Question

The mean of 8, 7, 5, 9, 4, 3, and `k+4` is 6. What is `k`?

Answer 1

`k=2`.

Explanation:

The formula for the mean `barx` of some numbers is the sum of those numbers, divided by how many there are. For a dataset of 7 numbers (like in this question), the formula is:

`barx = (x_1+x_2+x_3+x_4+x_5+x_6+x_7)/7`

Knowing the mean is 6 and the values of the dataset, we can substitute them in:

`6=(8+7+5+9+4+3+(k+4))/7`

This is now an equation with one unknown variable, `k`, and so we can solve for `k`:

`42=8+7+5+9+4+3+(k+4)`
`42=40+k`
`color(white)0 2=k`

We can also test this value of `k` to see if it works with the original question. In other words, if we take the mean of 8, 7, 5, 9, 4, 3, and `k+4` (when `k=2`) we should get a mean of 6. Do we?

`barx = (8+7+5+9+4+3+(color(blue)k+4))/7`

`color(white)(barx) = (8+7+5+9+4+3+(color(blue)2+4))/7`

`color(white)(barx) = (36+(6))/7`

`color(white)(barx) = 42/7" "=6`

We do! So we have verified that `k=2`.

Answer 2

Alternate way to find the unknown data point (and `k`). The answer is still `k=2`.

Explanation:

If you know all the data in a set except for one number, and you know the mean, you can find the last number by remembering that the sum of all the differences from the mean is always 0.

To illustrate, our data starts with an 8, and 8 is two more than 6 (the mean). Remember `+2.`

Next is 7, which is 1 more than the mean. Add `2+1` to get `3.`

Next: 5. One less than the mean. `3-1=2.`

Next: 9. Three more than the mean. `2+3=5.`

Next: 4. Two less. `5-2=3.`

Next: 3. Three less. `3-3=0`.

Now we're down to our last data point, and the sum of all the differences so far is 0. Thus, to end with a total difference of 0, the last difference must be the negative of the cumulative differences so far: that is, `-0`, which is just 0.

`k+4 -6= 0`

Which simplifies quite easily to

`k=2.`