Using and Interpreting Concepts


Finding and Discussing the Mean, Median, and Mode In Exercises 17–34, find the mean, the median, and the mode of the data, if possible. If any measure cannot be found or does not represent the center of the data, explain why.


College Credits The number of credits being taken by a sample of 14 full-time college students for a semester
12 14 16 15 13 14 15
18 16 16 12 16 15 17

Verified step by step guidance

1. Step 1: Organize the data in ascending order. This helps in calculating the median and identifying the mode more easily. The data set is: 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 18.
2. Step 2: Calculate the mean. The mean is the sum of all data values divided by the number of data points. Use the formula: $\mathrm{Mean}=\frac{\sum x}{n}$, where $\sum x$ is the sum of the data values and $n$ is the number of data points.
3. Step 3: Find the median. The median is the middle value when the data is ordered. If the number of data points is odd, the median is the middle value. If the number of data points is even, the median is the average of the two middle values. Since there are 14 data points (even), the median is the average of the 7th and 8th values in the ordered data set.
4. Step 4: Identify the mode. The mode is the value(s) that appear most frequently in the data set. Count the frequency of each value in the ordered data set to determine the mode. If no value repeats, there is no mode. If multiple values have the same highest frequency, the data set is multimodal.
5. Step 5: Interpret the results. Discuss whether the mean, median, and mode represent the center of the data effectively. Consider the distribution of the data (e.g., symmetric, skewed) and whether any outliers might affect the mean.

Key Concepts

Mean

The mean, or average, is calculated by summing all the values in a dataset and dividing by the number of values. It provides a central value that represents the dataset as a whole. However, the mean can be influenced by extreme values (outliers), which may skew the result and not accurately reflect the typical value in the data.

Median

The median is the middle value of a dataset 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 numbers. The median is a robust measure of central tendency, as it is not affected by outliers, making it a better representation of the center for skewed distributions.

Mode

The mode is the value that appears most frequently in a dataset. A dataset may 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 which is the most common category.