Question

Two samples of sizes 50 and 100 are given. The mean of these samples is 56 and 50 respectively. Find the mean of the combined samples.

Answer 1

Hint: Assume two samples A and B, and the elements of the sample A and sample B are \[{{x}_{1}},{{x}_{2}},..........,{{x}_{50}}\] and \[{{y}_{1}},{{y}_{2}},..........,{{y}_{100}}\] respectively. We know the formula of mean, \[Mean=\dfrac{Sum\,of\,the\,elements\,of\,the\,sample}{Sample\,size}\] . We have the mean and sample size of the samples A and B. Put the elements, mean, and sample size of x in the formula of the mean. Then, get the value of \[\left( {{x}_{1}}+{{x}_{2}}+..........+{{x}_{50}} \right)\] . Similarly, put the elements, mean, and sample size of y in the formula of the mean. Then, get the value of \[\left( {{y}_{1}}+{{y}_{2}}+..........+{{y}_{100}} \right)\] . For the combined sample, the elements are \[{{x}_{1}},{{x}_{2}},..........,{{x}_{50}},{{y}_{1}},{{y}_{2}},..........,{{y}_{100}}\] . Since we have 150 elements, we have the sample size of the combined sample equal to 150. Now, use the formula of the mean for the combined sample and put the values of \[\left( {{x}_{1}}+{{x}_{2}}+..........+{{x}_{50}} \right)\] and \[\left( {{y}_{1}}+{{y}_{2}}+..........+{{y}_{100}} \right)\] . Now, solve it further and get the value of the mean of the combined sample.

Complete step by step answer:
First of all, let us assume that A and B are two samples and the elements of sample A and sample B are \[{{x}_{1}},{{x}_{2}},..........,{{x}_{50}}\] and \[{{y}_{1}},{{y}_{2}},..........,{{y}_{100}}\] respectively.
Now, according to the question, we have
The mean of sample A = 56 …………………..(1)
The sample size of sample A = 50 ……………………(2)
The mean of sample B = 50 …………………..(3)
The sample size of sample B = 100 ……………………(4)
We know the formula of mean, \[\text{Mean}=\dfrac{\text{Sum of the elements of the sample}}{\text{Sample size}}\] ………………………..(5)
Now, from equation (1), equation (2), and equation (5), we get
\[\begin{align}
  & 56=\dfrac{{{x}_{1}}+{{x}_{2}}+.........+{{x}_{56}}}{50} \\
 & \Rightarrow 56\times 50={{x}_{1}}+{{x}_{2}}+.........+{{x}_{50}} \\
\end{align}\]
\[\Rightarrow 2800={{x}_{1}}+{{x}_{2}}+.........+{{x}_{50}}\] ……………………….(6)
Now, from equation (1), equation (2), and equation (5), we get
\[\begin{align}
  & 50=\dfrac{{{y}_{1}}+{{y}_{2}}+.........+{{y}_{100}}}{100} \\
 & \Rightarrow 50\times 100={{y}_{1}}+{{y}_{2}}+.........+{{y}_{100}} \\
\end{align}\]
\[\Rightarrow 5000={{y}_{1}}+{{y}_{2}}+.........+{{y}_{100}}\] ……………………….(7)
Now, we have to find the mean when the sample A and sample B are combined.
The sample size of the combined sample = Sample size of A + Sample size of B …………………(8)
From equation (2), equation (4), and equation (8), we get
The sample size of the combined sample = 50 + 100 = 150 ………………(9)
The elements of the combined sample are \[{{x}_{1}},{{x}_{2}},..........,{{x}_{50}},{{y}_{1}},{{y}_{2}},..........,{{y}_{100}}\] ………………..(10)
We know the formula of mean, \[\text{Mean}=\dfrac{\text{Sum of the elements of the sample}}{\text{Sample size}}\] …………………..(11)
From equation (9), equation (10), and equation (11), we get
\[Mean=\dfrac{\left( {{x}_{1}}+{{x}_{2}}+..........+{{x}_{50}} \right)+\left( {{y}_{1}}+{{y}_{2}}+..........+{{y}_{100}} \right)}{150}\] …………………………(12)
Now, putting the value of \[\left( {{x}_{1}}+{{x}_{2}}+..........+{{x}_{50}} \right)\] from equation (6) and \[\left( {{y}_{1}}+{{y}_{2}}+..........+{{y}_{100}} \right)\] from equation (7), in equation (12), we get
\[Mean=\dfrac{\left( 2800 \right)+\left( 5000 \right)}{150}=\dfrac{7800}{150}=52\]

Hence, the mean of the combined sample A and B is 52.

Note: In this question, one might calculate the mean of the combined sample as the average of the mean of the individual samples, \[\dfrac{56+50}{2}\] . This is wrong because it indicates that the sample size of the combined sample is 2 whereas, in reality, we have 156 elements in the combined sample.