Colin surveyed 12 teachers at his school to determine how much each person budgets for lunch. He recorded his results in the table.
| 10 | 5 | 8 | 10 | 12 | 6 |
|---|---|---|---|---|---|
| 8 | 10 | 15 | 6 | 12 | 18 |
What does the relationship between the mean and median reveal about the shape of the data?
A. The mean is less than the median, so the data is skewed left.
B. The mean is more than the median, so the data is skewed right.
C. The mean is equal to the median, so the data is symmetrical.
D. The mean is equal to the median, so the data is linear.
Answer (1)
Calculate the mean of the data: 12 10 + 5 + 8 + 10 + 12 + 6 + 8 + 10 + 15 + 6 + 12 + 18 = 10 .
Calculate the median of the data: Sort the data: 5, 6, 6, 8, 8, 10, 10, 10, 12, 12, 15, 18. The median is 2 10 + 10 = 10 .
Compare the mean and the median: The mean is 10 and the median is 10.
Since the mean is equal to the median, the data is symmetrical: T h e m e an i s e q u a l t o t h e m e d ian , so t h e d a t a i s sy mm e t r i c a l .
Explanation
Analyze the problem and data We are given a set of lunch budget values from 12 teachers and asked to determine the shape of the data based on the relationship between the mean and the median. The data values are: 10, 5, 8, 10, 12, 6, 8, 10, 15, 6, 12, 18.
Calculate the mean First, we need to calculate the mean of the data. The mean is the sum of all the values divided by the number of values.
Calculate the sum and the mean The sum of the values is 10 + 5 + 8 + 10 + 12 + 6 + 8 + 10 + 15 + 6 + 12 + 18 = 120 . There are 12 values, so the mean is 12 120 = 10 .
Calculate the median Next, we need to calculate the median of the data. The median is the middle value when the data is sorted. Since there are 12 values (an even number), the median is the average of the two middle values.
Sort the data and find the middle values First, sort the data in ascending order: 5, 6, 6, 8, 8, 10, 10, 10, 12, 12, 15, 18. The two middle values are the 6th and 7th values, which are both 10. Therefore, the median is 2 10 + 10 = 10 .
Compare the mean and median and determine the shape of the data Now, we compare the mean and the median. The mean is 10, and the median is 10. Since the mean is equal to the median, the data is symmetrical.
Examples
Understanding the shape of data distributions is crucial in many real-world scenarios. For instance, in finance, analyzing the distribution of stock returns helps assess risk. If the returns are symmetrically distributed, it indicates a balanced risk profile. However, if the distribution is skewed, it suggests a higher probability of extreme gains or losses. Similarly, in healthcare, understanding the distribution of patient recovery times can help hospitals allocate resources effectively and manage patient expectations. Knowing whether the data is symmetrical or skewed provides valuable insights for decision-making in various fields.
