Final answer:
In a symmetric data set, the mean and median have a close relationship, often being the same or nearly the same value, indicating that the data is evenly distributed around the central point without skewness.
Step-by-step explanation:
When a data set is symmetric, it often refers to a situation where the dataset shows balance around a central value. If the relationship between the mean and median in a symmetric data set is explored, one will find that they are either identical or very close to each other. This is due to the lack of skewness in the dataset, which means that the values are distributed evenly around the center. In the case of a perfectly symmetric data distribution, particularly the normal distribution, the mean, median, and mode all coincide at the same point on the graph, representing this point of symmetry.
For example, if the data points are arranged in ascending order, the median will be the middle value in a dataset with an odd number of observations, or the average of the two middle values in a dataset with an even number of observations. The mean is calculated by adding all the data points together and dividing by the number of points, and it also reflects the central point of a symmetric distribution.
Therefore, in a symmetric data set, the mean and median will have a very close relationship, often being the same number, indicating that the data is evenly distributed without skewness to the left or right.