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This is probably super simple to most of you on here, but I was chatted by a friend earlier with a question. It reads just like this:

A sample of n =7 scores has a mean of M =5. After one new score is added to the sample, 
the new mean is found to be M =6. What is the value of the new
score? (Hint: Compare the values for E X before and after the score was added.)

She said the E X looks like the greek symbols. Can someone please explain this problem to me and provide a solution or tell me how to solve it? I guess I don't understand the question or how to set up an equation to solve it. Thanks!

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migrated from May 23 '13 at 20:14

This question came from our site for users of Mathematica.

up vote 3 down vote accepted

The sample mean is given by:

$M=\dfrac{\sum_{i=1}^n x_i}{n}$

where $x$ is your individual observations and $n$ the number of observations.

Before you added the observation you have:

$5=\dfrac{\sum_{i=1}^7 x_i}{7}$

And after you add the observation you have:

$6=\dfrac{\sum_{i=1}^8 x_i}{8}$

Given these you can solve for the sums of $x_i$

These turn out to be:

$\sum_{i=1}^7 x_i=5 \times 7 = 35$

$\sum_{i=1}^8 x_i=6 \times 8 = 48$

We know that the above summations share the first $7$ observations. Therefore we can find the value of the new observation by taking the difference.

$x_8=\sum_{i=1}^8 x_i- \sum_{i=1}^7 x_i=48 - 35 = 13$

Therefore your new observation is $13$

Some further explanation:

We may also write $\sum_{i=1}^7 x_i=x_1 + x_2 + x_3 +x_4 + x_5 +x_6 +x_7$

As we, we may write: $x_8=\sum_{i=1}^8 x_i = x_1 + x_2 + x_3 +x_4 + x_5 +x_6 +x_7 +x_8$

We had shown that $\sum_{i=1}^7 x_i=x_1 + x_2 + x_3 +x_4 + x_5 +x_6 +x_7=35$

Therefore, we may substitute this into the equation for $\sum_{i=1}^8 x_i$:

$\sum_{i=1}^8 x_i = x_1 + x_2 + x_3 +x_4 + x_5 +x_6 +x_7 +x_8 = 35 +x_8$

Since using the formula for the mean we also shown:

$\sum_{i=1}^8 x_i= 45$

Then we have:

$45 = 35 + x_8$

and simply solving for $x_8$:

$x_8 = 13$

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We had came up with 13 in one of our attempts, but then also divided that by 2 to get 6.5 for our answer. We had also somehow came up with 48 as an answer in one of our attempts. Can you possibly explain why we ended with those two answers? Obviously math is not my strong point. – AndyWarren May 23 '13 at 20:33
@AndyWarren I added some further clarification. For the means themselves, I just used the arithmetic mean formula for the two individual means. – GovEcon May 23 '13 at 20:39
Thank you for the clarification, answer accepted! – AndyWarren May 23 '13 at 20:53

The sum of the first seven scores is $5 \cdot 7$. It's $7$ times the mean.

Similarly, you can find the sum of all eight scores - to find the last one, all you need to do is subtract.

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The mean is the sum $S$ of the scores divided by the number of scores. Here we have $M = S/n$. Plugging in our values gives $S = 7 \cdot 5 = 35$. Now the new mean is $6 = (S + X)/(n+1) = (35 + X)/8$. Then just solve for $X$.

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