hand, the median equals {\color{blue}\underline{1}}, which is a reliable measure of the data set's center despite the outlier.

0, \quad 0, \quad 1, \quad {\color{blue}\underline{1}}, \quad 2, \quad 2, \quad 134

In general, the median is a reliable measure of center even when data is skewed or contains outliers. We say the median is resistant to skew and outliers.

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Example: Determining When to Use Mean Versus the Median

In which of the following distributions might it be preferable to use the median instead of the mean to measure the center?



EXPLANATION

The mean is sensitive to skew and outliers, whereas the median is resistant to skew and outliers.

So, we should use the median instead of the mean when we are measuring the center of a skewed distribution or a distribution with outliers.

Among the given options, all distributions are symmetric except for the following, which is left-skewed.


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Practice: Determining When to Use Mean Versus the Median

In which of the following distributions might it be preferable to use the median instead of the mean to measure the center?

a
b
c
d
e
Practice: Determining When to Use Mean Versus the Median

In which of the following distributions might it be preferable to use the median instead of the mean to measure the center?

a
b
c
d
e
Example: Identifying Reliable Measures of Center Given a Dot Plot, Box Plot, or Histogram

Which of the following is true regarding the data set represented by the histogram below?



  1. The distribution has outliers.
  2. The median is not sensitive to the skew of the distribution.
  3. The mean is the best measure of the center for the given distribution.
EXPLANATION

First, let's reсall the following facts.

  • The mean is sensitive to skew and outliers.

  • The median is not sensitive to skew nor outliers.

With that in mind, let's examine each of the statements.

  • Statement I is true. Our distribution has outliers.

  • Statement II is true. Our distribution is right-skewed, but the median is not sensitive to the skew.

  • Statement III is false. Since the distribution is skewed and has outliers, the median is the best measure of the center.

Therefore, the correct answer is "I and II only."

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Practice: Identifying Reliable Measures of Center Given a Dot Plot, Box Plot, or Histogram

The histogram above shows the age distribution of Anna's streaming channel subscribers.

Which of the following are true?

  1. The distribution has outliers.
  2. The mean is sensitive to the skew of the distribution.
  3. The mean is the best measure of the center of the distribution.
a
 I and II only
b
 II only
c
 III only
d
 I only
e
 I and III only
Practice: Identifying Reliable Measures of Center Given a Dot Plot, Box Plot, or Histogram

Why might the median be a more reliable measure of the center of the data presented in the histogram above than the mean?

  1. The distribution has outliers.
  2. The distribution is right-skewed, and the median is resistant to skew.
  3. The distribution is left-skewed, and the median is resistant to skew.
a
 I only
b
 II only
c
 III only
d
 I and III only
e
 I and II only
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