To calculate a new average when you have the old average and a new value, you don't need to re-calculate everything from scratch. You can use the following method:
Calculating the New Average
Here’s the process for updating an average when you get a new data point:
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Multiply the old average by the number of data points that went into the old average (let's call this 'n-1'). This gives you the sum of the original numbers.
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Add the new number to this sum. This gives you the sum of all the numbers including the new one.
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Divide the total from step 2 by the total number of numbers (now 'n'). This provides you with the new average.
Formula
The above steps can be summarized with the following formula:
*New Average = ((Old Average (n - 1)) + New Number) / n**
Where:
- Old Average is the average you started with.
- n is the total number of data points including the new value.
- n - 1 represents the number of data points used to calculate the original average.
- New Number is the new value you are adding.
Example:
Let's say you have an average of 10 from 4 numbers (i.e. n-1 = 4), and you add a new number: 18.
- Step 1: 10 (old average) * 4 (n-1) = 40
- Step 2: 40 + 18 (new number) = 58
- Step 3: 58 / 5 (n) = 11.6
Therefore, the new average is 11.6.
Table Summary
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1. Sum of Old Data | Multiply the old average by the number of data points. (n-1) | Old Average * (n-1) | Sum |
| 2. Add New Number | Add the new number to the sum of old numbers. | Sum + New Number | New Sum |
| 3. New Average | Divide the sum by the new number of data points (n). | New Sum / n | New Average |
Practical Insights
- This method is useful for tracking averages on the fly without needing to keep all the original data.
- It's particularly valuable in situations where you are continuously receiving new values and want to maintain an updated average efficiently.
- The method is outlined in reference materials as: to calculate the new average after then nth number, you multiply the old average by n−1, add the new number, and divide the total by n.
