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.


Cholesterol The cholesterol levels of a sample of 10 female employees
154 240 171 188 235 203 184 173 181 275

Verified step by step guidance

1. Step 1: Organize the data in ascending order to make calculations easier. The cholesterol levels are: 154, 171, 173, 181, 184, 188, 203, 235, 240, 275.
2. Step 2: Calculate the mean (average). Add all the cholesterol levels together and divide by the total number of data points (10). Use the formula: $\frac{\sum x}{n}$, where $\sum x$ is the sum of the data points and $n$ is the number of data points.
3. Step 3: Find the median. Since there are 10 data points (an even number), the median is the average of the two middle values. Identify the two middle values in the ordered data set and calculate their average using the formula: $\frac{x}{}$₅+x₆2, where $x$₅ and $x$₆ are the fifth and sixth values in the ordered data set.
4. Step 4: Determine the mode. The mode is the value that appears most frequently in the data set. Check the frequency of each cholesterol level in the ordered data set. If no value repeats, then the data set has no mode.
5. Step 5: Interpret the results. Discuss whether the mean, median, and mode represent the center of the data effectively. Consider factors such as the presence of outliers (e.g., 275) and whether the measures of central tendency provide a meaningful summary of the data distribution.

Key Concepts

Mean

The mean, often referred to as the 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 but can be influenced by extreme values (outliers). For example, in the cholesterol levels provided, the mean gives a general idea of the average cholesterol level among the employees.

Median

The median is the middle value of a dataset when it is ordered from least to greatest. If there is an even number of observations, the median is the average of the two middle numbers. This measure is particularly useful for understanding the center of a dataset that may have outliers, as it is not affected by extreme values. In the cholesterol data, finding the median helps to identify a central tendency that is more representative of the majority of the values.

Mode

The mode is the value that appears most frequently in a dataset. A dataset may have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values are unique. In the context of cholesterol levels, identifying the mode can highlight the most common cholesterol level among the employees, providing insight into typical health metrics within the group.