The shoe sizes of a group of middle school girls are shown.

If a shoe size of 9 is added to the data, how does the median change?

Sort the original data set: 5, 5.5, 6, 6.5, 6.5, 7, 7.5, 8, 8, 8.5.
Calculate the original median: 2 6.5 + 7 ​ = 6.75 .
Add the new data point and sort: 5, 5.5, 6, 6.5, 6.5, 7, 7.5, 8, 8, 8.5, 9.
Calculate the new median: 7. The change in median is 7 − 6.75 = 0.25 . The median increases by 0.25 ​ .

Explanation

Understanding the Problem We are given a set of shoe sizes and asked to determine how the median changes when a new shoe size is added to the data set. The median is the middle value of a data set when it is sorted in ascending order. If there is an even number of data points, the median is the average of the two middle values.

Sorting the Original Data First, we need to sort the original data set: 5.5, 6, 7, 8.5, 6.5, 6.5, 8, 7.5, 8, 5. Sorting this data set in ascending order gives us: 5, 5.5, 6, 6.5, 6.5, 7, 7.5, 8, 8, 8.5.

Calculating the Original Median Next, we calculate the median of the original data set. Since there are 10 data points (an even number), the median is the average of the 5th and 6th values. In this case, the 5th value is 6.5 and the 6th value is 7. Therefore, the original median is 2 6.5 + 7 ​ = 6.75 .

Adding the New Data Point Now, we add the shoe size of 9 to the data set. The new data set is: 5, 5.5, 6, 6.5, 6.5, 7, 7.5, 8, 8, 8.5, 9.

Calculating the New Median We calculate the median of the new data set. Since there are now 11 data points (an odd number), the median is the 6th value, which is 7.

Determining the Change in Median Finally, we determine the change in the median by subtracting the original median from the new median: 7 − 6.75 = 0.25 . Therefore, the median increases by 0.25.


Examples
Understanding how the median changes when new data is added is useful in many real-world scenarios. For example, consider a teacher who wants to know how adding a new student's test score will affect the class's median score. Or, a real estate agent might want to know how adding a new house price to a neighborhood's data set will affect the median house price. These kinds of analyses help in understanding the central tendency of the data and how it shifts with new information. The median is less sensitive to outliers than the mean, making it a robust measure in such situations.

Answered by GinnyAnswer | 2025-07-03

When a shoe size of 9 is added to the original data set, the median changes from 6.75 to 7. Therefore, the median increases by 0.25.
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Answered by Anonymous | 2025-07-04