Final answer:
In a symmetrical and bell-shaped (normal) distribution, the Mean, Median, and Mode are all equal. This reflects the defining characteristics of a normal distribution, where these measures of central tendency coincide.
Step-by-step explanation:
If your data is symmetrical and bell-shaped, this commonly describes what is known as a normal distribution. In such a distribution, the Mean, Median, and Mode are all equal. This occurs because the symmetry ensures that the central point (the peak of the bell) represents all three measures of central tendency; the mean is the arithmetic average of the data, the median is the middle value when the data is ordered, and the mode is the most frequently occurring value. In a perfectly symmetrical (normal) distribution, due to the data being evenly spread around a central value, no single measure of central tendency is pulled away from the center by extreme values or skewness. Thus, Option 1: The Mean, Median, and Mode are all equal is the correct answer.
When data is skewed, this relationship changes. For example, in a right-skewed (positive skew) distribution, the mean is typically greater than the median due to the longer tail pulling the mean to the right. Conversely, in a left-skewed (negative skew) distribution, the mean would be less than the median. Skewness impacts the mean more than the median or mode, as the mean takes into account every value in the dataset, while the median and mode are more robust against the effects of outliers.