Averages and range from a list
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The mode
The most frequent value in a data set, including sets with two modes or none.
The median
The middle value of ordered data, and the halfway pair when the count is even.
The mean
The total shared equally across the count, and why one extreme drags the mean.
The range
Largest minus smallest as a measure of spread, not an average.
Choosing which average to use
Mode for categories, median against extremes, mean when every value counts.
Working backwards from the mean
Recovering the total from a stated mean to find a missing value.
The mean of a combined group
Combining two groups through their totals, not by averaging the two means.