Script
Let’s say that you’re building a mobile.
You attach a string to the center of the mobile arm.
Then, you hang an object from the arm one unit to the left of the attachment point.
With just this one object, the mobile is out of balance. It tilts in the direction of this object.
Next, you hang a second object of equal mass from the arm. You place this object three units to the right of the attachment point.
The mobile arm is again out of balance. Now, the arm tilts in the direction of this new object.
Why is the arm out of balance, even though we have one object hanging from each side of the attachment?
In physics terms, think of the relationship between torque and force. Torque = Force times distance. The objects have the same mass, but are hung at different distances from the attachment. So, they don’t exert the same torque on the arm, or beam. The object that is three units to the right of the attachment exerts more torque on the beam than the object that is one unit to the left of the attachment. Because of this, the beam tilts to the right.
In statistical terms, you can think of these distances as deviations. The mobile is out of balance because the deviation on the left, below the attachment, is smaller than the deviation on the right, above the attachment.
Let’s say that we want to hang a third object from the mobile. Where would we need to hang this object to balance the mobile?
We need to hang the object so that the deviations or the sum of the deviations, below the attachment are the same as the deviations above the attachment.
Or, if you think of the deviations below the attachment as negative and the deviations above the attachment as positive, the sum of all of the deviations must equal zero.
The sum of the deviations for our two objects is -1 + 3 = 2.
So, to balance the mobile, we need to hang the new object two units to the left of the attachment, at -2.
Now, let’s consider this from another perspective. Instead of starting with and empty mobile arm, let’s say we have 5 objects of equal mass, and we hang the objects left to right at these positions.
[On Screen] 4.8, 5.4, 6.1, 6.5, 7.0
Where should we put the attachment in order to balance the mobile?
As we slide the attachment towards the lowest value and then towards the highest value, notice how the slope of the mobile arm changes.
To find the attachment position that balances the mobile arm, we need to again look at the deviations.
Notice how the deviations change as we move the attachment position.
To balance the mobile, we need to find the attachment position in which the sum of the deviations is zero.
This value, in statistical terms, is the mean.
The mean, or the arithmetic average, is the balancing point of the data. For these data, the mean is 5.96.
The mean is a measure of the central tendency of the data.
Another measure of the central tendency is the median. The median is simply the middle value.
For our example, the median is 6.1
What if our last value is 8.5 instead of 7.0? How would the mean and the median values change?
When we move the last point to 8.5, the beam is out of balance. The mean has to shift in the direction of this point to rebalance the beam.
The new mean is 6.26.
What about the median?
Since the median is the middle value, it doesn’t change. The median is still 6.1.
The mean is influenced by extreme data values or outliers.
However, the median is a measure of central tendency that is robust to extreme data values.
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