Median: Videos & Practice Problems

The concept of the median serves as a vital measure of center in a data set, providing a central value that summarizes the data effectively. To determine the median, the first step is to sort the data in ascending order, from the smallest to the largest value. This process is crucial 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. After sorting these values, we obtain the ordered set: 3, 5, 10, 12, and 14. With a total of five numbers (n = 5), which is an odd count, the median is simply the middle value. In this case, the median is 10, as it is the third number in the sorted list.

However, when the data set includes an even number of values, the process changes slightly. 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 still have six values (n = 6), which is even. In this scenario, the median is calculated by finding the mean of the two middle numbers. The two middle numbers in this sorted list are 10 and 12. To find the median, we add these two values together and divide 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 set. Understanding how to find the median is essential for analyzing data, as it provides a clear representation of the central tendency, especially in cases where the data may be skewed by outliers.

In this example, we explore how to find the median number of college credits from a histogram, which visually represents the distribution of credits taken by a sample of students. The histogram provides frequency data for different credit categories, allowing us to reconstruct the original data set without having a direct list of numbers.

To determine the median, we first interpret the histogram. For instance, if the histogram indicates that one student is taking 11 credits, we note that there is one occurrence of the number 11. If four students are taking 12 credits, we recognize that the number 12 appears four times in our data set. This process continues for each category, allowing us to recreate the full data set:

• 1 occurrence of 11 credits
• 4 occurrences of 12 credits
• 3 occurrences of 13 credits
• 2 occurrences of 14 credits
• 1 occurrence of 15 credits

Thus, the reconstructed data set is: 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 in this ordered list.

To find the median, we can eliminate numbers from both ends of the list until we reach the center. In this case, we pair off the numbers from the outer edges towards the center:

11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 15

After removing pairs, we find that the middle number is 13. Therefore, the median number of credits taken by the students is 13. While the most common number of credits (the mode) is 12, the median provides a different perspective on the data distribution, highlighting the central tendency of the credit hours taken.

This method of deriving the median from a histogram is a valuable skill, as it allows for the analysis of data distributions even when direct numerical data is not available.

Understanding the concepts of mean and median is essential for summarizing data effectively. Both are measures of center that provide a single value representing 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 we have a dataset of five numbers, the mean can be calculated as:

\[\text{Mean} = \frac{\sum \text{values}}{n}\]

where \( n \) is the number of values. While the mean utilizes all data points, making it a comprehensive measure, it is sensitive to extreme values, known as outliers. For instance, introducing an outlier like 76 into a dataset can significantly skew the mean, changing it drastically compared to the other values. Therefore, the mean is most effective when the data is symmetric and free from outliers.

On the other hand, the median is determined by organizing the data from smallest to largest and identifying the middle value. If there is an even number of observations, the median is the average of the two middle numbers. This method provides a more stable measure of center when outliers are present. For example, even with the outlier of 76, the median remains largely unchanged, demonstrating its resistance to extreme values. The median is calculated as:

\[\text{Median} = \begin{cases} \text{middle value} & \text{if odd number of values} \\ \frac{\text{value}_k + \text{value}_{k+1}}{2} & \text{if even number of values} \end{cases}\]

In scenarios where data is roughly symmetric but contains outliers, the median is often the better measure of center. This resistance to outliers makes the median particularly useful in real-world applications, such as reporting salaries, where a few high earners can distort the mean significantly. In such cases, the median provides a more accurate reflection of what most individuals earn.

In summary, while both mean and median serve as measures of center, their effectiveness varies based on the dataset's characteristics. The mean is advantageous for symmetric data without outliers, while the median excels in datasets with outliers, providing a more reliable representation 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, we find that the mean price is approximately $354.8 thousand.

Next, we calculate the median price, which requires sorting the home prices from smallest to largest. The sorted prices are: $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 numbers. Thus, we calculate:

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

This indicates that the median price is $296 thousand. Comparing the mean and median, we observe that the mean is significantly influenced by the outlier price of $850 thousand, which skews the mean higher than most of the other values. In contrast, the median, being less affected by extreme values, provides a better representation of the central tendency of the data set.

In conclusion, while both the mean and median are measures of central tendency, the median is more representative of this particular data set due to the presence of an outlier that distorts the mean. This highlights the importance of understanding the characteristics of data when choosing which measure to use for analysis.

In analyzing a histogram that represents the annual salaries of CEOs, typically measured in billions, we can estimate both the mean and the median to understand the central tendency of the data. The median serves as a value that divides the dataset into two equal halves, effectively separating the lower 50% from the upper 50%. In this case, the distribution of salaries shows a concentration of lower values in the tens, twenties, and thirties, with fewer higher values skewing the data to the right. Thus, the median is estimated to be around 30, reflecting the central point of the lower half of the data.

On the other hand, the mean is influenced by all data points, including the extreme high values present in the dataset. These outliers, which may be in the eighties, nineties, or even hundreds, pull the mean towards the higher end of the scale. Consequently, the mean is estimated to be around 50, indicating a shift due to these higher salaries.

When determining which measure of central tendency is more appropriate for this skewed dataset, it is essential to recognize that in a right-skewed distribution, the median will always be less than the mean. This characteristic makes the median a more reliable measure of the center of the data, as it better represents the typical CEO salary amidst the influence of extreme values. Therefore, in this scenario, the median is favored as it provides a clearer picture of the central tendency, reflecting the more common salaries rather than being distorted by the few exceptionally high salaries.