Question

Abby has the following data:

81 q 86 78 62 67 86
If the mean is 78, which number could q be?
93 or 86

Answer 1

To determine the possible value for \( q \) based on the given mean of 78, we first need to understand how to calculate the mean.

The mean is calculated by dividing the sum of all numbers by the total number of numbers. In this case, Abby has 7 data points: \( 81, q, 86, 78, 62, 67, 86 \).

The formula for the mean is:

\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\]

Given that the mean is 78 and there are 7 data points, we can set up the equation:

\[
78 = \frac{81 + q + 86 + 78 + 62 + 67 + 86}{7}
\]

First, we calculate the sum of the known numbers:

\[
81 + 86 + 78 + 62 + 67 + 86 = 460
\]

Now we can substitute this back into the mean equation:

\[
78 = \frac{460 + q}{7}
\]

To eliminate the fraction, we multiply both sides by 7:

\[
78 \times 7 = 460 + q
\]

Calculating the left side:

\[
546 = 460 + q
\]

Now, we can solve for \( q \) by subtracting 460 from both sides:

\[
q = 546 - 460 = 86
\]

Now, let’s check which of the options (93 or 86) could be \( q \):

1. **If \( q = 86 \)**:

The data set becomes: \( 81, 86, 86, 78, 62, 67, 86 \).

The sum is: \( 81 + 86 + 86 + 78 + 62 + 67 + 86 = 546 \).

The mean is:

\[
\frac{546}{7} = 78
\]

So, \( q = 86 \) works.

2. **If \( q = 93 \)**:

The data set becomes: \( 81, 93, 86, 78, 62, 67, 86 \).

The sum is: \( 81 + 93 + 86 + 78 + 62 + 67 + 86 = 573 \).

The mean is:

\[
\frac{573}{7} \approx 81.86
\]

This mean does not equal 78, so \( q = 93 \) does not work.

Thus, the only possible value for \( q \) that maintains a mean of 78 is \( \boxed{86} \).