Descriptive Statistics for Rental Properties
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This document provides a comprehensive analysis of rental properties, focusing on descriptive statistics such as mean, median, mode, standard deviation, and variance. It includes frequency distribution tables and histograms for different price ranges. The document also compares rental properties near the university campus and in other areas, highlighting differences in central tendencies and variations.
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BUGEN1502 Business Statistics
Semester 1, 2019
Assignment 1: Descriptive statistics
1
Semester 1, 2019
Assignment 1: Descriptive statistics
1
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Answer 1
Table 1: Frequency Distribution of rent prices ($)
LL UL Intervals
(dollars) Frequency Percentage
0 200 0 to 200 59 14.75%
200 250 Over 200 to 250 164 41.00%
250 300 Over 250 to 300 128 32.00%
300 350 Over 300 to 350 36 9.00%
350 --- Over 350 13 3.25%
Figure 1: Bar Diagram for Rent Prices ($) near and other localities of The University
Figure 2: Pie Chart for percentage distribution of properties corresponding to rent prices ($)
2
Table 1: Frequency Distribution of rent prices ($)
LL UL Intervals
(dollars) Frequency Percentage
0 200 0 to 200 59 14.75%
200 250 Over 200 to 250 164 41.00%
250 300 Over 250 to 300 128 32.00%
300 350 Over 300 to 350 36 9.00%
350 --- Over 350 13 3.25%
Figure 1: Bar Diagram for Rent Prices ($) near and other localities of The University
Figure 2: Pie Chart for percentage distribution of properties corresponding to rent prices ($)
2
Brief Discussion
Distribution of rent prices or costs is skewed towards left. The bar diagram in figure 1 reveals
that 164 rental properties are priced between $ 200 to $ 250, which constitutes 41% (from
figure 2) overall rental properties. Pie chart in figure 2 indicates that almost 73% properties
are prices between $ 200 and$ 300. Also, it can be noted from the distribution Table 1 that 13
unusually higher rental properties are also available, which constitutes of 3% rental properties
and can be termed as extraordinarily priced rentals.
Answer 2
Table 2: Frequency distribution of rental properties according to rental prices ($)
LL UL Midpoint Intervals Frequency
0 50 25 0 to 50 0
50 10
0 75 Over 50 to 100 0
100 15
0 125 Over 100 to
150 5
150 20
0 175 Over 150 to
200 54
200 25
0 225 Over 200 to
250 164
250 30
0 275 Over 250 to
300 128
300 35
0 325 Over 300 to
350 36
350 40
0 375 Over 350 to
400 11
400 45
0 425 Over 400 to
450 2
450 50
0 475 Over 450 0
3
Distribution of rent prices or costs is skewed towards left. The bar diagram in figure 1 reveals
that 164 rental properties are priced between $ 200 to $ 250, which constitutes 41% (from
figure 2) overall rental properties. Pie chart in figure 2 indicates that almost 73% properties
are prices between $ 200 and$ 300. Also, it can be noted from the distribution Table 1 that 13
unusually higher rental properties are also available, which constitutes of 3% rental properties
and can be termed as extraordinarily priced rentals.
Answer 2
Table 2: Frequency distribution of rental properties according to rental prices ($)
LL UL Midpoint Intervals Frequency
0 50 25 0 to 50 0
50 10
0 75 Over 50 to 100 0
100 15
0 125 Over 100 to
150 5
150 20
0 175 Over 150 to
200 54
200 25
0 225 Over 200 to
250 164
250 30
0 275 Over 250 to
300 128
300 35
0 325 Over 300 to
350 36
350 40
0 375 Over 350 to
400 11
400 45
0 425 Over 400 to
450 2
450 50
0 475 Over 450 0
3
Figure 3: Histogram for the distribution of rental prices in Table 2
Brief description of Histogram
The histogram reveals that the frequency distribution of rents is almost normal with slight
positive skewness. The rents are highly accumulated within the $ 150 to $ 300 price range.
Some unusual or extraordinarily high rental properties can also be noted, which have
impacted positively on the skewness of the distribution. The median is expected to hover
within the $ 225 and $ 275 price range.
Answer 3
Table 3: Descriptive Values for weekly rental prices ($)
Descriptive Values price ($)
Mean 248.21
Standard Error 2.37
Median 240
Mode 225
Standard Deviation 47.43
Sample Variance 2249.49
Kurtosis 0.41
Skewness 0.64
Range 285
Minimum 135
Maximum 420
Sum 99285
Count 400
4
Brief description of Histogram
The histogram reveals that the frequency distribution of rents is almost normal with slight
positive skewness. The rents are highly accumulated within the $ 150 to $ 300 price range.
Some unusual or extraordinarily high rental properties can also be noted, which have
impacted positively on the skewness of the distribution. The median is expected to hover
within the $ 225 and $ 275 price range.
Answer 3
Table 3: Descriptive Values for weekly rental prices ($)
Descriptive Values price ($)
Mean 248.21
Standard Error 2.37
Median 240
Mode 225
Standard Deviation 47.43
Sample Variance 2249.49
Kurtosis 0.41
Skewness 0.64
Range 285
Minimum 135
Maximum 420
Sum 99285
Count 400
4
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First Quartile 210
Third Quartile 270
Interquartile Range 60
Coefficient of
Variation 19.11
Answer 4
Mean rental price is noted to be $ 248.21, which is also observed to be affected by some high
rental observations. A slight positive skewness can be noted due to that. Median is noted to
be $ 240. Mean is greater than the median confirming that the distribution is slightly right
skewed. Mode of the distribution is $ 225, which implies that the highest several rental (N =
18) properties are priced at $ 225. Again, mode less than the median reconfirm the positive or
right skewness of the distribution.
Answer 5
Standard deviation of $ 47.43 reflected the measure of dispersion from average rental price of
the properties. The three sigma limits can be evaluated as μ±3 σ =[ 105 . 92, 390 .5 ] . Hence, any
observation outside this range can be considered as unusual or outlier observation.
Variance implied that the square of average difference from the mean rental price is around
$2249.49.
Range of $ 285 reflected the difference between the minimum and maximum rents. This
measure implied availability of rents in variable scales and locations.
5
Third Quartile 270
Interquartile Range 60
Coefficient of
Variation 19.11
Answer 4
Mean rental price is noted to be $ 248.21, which is also observed to be affected by some high
rental observations. A slight positive skewness can be noted due to that. Median is noted to
be $ 240. Mean is greater than the median confirming that the distribution is slightly right
skewed. Mode of the distribution is $ 225, which implies that the highest several rental (N =
18) properties are priced at $ 225. Again, mode less than the median reconfirm the positive or
right skewness of the distribution.
Answer 5
Standard deviation of $ 47.43 reflected the measure of dispersion from average rental price of
the properties. The three sigma limits can be evaluated as μ±3 σ =[ 105 . 92, 390 .5 ] . Hence, any
observation outside this range can be considered as unusual or outlier observation.
Variance implied that the square of average difference from the mean rental price is around
$2249.49.
Range of $ 285 reflected the difference between the minimum and maximum rents. This
measure implied availability of rents in variable scales and locations.
5
The first quartile reflected the cheapest 25% rental properties are price below $ 210, and third
quartile indicated that the costly top 25% properties are placed above $ 270. The middle 50%
properties are placed between $ 210 and $ 270 (the interquartile range).
Answer 6
The z-scores have been calculated using excel and have presented in the appendix section.
The descriptive values have been presented in table 4.
Table 4: Descriptive value of the z-scores
Descriptive Values Z-Scores
Mean 0
Standard Error 0.05
Median -0.17
Mode -0.49
Standard Deviation 1
Sample Variance 1
Kurtosis 0.41
Skewness 0.64
Range 6.01
Minimum -2.39
Maximum 3.62
Sum 0
Count 400
The mean of the z-scores is zero, which implies that the average of standard normalised rental
price is zero. The standard deviation is 1, which implied that the root mean square deviation
of standard normalised rental prices are spread with one sigma limit. These results are in line
with the mean of the standard normal distribution.
6
quartile indicated that the costly top 25% properties are placed above $ 270. The middle 50%
properties are placed between $ 210 and $ 270 (the interquartile range).
Answer 6
The z-scores have been calculated using excel and have presented in the appendix section.
The descriptive values have been presented in table 4.
Table 4: Descriptive value of the z-scores
Descriptive Values Z-Scores
Mean 0
Standard Error 0.05
Median -0.17
Mode -0.49
Standard Deviation 1
Sample Variance 1
Kurtosis 0.41
Skewness 0.64
Range 6.01
Minimum -2.39
Maximum 3.62
Sum 0
Count 400
The mean of the z-scores is zero, which implies that the average of standard normalised rental
price is zero. The standard deviation is 1, which implied that the root mean square deviation
of standard normalised rental prices are spread with one sigma limit. These results are in line
with the mean of the standard normal distribution.
6
Answer 7
Two outlier observations are identified from the z-scores (rent having z-score less than -3.0
or higher than 3.0). The two outlier rental prices are $ 405 and $ 420.
Table 5: Descriptive values of rental prices with and without outlier prices
Descriptive Values Without OL price ($) With OL price ($)
Mean 247.39 248.213
Standard Error 2.31 2.371
Median 240 240
Mode 225 225
Standard Deviation 46.09 47.43
Sample Variance 2123.89 2249.49
Kurtosis 0.05 0.41
Skewness 0.52 0.64
Range 255 285
Minimum 135 135
Maximum 390 420
Sum 98460 99285
Count 398 400
First Quartile 210 210
Third Quartile 270 270
Interquartile Range 60 60
Coefficient of Variation 18.63% 19.11%
7
Two outlier observations are identified from the z-scores (rent having z-score less than -3.0
or higher than 3.0). The two outlier rental prices are $ 405 and $ 420.
Table 5: Descriptive values of rental prices with and without outlier prices
Descriptive Values Without OL price ($) With OL price ($)
Mean 247.39 248.213
Standard Error 2.31 2.371
Median 240 240
Mode 225 225
Standard Deviation 46.09 47.43
Sample Variance 2123.89 2249.49
Kurtosis 0.05 0.41
Skewness 0.52 0.64
Range 255 285
Minimum 135 135
Maximum 390 420
Sum 98460 99285
Count 398 400
First Quartile 210 210
Third Quartile 270 270
Interquartile Range 60 60
Coefficient of Variation 18.63% 19.11%
7
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Comparison of data analysis results
After removal of the outliers, the average rental price reduced to $ 247.39 and came near to
median rental price of $ 240. Median is not affected by outlier values, and here also a similar
reflexion is noted.
The standard deviation reduced to $ 46.09 due to outlier removal. Hence, the variability of
the prices reduced approximately by $ 1.34.
The distribution of rental prices after outlier removal is still positively skewed, but the
dissemination became more close to normal distribution. The coefficient of variance also
reduced to 18.63%. It is also to be noted that the quartiles remain unchanged.
Answer 8
Table 6: Frequency distribution of Near Prices
LL UL Midpoint Frequency
0 180 90 5
180 230 205 41
230 280 255 66
280 330 305 68
330 380 355 17
380 430 405 3
8
After removal of the outliers, the average rental price reduced to $ 247.39 and came near to
median rental price of $ 240. Median is not affected by outlier values, and here also a similar
reflexion is noted.
The standard deviation reduced to $ 46.09 due to outlier removal. Hence, the variability of
the prices reduced approximately by $ 1.34.
The distribution of rental prices after outlier removal is still positively skewed, but the
dissemination became more close to normal distribution. The coefficient of variance also
reduced to 18.63%. It is also to be noted that the quartiles remain unchanged.
Answer 8
Table 6: Frequency distribution of Near Prices
LL UL Midpoint Frequency
0 180 90 5
180 230 205 41
230 280 255 66
280 330 305 68
330 380 355 17
380 430 405 3
8
Figure 4: Histogram of Rents for Near Properties
Shape of the distribution
The histogram of rental prices for the near the university properties is almost normal with
slight positive skewness due to presence of few outlier values. Rents above $ 390 are the two
outlier values present in the distribution. The majority of the rental properties are
concentrated within price range of $ 230 to $ 330.
Table 7: Frequency distribution of other rental properties
LL UL Midpoint Frequency
0 140 70 2
140 180 160 18
180 220 200 61
220 260 240 102
260 300 280 16
300 340 320 1
9
Shape of the distribution
The histogram of rental prices for the near the university properties is almost normal with
slight positive skewness due to presence of few outlier values. Rents above $ 390 are the two
outlier values present in the distribution. The majority of the rental properties are
concentrated within price range of $ 230 to $ 330.
Table 7: Frequency distribution of other rental properties
LL UL Midpoint Frequency
0 140 70 2
140 180 160 18
180 220 200 61
220 260 240 102
260 300 280 16
300 340 320 1
9
Figure 5: Histogram of Rents for other Properties
Shape of the distribution
The histogram of rental prices for the other properties is almost normal with slight negative
skewness due to presence of few low rental properties. Rents below $ 180 are the two lower
rental values present in the distribution. The majority of the rental properties are concentrated
within price range of $ 180 to $ 260 (Chatfield, 2018).
Answer 9
The descriptive values have been evaluated for the rental properties, separately for the “near”
and “other” properties. Table 8 represents the descriptive summary for properties at the two
locations (Spiegel, Schiller, & Srinivasan, 2013).
Table 8: Descriptive Values of "near" and "other" properties
Descriptive Values Near Price($) Other Price ($)
Mean 272.78 223.65
Standard Error 3.40 2.22
Median 270 225
10
Shape of the distribution
The histogram of rental prices for the other properties is almost normal with slight negative
skewness due to presence of few low rental properties. Rents below $ 180 are the two lower
rental values present in the distribution. The majority of the rental properties are concentrated
within price range of $ 180 to $ 260 (Chatfield, 2018).
Answer 9
The descriptive values have been evaluated for the rental properties, separately for the “near”
and “other” properties. Table 8 represents the descriptive summary for properties at the two
locations (Spiegel, Schiller, & Srinivasan, 2013).
Table 8: Descriptive Values of "near" and "other" properties
Descriptive Values Near Price($) Other Price ($)
Mean 272.78 223.65
Standard Error 3.40 2.22
Median 270 225
10
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Mode 270 225
Standard Deviation 48.05 31.44
Sample Variance 2308.97 988.62
Kurtosis -0.25 0.63
Skewness 0.34 0.02
Range 240 195
Minimum 180 135
Maximum 420 330
Sum 54555 44730
Count 200 200
First Quartile 240 195
Third Quartile 300 240
Interquartile Range 60 45
Coefficient of
Variation 17.62% 14.06%
Answer 10
Comparison of the central tendencies
Average rental price is considerably low in other location compared to near university
campus. Students with budget constraint can look for comparatively cheap properties in the
other region. Two unusually high rental properties are also present near the university
campus. Mean, median and mode are almost same in both regions, which reflected normally
distributed property prices in both regions. Average rents are comparatively cheaper in other
region compared to near university campus.
Comparison of variations
Rental prices in other region have less price variation compared to near region. Standard
deviation of price is $ 31.44 in other region, whereas standard deviation is $ 48.05 in near
region. Coefficient of variation for rental prices in near area is 17.62%, which is considerably
higher than 14.06% of the other area.
Considering the quartiles, the top 25% rental properties are placed above $ 240 in other area,
whereas for near area rents of top 25% properties start from $ 300. Even the cheapest 25%
11
Standard Deviation 48.05 31.44
Sample Variance 2308.97 988.62
Kurtosis -0.25 0.63
Skewness 0.34 0.02
Range 240 195
Minimum 180 135
Maximum 420 330
Sum 54555 44730
Count 200 200
First Quartile 240 195
Third Quartile 300 240
Interquartile Range 60 45
Coefficient of
Variation 17.62% 14.06%
Answer 10
Comparison of the central tendencies
Average rental price is considerably low in other location compared to near university
campus. Students with budget constraint can look for comparatively cheap properties in the
other region. Two unusually high rental properties are also present near the university
campus. Mean, median and mode are almost same in both regions, which reflected normally
distributed property prices in both regions. Average rents are comparatively cheaper in other
region compared to near university campus.
Comparison of variations
Rental prices in other region have less price variation compared to near region. Standard
deviation of price is $ 31.44 in other region, whereas standard deviation is $ 48.05 in near
region. Coefficient of variation for rental prices in near area is 17.62%, which is considerably
higher than 14.06% of the other area.
Considering the quartiles, the top 25% rental properties are placed above $ 240 in other area,
whereas for near area rents of top 25% properties start from $ 300. Even the cheapest 25%
11
rentals are priced below $ 195 in other area, whereas the same is price below $ 240 in near
area. Clearly, prices in near university campus are higher than other area.
References
Spiegel, M. R., Schiller, J. J., & Srinivasan, R. A. (2013). Probability and statistics (Fourth
edition). In Schaum’s Outlines (Fourth edition). New York: McGraw-Hill.
Chatfield, C. (2018). Statistics for Technology, A Course in Applied Statistics. Third edition.
New York, USA: Routledge. https://doi.org/10.1201/9780203738467
Appendix
Table 9: Z-scores of prices (related to Answer 6 and 7)
price
($) Z-Score
135 -2.386999038
135 -2.386999038
150 -2.070735502
150 -2.070735502
150 -2.070735502
165 -1.754471966
165 -1.754471966
12
area. Clearly, prices in near university campus are higher than other area.
References
Spiegel, M. R., Schiller, J. J., & Srinivasan, R. A. (2013). Probability and statistics (Fourth
edition). In Schaum’s Outlines (Fourth edition). New York: McGraw-Hill.
Chatfield, C. (2018). Statistics for Technology, A Course in Applied Statistics. Third edition.
New York, USA: Routledge. https://doi.org/10.1201/9780203738467
Appendix
Table 9: Z-scores of prices (related to Answer 6 and 7)
price
($) Z-Score
135 -2.386999038
135 -2.386999038
150 -2.070735502
150 -2.070735502
150 -2.070735502
165 -1.754471966
165 -1.754471966
12
165 -1.754471966
165 -1.754471966
165 -1.754471966
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
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195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
13
165 -1.754471966
165 -1.754471966
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
180 -1.43820843
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
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195 -1.121944894
195 -1.121944894
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195 -1.121944894
195 -1.121944894
195 -1.121944894
195 -1.121944894
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195 -1.121944894
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210 -0.805681358
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210 -0.805681358
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