Video: Calculating the Mean of a Whole Data Set given the Means of Some of Its Subsets

A student takes five tests: ๐ด, ๐ต, ๐ถ, ๐ท, and ๐ธ. Their mean score for tests ๐ด, ๐ต, and ๐ถ is 48. Their mean score for tests ๐ท and ๐ธ is 38. Calculate the mean score of all 5 tests.

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Video Transcript

A student takes five tests ๐ด, ๐ต, ๐ถ, ๐ท, and ๐ธ. Their mean score for tests ๐ด, ๐ต, and ๐ถ is 48. Their mean score for tests ๐ท and ๐ธ is 38. Calculate the mean score of all five tests.

To answer this question, we can recall that the mean of a set of numbers is the sum of all the numbers divided by how many numbers there are. Weโ€™re told that a student takes five tests, ๐ด through ๐ธ, and that the mean of ๐ด, ๐ต, and ๐ถ is 48 and the mean of ๐ท and ๐ธ is 38. What we need to do here then is worked out the mean of all five. Letโ€™s start by looking at the first three tests where the mean is 48. Using the formula then, we know that the mean is 48. We donโ€™t know the sum of the numbers, but we do know that there are three numbers, since thereโ€™re three tests. To rearrange this then, we can multiply both sides of our equation by three, giving us 48 times three on the left-hand side, which we can evaluate as 144. So now, we know that the sum of the scores in ๐ด, ๐ต, and ๐ถ is 144.

In the same way, we can find the sum of the scores for tests ๐ท and ๐ธ. Weโ€™re given that there mean is 38 and since weโ€™re talking about two tests, then we know that we divide by two. Multiplying both sides of our equation by two will give us that 38 times two is equal to the sum of the numbers. And since this evaluates to 76 on the left-hand side, then we now know that the sum of the scores in test ๐ท and ๐ธ is 76. Itโ€™s worth noting at this point that we donโ€™t know the actual individual scores of each of the tests. We just know the sum. However, we can still continue with our calculation for the mean of all five tests.

Therefore, to find the mean of ๐ด, ๐ต, ๐ถ, ๐ท, and ๐ธ, we calculate the sum of all the numbers which would be 144 plus 76 and divide by how numbers there are. In this case, there are five tests. So there must be five values. Simplifying then, we have 220 over five or 220 divided by five, which is equal to 44. And so our final answer is that the mean score of all five tests is 44.

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