The monthly rents for the apartments in a building are listed below. $425, $550, $550, $550, $650, $650, $650, $650, $800, $900
Part B: The apartment manager considers raising the rent for every apartment by $50. Find the mean, median, mode, range, and standard deviation of the rents after they are raised by $50. Explain your reasoning.
The mean, median, mode are shifted by $50, while the range does not change, from the values of part 1. This is just a shift of origin, and the properties of the given parameters are as such.
To find the mean, median, mode, range, and standard deviation of the rents after they are raised by $50, we need to follow these steps:
1. Add $50 to each rent value to account for the rent increase. This is because the apartment manager is considering raising the rent for every apartment by $50.
New rent values: $475, $600, $600, $600, $700, $700, $700, $700, $850, $950
2. Find the mean:
To find the mean, we sum up all the new rent values and then divide the sum by the total number of rent values.
Sum of the new rent values: $475 + $600 + $600 + $600 + $700 + $700 + $700 + $700 + $850 + $950 = $6875
The total number of rent values: 10
Mean = Sum of new rent values / Total number of rent values = $6875 / 10 = $687.5
Therefore, the mean rent after the increase is $687.5.
3. Find the median:
The median is the middle value when the rent values are arranged in numerical order. Let's reorder the new rent values:
$475, $600, $600, $600, $700, $700, $700, $700, $850, $950
Since there are 10 values, the middle two values are the fifth and sixth values, which are both $700.
Therefore, the median rent after the increase is $700.
4. Find the mode:
The mode is the most frequently occurring value. From the new rent values, $700 occurs four times, which is more frequent than any other value.
Therefore, the mode rent after the increase is $700.
5. Find the range:
The range is the difference between the highest and lowest values.
Highest value: $950
Lowest value: $475
Range = Highest value - Lowest value = $950 - $475 = $475
Therefore, the range of rents after the increase is $475.
6. Find the standard deviation:
The standard deviation measures the spread of the data. To calculate it, follow these steps:
a. Find the mean of the new rent values, which is $687.5 (calculated earlier).
b. Subtract the mean from each new rent value and square the result.
c. Find the mean of the squared differences.
d. Take the square root of the mean calculated in step c.
Here are the squared differences from the mean:
($475 - $687.5)^2 = $4356.25
($600 - $687.5)^2 = $7656.25
($600 - $687.5)^2 = $7656.25
($600 - $687.5)^2 = $7656.25
($700 - $687.5)^2 = $156.25
($700 - $687.5)^2 = $156.25
($700 - $687.5)^2 = $156.25
($700 - $687.5)^2 = $156.25
($850 - $687.5)^2 = $26562.5
($950 - $687.5)^2 = $69206.25
Now, sum up the squared differences:
$4356.25 + $7656.25 + $7656.25 + $7656.25 + $156.25 + $156.25 + $156.25 + $156.25 + $26562.5 + $69206.25 = $184161.25
Calculate the mean of the squared differences:
Mean = Sum of squared differences / Total number of rent values = $184161.25 / 10 = $18416.125
Finally, take the square root of the mean:
Standard Deviation = √Mean = √$18416.125 ≈ $135.75
Therefore, the standard deviation of the rents after the increase is approximately $135.75.