Final answer:
In a perfectly symmetrical distribution, the correct attribute is that the arithmetic mean equals the median. This is due to the data being evenly spread around the central point of the distribution, which is a characteristic of symmetry in statistics. Therefore, the correct answer is C: The arithmetic mean equals the median.
Step-by-step explanation:
What is true about a perfectly symmetrical distribution? This question refers to a statistical property of data distributions. In a perfectly symmetrical distribution, which is often represented by a bell-shaped curve, certain characteristics hold. The most relevant characteristic to this question is that the arithmetic mean equals the median. This is because symmetry around the central point ensures that the average (mean) value of the data is the same as the middle value when all data points are sorted (median).
Let's address the given options based on statistical principles:
- The variance does not equal the standard deviation. Variance is the square of the standard deviation.
- The range does not equal the interquartile range. The range is the difference between the largest and smallest values, while the interquartile range is the range between the first and third quartiles.
- The arithmetic mean equals the median in a perfectly symmetrical distribution due to the data being evenly distributed about the center.
- The interquartile range does not equal the arithmetic mean. These are different measures; the IQR measures the middle 50% of data, whereas the mean is an average of all data points.
Therefore, the correct answer is C: The arithmetic mean equals the median.