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
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
to achieve balance.
Snapshot 5: The vector
represents the dataset,
-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
.To find the projection, choose
so that the angle between
is a right angle.