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# Different Perspectives on Achieving the Mean of a Distribution

Using a small dataset , we can determine the mean value corresponding to five different perspectives on an equitable distribution. Within each perspective, use the slider to accomplish the desired goal.

### DETAILS

For each perspective, the choice of achieves a particular goal. By sliding the control for , you can explore different properties of the mean.
Snapshot 1: Colored bars represent data values. Choose to divide the total into equal parts to show that the mean represents a "fair share."
Snapshot 2: Towers represent data values. By varying , remove area from the taller bars and add it to the shorter bars. Choose so that all the bars are leveled off at the same height.
Snapshot 3: Circles on the number line represent data values. Each square represents the squared distance from an observation to . Choose to minimize the sum of the areas of the squares.
Snapshot 4: Circles with unit mass on the "teeter-totter" number line represent data values. The position of the fulcrum is given by . Choose to achieve balance.
Snapshot 5: The vector represents the dataset, is the -vector of all ones, and is the vector of differences. The different colors of the segments forming the vector represent the contributions of the coordinates to the norm of the vector. For example, since each component is equal, is made of five equal-length segments. The mean is determined by projecting onto .To find the projection, choose so that the angle between and is a right angle.

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