Median: Videos & Practice Problems

In statistics, the median serves as a vital measure of central tendency, providing a central value that summarizes a data set. To determine the median, one must first organize the data in ascending order, from the smallest to the largest value. This process is essential because simply selecting the middle number without sorting can lead to incorrect conclusions.

For example, consider the data set consisting of the numbers 5, 10, 14, 12, and 3. The first step is to sort these numbers, resulting in the ordered set: 3, 5, 10, 12, and 14. With this sorted list, we can identify the median. Since there are five numbers (an odd count), the median is the middle value, which in this case is 10.

When dealing with an even number of data points, the process slightly changes. For instance, if we add a sixth number, 76, to the previous set, we now have the numbers 3, 5, 10, 12, 14, and 76. After sorting, we find the two middle numbers are 10 and 12. To find the median in this scenario, we calculate the mean of these two middle values. The mean is computed by adding the two numbers together and dividing by 2:

Median = \(\frac{10 + 12}{2} = 11\)

Thus, the median of this data set is 11, even though this value does not appear in the original data. Understanding how to find the median is crucial for accurately summarizing data sets, especially when they contain an even number of values.

In this example, we explore how to find the median number of college credits taken by a sample of students using a histogram. A histogram visually represents the frequency of data points within specified ranges, but it does not provide a direct list of values. To determine the median from a histogram, we can reconstruct the original data values based on the frequencies indicated in the histogram.

For instance, if the histogram shows that one student is taking 11 credits, we note that there is one occurrence of the value 11. If four students are taking 12 credits, we represent this as four occurrences of the value 12. Continuing this process, we can summarize the data as follows: one student taking 11 credits, four students taking 12 credits, three students taking 13 credits, two students taking 14 credits, and one student taking 15 credits. This gives us the complete dataset: 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 15.

To find the median, we arrange the data in ascending order, which is already done through the histogram. The median is the middle value of a sorted dataset. In this case, with 11 total data points, the median is the sixth value when counting from either end. By eliminating values from both ends, we find that the middle number is 13. Thus, the median number of credits taken by the students is 13.

It's important to note that while the mode, or the most frequently occurring value, is 12, the median provides a different measure of central tendency, reflecting the middle of the dataset. Understanding how to extract and analyze data from a histogram is a valuable skill in statistics, allowing for deeper insights into the distribution of data.

Understanding the concepts of mean and median is essential for summarizing data effectively. Both are measures of center that provide insights into the distribution of values within a dataset, but they have distinct advantages and disadvantages, particularly in the presence of outliers.

The mean is calculated by adding all values in a dataset and dividing by the total number of values. For example, if a dataset consists of the numbers 5, 8, 10, and 12, the mean would be calculated as follows:

Mean = \(\frac{5 + 8 + 10 + 12}{4} = 8.75\)

However, introducing an outlier, such as 76, significantly alters the mean:

New Mean = \(\frac{5 + 8 + 10 + 12 + 76}{5} = 22.2\)

This demonstrates that the mean is sensitive to extreme values, which can skew the representation of the dataset. Therefore, the mean is most effective when the data is symmetric and free from outliers.

In contrast, the median is the middle value when the data is organized in ascending order. For the same dataset, the median remains unaffected by the outlier:

Median = 10 (for the original dataset) and remains 10 even with the outlier included. This stability illustrates that the median is resistant to outliers, making it a better measure of center when the data includes extreme values.

Resistance refers to the ability of a measure to remain unaffected by outliers. The mean is not resistant, as demonstrated by the significant change when an outlier is present, while the median is resistant, showing minimal change.

When evaluating which measure of center to use, consider the distribution of the data. For instance, in a histogram representing graduate student salaries, if most salaries cluster around a certain range but a few individuals earn significantly more, the median salary provides a more accurate representation of what a typical graduate earns. The mean, influenced by those high earners, would not reflect the majority's experience.

In summary, while both mean and median serve as valuable tools for data analysis, the choice between them should be guided by the nature of the dataset. The median is often preferred in cases with outliers or skewed distributions, as it offers a more reliable measure of central tendency.

In this analysis, we explore the concepts of mean and median using the prices of eight homes, expressed in thousands of dollars. To find the mean price, we sum all the home prices and divide by the total number of homes. The formula for the mean (\( \bar{x} \)) is given by:

\( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \)

In this case, the total number of homes (\( n \)) is 8. After calculating the sum of the prices (275, 229, 850, 240, 305, 287, 310, and 342), we find that the mean price is approximately $354,800.

Next, we calculate the median price, which requires sorting the home prices in ascending order: 229, 240, 275, 287, 305, 310, 342, and 850. Since there is an even number of observations, the median is the average of the two middle values (287 and 305). Thus, the median price is:

\( \text{Median} = \frac{287 + 305}{2} = 296 \)

This results in a median price of $296,000. Comparing the mean and median, we observe that the mean is significantly higher than most of the other values, primarily due to the outlier (850). This outlier skews the mean, making it less representative of the overall data set.

In contrast, the median, being less affected by extreme values, provides a better measure of central tendency for this data set. Therefore, in this scenario, the median is the more representative statistic for the home prices, as it reflects a typical value more accurately than the mean.