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.


Video Durations The lengths (in minutes) of seven educational videos from the Public Broadcasting Service (PBS) (Source: PBS)
83 67 90 55 56 119 52

Verified step by step guidance

1. Step 1: Organize the data in ascending order. The given data is: 83, 67, 90, 55, 56, 119, 52. Arrange it as: 52, 55, 56, 67, 83, 90, 119.
2. Step 2: Calculate the mean. The mean is the sum of all data points 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 points 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. Since there are 7 data points (an odd number), the median is the 4th value in the ordered list: 52, 55, 56, 67, 83, 90, 119.
4. Step 4: Determine the mode. The mode is the value that appears most frequently in the data set. Check the ordered data: 52, 55, 56, 67, 83, 90, 119. If no value repeats, then the mode does not exist.
5. Step 5: Interpret the results. Discuss whether the mean, median, or mode best represents the center of the data. Consider factors such as the presence of outliers (e.g., 119) and whether the data is skewed.

Key Concepts

Mean

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

Median

The median is the middle value of a dataset when the values are arranged in ascending or descending order. If there is an even number of observations, the median is the average of the two middle numbers. The median is particularly useful for understanding the center of a dataset that may contain outliers, as it is not affected by extreme values.

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 useful for identifying the most common value in a dataset, but it may not always provide a clear representation of the center, especially in continuous data.