The frequency distribution of a data set is symmetric and the mean is 20. Which statement is true about this data set?

In a symmetric frequency distribution, the mean, median, and mode all coincide at the center of the distribution. When the distribution is perfectly symmetric, it indicates that the data is evenly distributed on both sides of the center point. Since the mean of the data set is given as 20, it follows that the distribution's center—where both halves balance—is also at 20.

Therefore, the median, which represents the middle value in a sorted list of data, must also be 20 in this case. This direct relationship between the mean and median in a symmetric distribution confirms that statement about the median being 20 as true.

It is important to note that in a symmetric distribution, the standard deviation is always a non-negative value, reflecting the spread of data around the mean. Thus, the claim that the standard deviation could be negative is incorrect, as standard deviation cannot be negative by definition. Also, the median cannot be 10 or 40, as those values lie outside the center point of the distribution, contradicting the relationship established in a symmetric set. Therefore, the median being 20 aligns perfectly with the properties of a symmetric frequency distribution.