Which measure of center is best for a symmetric distribution?

For a symmetric distribution, the mean (arithmetic average) is typically the best measure of center to use. The reason for this is that in a symmetric distribution, the mean, median, and mode are all equal, or very close to each other. Since the mean takes into account every value in the data set, it provides a measure of center that is sensitive to all values, which can give a more accurate representation of the center of the data set for symmetric distributions.

Additionally, the mean has the advantage of being used in further statistical analyses, like computing standard deviation and variance, which are measures of spread; this is because these measures rely on the arithmetic mean as a reference point.

There are several measures of central tendency that can be used to describe the center of a data set: the mean, median, and mode are the most common. The choice among these measures can depend on the shape of the distribution and other factors.

- The mean is the sum of all the data points divided by the total number of points. It is sensitive to extreme values (outliers), which can pull the mean away from the center in skewed distributions.

- The median is the middle value when the data points are ordered from smallest to largest (or the average of the two middle values if there's an even number of data points). The median is less sensitive to outliers and skewed data because it only depends on the position of the values, not their magnitude.

- The mode is the most frequently occurring value in a data set. A data set may have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if no number repeats.

For skewed distributions, the median is often the preferred measure of center, as it is not affected by extreme values. If a distribution is perfectly symmetric, the mean and median will be the same. However, if there is some skewness or outliers, the median may be a better measure of the central tendency. The mode can be useful for categorical data or for describing the shape of the distribution, but it is not often used as the primary measure of center for numerical data.