1
The balance point of a data set
Discover Mathematics P6: Term 1, Unit 7, Lessons 1–2
GoalExplain the mean as the balance point of a dot plot and find it from the distances.
New words
mean · the number that balances the data; the sum divided by the number of valuesbalance point · the spot on the number line where the dot plot would not tipdistance · how far a value is from the balance point
Explanation
Imagine a dot plot as a plank on a number line, with one equal weight at every dot. The mean is the point where the plank balances. At that point the total distance of the dots on the left equals the total distance of the dots on the right. A dot that is far from the balance point pulls strongly, so a single dot far away drags the balance point towards it. To find the balance point, you do not need the plank: the mean equals the sum of the values divided by the number of values.
★Remember
Mean = balance point. Distances on the left add up to the distances on the right. Mean = sum ÷ number of values.
Worked examples
Data: 2, 3, 3, 4, 8. The sum is 20 and there are 5 values, so the mean is 20 ÷ 5 = 4. Look at the distances from 4: on the left 2, 1 and 1 (total 4); on the right 0 and 4 (total 4). The plank balances at 4.
A symmetric dot plot: 4, 5, 5, 6, 6, 6, 7, 7, 8. The two sides are mirror images around 6, so the balance point is the centre, 6. Check: the sum is 54 and 54 ÷ 9 = 6.
A skewed dot plot: 1, 1, 2, 2, 3, 9. The sum is 18 and 18 ÷ 6 = 3. The single dot at 9 drags the balance point to the right of the middle dots: the median is 2, but the mean is 3.
Class activity
The plank game. Take a ruler, put a pencil under it as a pivot, and put coins of the same size on the numbers 1, 2 and 9 of the ruler. Where must the pencil be so that the ruler stays level? Check with the mean of 1, 2 and 9.
Practice
1
The dot plot has dots at 2, 4 and 6. Where would the plank balance?
At 4, because (2 + 4 + 6) ÷ 3 = 4.
2
Find the mean of 1, 5, 9.
(1 + 5 + 9) ÷ 3 = 15 ÷ 3 = 5.
3
Show with distances that 6 is the balance point of 3, 5, 6, 10.
Left of 6: 3 is 3 away and 5 is 1 away, total 4. Right of 6: 10 is 4 away, total 4. The two totals are equal. Also (3 + 5 + 6 + 10) ÷ 4 = 24 ÷ 4 = 6.
4
One more dot is added far to the right of a dot plot. Does the balance point move left or right?
To the right, towards the new dot.
5
Find the balance point (the mean) of this dot plot.
The values are 5, 5, 6, 8, 11. The sum is 35 and 35 ÷ 5 = 7.