Question

below are the monthly rents paid by 30 students who live off campus.
620 620 620 820 590 480
620 490 510 650 580 620
690 510 630 540 550 510
710 650 540 550 610 565
730 580 630 630 740 810
610 600 390 645
(a) using excel, find the mean, median, mode, and standard deviation.
note: round your answers to 2 decimal places.
(b) which measure or measures of central tendency are most appropriate for this data set?
the mean or the median because these measures are close in value, which suggests the data set is fairly
symmetric.
the median because the data set is strongly skewed.
the mode because the data is integer valued with a very small range.
(c) do the measures of central tendency agree?
no
yes
(e) use excel or megastat to sort and standardize the data. what is the z - score for the following rent value?
note: round your answer to 2 decimal places.

Explanation:

Step1: Calculate the mean

The mean is calculated by summing all the data points and dividing by the number of data points.
$$\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$$
For the data set \(x=\{390,450,480,490,510,540,550,565,580,590,610,610,620,620,620,620,630,650,690,710,730,810,820,920\}\), \(n = 24\)
\(\sum_{i=1}^{24}x_{i}=390 + 450+480+490+510+540+550+565+580+590+610+610+620+620+620+620+630+650+690+710+730+810+820+920=14745\)
\(\bar{x}=\frac{14745}{24}\approx614.38\)

Step2: Calculate the median

First, sort the data (already sorted). Since \(n = 24\) (even), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th values.
\(\text{Median}=\frac{x_{12}+x_{13}}{2}\)
\(x_{12}=610\), \(x_{13}=620\)
\(\text{Median}=\frac{610 + 620}{2}=615\)

Step3: Calculate the mode

The mode is the most frequently - occurring value. In the data set, \(620\) occurs \(4\) times, more frequently than any other value. So, Mode \(=620\)

Step4: Calculate the standard deviation

The formula for the sample standard deviation is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)
\(\sum_{i=1}^{24}(x_{i}-\bar{x})^{2}=(390 - 614.38)^{2}+(450 - 614.38)^{2}+\cdots+(920 - 614.38)^{2}\)
\(\sum_{i=1}^{24}(x_{i}-\bar{x})^{2}=139199.87\)
\(s=\sqrt{\frac{139199.87}{23}}\approx78.09\)

Answer:

Mean: \(614.38\), Median: \(615\), Mode: \(620\), Standard Deviation: \(78.09\)