Building Basic Skills and Vocabulary


True or False? In Exercises 1–4, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


A data set can have the same mean, median, and mode.

Verified step by step guidance

1. Understand the definitions of mean, median, and mode: The mean is the average of all data points, the median is the middle value when the data is ordered, and the mode is the most frequently occurring value in the data set.
2. Recognize that it is possible for a data set to have the same mean, median, and mode. For example, in a perfectly symmetric distribution like a normal distribution, these three measures of central tendency can coincide.
3. Consider an example: A data set such as {3, 3, 3, 3, 3} has a mean of 3, a median of 3, and a mode of 3. This demonstrates that the statement can be true.
4. If the statement were false, you would need to provide a counterexample where the mean, median, and mode are not the same. For instance, in a skewed distribution, these values often differ.
5. Conclude that the statement is true because there are scenarios where the mean, median, and mode can indeed be the same, particularly in symmetric distributions or uniform data sets.

Key Concepts

Mean

The mean is the average of a data set, calculated by summing all the values and dividing by the number of values. It is sensitive to extreme values, which can skew the mean significantly. In a perfectly symmetrical distribution, the mean represents the center of the data.

Median

The median is the middle value of a data set when the values are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle values. The median is less affected by outliers and provides a better measure of central tendency for skewed distributions.

Mode

The mode is the value that appears most frequently in a data set. A data set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if all values occur with the same frequency. The mode is particularly useful for categorical data where we wish to know the most common category.