Balance board: the mean as centre of mass
About this app
Building blocks on a weightless polystyrene board: it balances exactly above the mean, because the deviations Σ(xᵢ − x̄) cancel out. Drag the support, move the outlier, show the median and lever arms; plus the overall mean of two groups instead of the mean of the means.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The statistics
The arithmetic mean x̄ is the centre of mass of a distribution. Picture every observation as an equally heavy block on a weightless board: the board is level exactly when the support sits under x̄. The reason is that the deviations sum to zero, Σ(xᵢ − x̄) = 0. Deviations to the left and right of the mean cancel like lever arms on a seesaw. The median, by contrast, only counts how many values lie on each side, not how far away they are.
What you see
The app recreates the lecture experiment: 16 families with 1, 2, 3, 4 and 7 children stand as blocks above their number of children. The board balances at 2.5, the mean 40/16. At the median 2 and at the middle of the scale it tips. On request the deviations appear as arrows from x̄ to every block and below as a balance: 9 on the left, 9 on the right. A second example combines 5 women with a mean of 173 cm and 13 men with 175 cm; the overall mean is 174.44 cm, not 174 cm. A values panel shows the frequency table, x̄, median and torque.
Things to try
Drag the support under the board until the spirit level shows level, and compare the spot with x̄. Move the outlier from 7 to 11 children: the board tips right, the finger has to follow to 2.75, and the median stays 2. Drag single blocks to other numbers and watch the balance point and the table change. In the groups scene, change group sizes and means and see the larger group pull the balance point towards it.
Frequently asked questions
- Why is the arithmetic mean the centre of mass?
- The deviations of all values from the mean sum to zero: Σ(xᵢ − x̄) = 0. If every value is an equal weight on a scale, that is exactly the condition for the torques on the left and right to cancel. So a board carrying these weights balances above x̄.
- Why does the board not balance at the median?
- The median only splits the values into two halves by count. Balance also depends on distance. An outlier far to the right has a long lever arm and pulls the balance point right, while the median does not change.
- How do you compute the overall mean of two groups?
- Weight the group means by the group sizes: x̄ = (n₁·x̄₁ + n₂·x̄₂)/(n₁ + n₂). For 5 women at 173 cm and 13 men at 175 cm that gives 3140/18 = 174.44 cm. The plain mean of the means, 174 cm, is only right when both groups have the same size.
- What can I change in the balance board app?
- You move the support, the outlier and single blocks, show the mean, median and deviations, and switch to a scene with two groups whose size and mean you set. The values panel shows the frequency table, x̄, median, torque and tilt.
Subject: Statistics
More from Statistics
- Anscombe's quartet
- Bias-variance / overfitting
- Law of large numbers
- Histogram workshop
- Hypothesis test & p-value
- Confidence interval
- Measures of location & spread
- Measures of location & spread 2.0
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