Question

One sample has a mean of M = 4 and a second sample has a mean of M = 8. The two samples are combined into a single set of scores.

What is the mean for the combined set if both of the original samples have n = 7 scores?
What is the mean for the combined set if the first sample has n = 3 and the second sample has n = 7?
What is the mean for the combined set if the first sample has n = 7 and the second sample has n = 3?

Answer 1

Answer on question 36529 – Math – Probability and Statistics

One sample has a mean of $M1 = 4$ and a second sample has a mean of $M2 = 8$. The two samples are combined into a single set of scores.

What is the mean for the combined set if both of the original samples have $n = 7$ scores?

What is the mean for the combined set if the first sample has $n = 3$ and the second sample has $n = 7$?

What is the mean for the combined set if the first sample has $n = 7$ and the second sample has $n = 3$?

Solution

Let we have a samples $x_{1}, x_{2}, \ldots, x_{n}$ and $y_{1}, y_{2}, \ldots, y_{m}$ than the mean is $M1 = \frac{x_{1} + x_{2} + \cdots + x_{n}}{n}$ and $M2 = \frac{y_{1} + y_{2} + \cdots + y_{m}}{m}$.

the mean for the combined set is

$$M = \frac{x_{1} + \cdots + x_{n} + y_{1} + \cdots + y_{m}}{m + n}.$$

If $n = m = 7$ we get

$$M = \frac{x_{1} + \cdots + x_{7} + y_{1} + \cdots + y_{7}}{14} = \frac{x_{1} + x_{2} + \cdots + x_{7}}{14} + \frac{y_{1} + y_{2} + \cdots + y_{7}}{14} = \frac{M1}{2} + \frac{M2}{2} = 2 + 4 = 6.$$

If $n = 3$ and $m = 7$, we get

$$M = \frac{x_{1} + x_{2} + x_{3} + y_{1} + \cdots + y_{7}}{10} = \frac{3M1 + 7M2}{10} = \frac{12 + 56}{10} = 6.8.$$

If $n = 7$ and $n = 3$, we get

$$M = \frac{x_{1} + \cdots + x_{7} + y_{1} + \cdots + y_{3}}{10} = \frac{7M1 + 3M2}{10} = \frac{28 + 24}{10} = 5.2.$$

Answer: 6; 6.8; 5.2.