Solution
The question tells us the rent of 9 people is given as:
940, 945, 975, 990, 1000, 1030, 1055, 1095, 1150.
- We are told that the rent of 940 changes to 985 and we are asked by how much the mean and median rent change.
- In order to know how much the mean and median rents changed, we need to know the initial mean and median rent values.
- The mean of the data can be calculated using the formula below:
[tex]\begin{gathered} \mu=\frac{\sum ^n_{k=1}x_k}{n} \\ \text{where,} \\ k\text{ is the position of an individual data point.} \\ n\text{ is the number of people or total number of data points} \end{gathered}[/tex]- The median is the middle number of the data points when arranged in ascending or descending order.
- With these two definitions, we can proceed to find the initial mean and median.
Initial Mean:
[tex]\begin{gathered} \mu_1=\frac{940+945+975+990+1000+1030+1055+1095+1150}{9} \\ \\ \mu_1=\frac{9180}{9} \\ \\ \mu_1=1020 \end{gathered}[/tex]Initial Median:
The middle number is the 5th number. The 5th number is 1000. Thus, 1000 is the initial median.
Final Mean:
- We are told that the rent of 940 was replaced with 985. This change would definitely change the mean. Thus, we can calculate the new mean by simply replacing 940 with 985 in our mean calculation. This is done below:
[tex]\begin{gathered} \mu_2=\frac{985+945+975+990+1000+1030+1055+1095+1150}{9} \\ \mu_2=\frac{9225}{9} \\ \\ \mu_2=1025 \end{gathered}[/tex]Final Median
- We know that 940 has been replaced with 985. This means that we should rearrange the dataset to get the new Median.
- Arranging the dataset in ascending order, we have:
945, 975, 985, 990, 1000, 1030, 1055, 1095, 1150
- The new median is the number in the 5th position and the new median is 1000
- Thus, we can answer the questions asked of us.
Question 1:
[tex]\begin{gathered} Old\text{ Mean - New Mean} \\ \mu_1-\mu_2=1020-1025=-5 \\ \\ \text{Thus, the Mean increased by 5} \end{gathered}[/tex]Question 2:
[tex]\begin{gathered} \text{Old median - New Median} \\ 1000-1000=0 \\ \\ \text{Thus, the Median does not change} \end{gathered}[/tex]Final Answer
The Mean increased by 5
The Median does not change
