Problem 84 - Algebra 1

Question

Find the mean, the median, and the mode of the collection of numbers. $$17,9,11,15,4,15,8,3,11$$

Step-by-Step Solution

Answer

The mean is 9.33, the median is 11, and the modes are 11 and 15.

Step 1: Arrange numbers in ascending order

Rearrange the given numbers in ascending order: \(3, 4, 8, 9, 11, 11, 15, 15, 17\).

Step 2: Compute the mean

The mean is computed by adding all numbers and dividing by the count of numbers. \[\text{Mean} = \frac{3 + 4 + 8 + 9 + 11 + 11 + 15 + 15 + 17}{9} = 9.33\].

Step 3: Calculate the median

The median is the middle value of an ordered set. In this case, the median can be found by taking the 5th number in the ordered set, which is 11.

Step 4: Find the mode

The mode is the number that appears most often in the set. In this case, the numbers 11 and 15 appear twice, so there are two modes, 11 and 15.

Key Concepts

Mean Calculation

Understanding the concept of the mean—often referred to as the average—is central to grasping basic statistics. The mean represents the sum of all values in a dataset divided by the quantity of values. In our exercise, the mean is calculated by adding together each number in the set:
\(3 + 4 + 8 + 9 + 11 + 11 + 15 + 15 + 17\) and then dividing by the number of values, which is 9. This results in a mean of \(9.33\), giving us a central value around which all other numbers in our set can be compared. While the mean is useful, it's important to note that it can be influenced by extreme values or outliers in the dataset, which can skew the result.

Median Calculation

The median is the value sitting in the middle of a dataset when it is arranged in numerical order. If the set has an odd number of values, the median is the one directly at the center. For an even number of values, you take the mean of the two central numbers. In the given problem, our arranged set \(3, 4, 8, 9, 11, 11, 15, 15, 17\) has nine numbers, so the median is the fifth number—\(11\). The median is particularly helpful as it is not affected by outliers like the mean, providing a better 'central' value for highly skewed datasets.

Mode Identification

The mode of a dataset is the number that appears most frequently. It's possible for a dataset to have one mode (unimodal), more than one mode (multimodal), or no mode at all if no number repeats. In our example, both \(11\) and \(15\) repeat twice. This makes our set bimodal with two modes: \(11\) and \(15\). When datasets have multiple modes, it suggests that there are several values around which data points cluster, potentially indicating different groupings or types within the data.

Data Organization

Organizing data is a fundamental first step before proceeding with any statistical analysis. Sequentially arranging data from smallest to largest, or vice versa, allows you to easily identify patterns, outliers, and the measures of central tendency like the mean, median, and mode. In the given exercise, data is organized in ascending order: \(3, 4, 8, 9, 11, 11, 15, 15, 17\). This neatly arranged set not only simplifies the process of finding the median but also aids in quickly identifying modes and calculating the mean. Effective data organization can reveal trends and patterns that make it easier to understand and interpret statistical information.