why is the population shape a concern when estimating a mean

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If the shape of the population distribution follows the Normal curve, the mean coincides with the mode and median, and hence it's is meaningful as an indicator.

On the other hand, if it is skewed, i.e. does not follow the normal curve, other indicators may be more meaningful, depending on the "shape" of the distribution curve.

The shape of the population distribution is a concern when estimating a mean because it affects the accuracy and reliability of the estimate. The mean is a measure of central tendency that represents the average of a set of values, so it is influenced by the values in the population distribution.

When the population distribution is symmetric and bell-shaped (like a normal distribution), the mean is an accurate and precise estimate of the central tendency. In this case, the mean represents the midpoint of the distribution, and the values on either side of the mean are approximately equal.

However, when the population distribution is not symmetric or is skewed (meaning it has a longer tail on one side), using the mean as an estimate may not be appropriate. Skewed distributions tend to have outliers or extreme values in one direction, which can pull the mean towards that direction, causing it to be an inaccurate representation of the typical values.

If the population distribution is heavily skewed or has multiple peaks (a bimodal or multimodal distribution), the mean may not accurately reflect the central tendency of the data. In such cases, alternative measures of central tendency like the median (the midpoint of the sorted data) or mode (the most frequent value) might be more appropriate.

In summary, the population shape is a concern when estimating a mean because it can significantly affect the accuracy and appropriateness of using the mean as an estimate of the central tendency. Understanding the population distribution is crucial to selecting the appropriate measure of central tendency for estimation.