Mean

An SAT Math micro-topic under Center, spread, and shape of distributions (Problem solving and data analysis). Free to read — no account needed.

The mean is the total of all the values divided by the number of values.
Add everything up, then divide by the count.
That single number represents the center of the data.
You can run this backward to find the total.
Since sum=mean×count\text{sum} = \text{mean} \times \text{count}, a set of 1010 numbers with a mean of 2020 has a sum of 20×10=20020 \times 10 = 200.
This helps when a problem gives you the mean but not the individual numbers.
When a new value is added, update both the sum and the count.
Add the value to the old sum, then divide by the new count.
Adding 4242 to a set of 1010 numbers with sum 200200 gives 200+4211=24211=22\frac{200 + 42}{11} = \frac{242}{11} = 22.
When values repeat, multiply each value by how many times it occurs before adding.
For a 11 scored 1010 times and a 22 scored 1515 times, the sum is 1×10+2×15=401 \times 10 + 2 \times 15 = 40.
Then divide by the total count as usual.

Worked examples

What is the mean of −8,−2,0,6,14-8, -2, 0, 6, 14?
Add them: −8−2+0+6+14=10-8 - 2 + 0 + 6 + 14 = 10.
Divide by the count 55: 105=2\frac{10}{5} = 2.
A set of 100100 numbers has a mean of 5252. What is their sum?
Multiply the mean by the count: 52×100=520052 \times 100 = 5200.
So the sum is 52005200.
A set of 1010 numbers has a sum of 200200. An 1111th number, 4242, is added. What is the new mean?
The new sum is 200+42=242200 + 42 = 242, and the new count is 1111.
So the new mean is 24211=22\frac{242}{11} = 22.

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