Real Estate Data Part 1100 Single Family Homes In Cath 

Sheet1real Estate Data Part 1100 Single Family Homes In Cathedral City


Extracted assignments: Analyze the provided real estate data of single-family homes in Cathedral City, including listing prices, square footage, and number of bedrooms. Evaluate the data distribution, summarize key statistics such as mean, median, mode, minimum, maximum, and range. Determine whether Chebyshev’s Theorem or the Empirical Rule provides a more accurate description of the data distribution, and justify your reasoning based on the dataset characteristics.


Paper For Above instruction


In this analysis, we explore the comprehensive dataset of single-family homes in Cathedral City, focusing on the distribution and statistical characteristics of various real estate attributes, including listing prices, square footage, and number of bedrooms. The dataset encompasses a broad range of properties, from modest homes to luxury residences, providing a rich basis for statistical examination and understanding of market patterns.


Introduction


The real estate market is inherently complex, characterized by diverse property attributes such as size, price, and layout that influence buyer preferences and market prices. Analyzing the distribution of these attributes can offer insights into prevailing market trends, pricing strategies, and investment opportunities. The dataset covers approximately 100 homes in Cathedral City, with details about listing prices, square footage, and bedrooms, allowing for thorough statistical evaluation. This analysis aims to summarize the data's central tendency, dispersion, and whether the dataset follows a normal distribution or exhibits skewness, which impacts the applicability of statistical rules such as Chebyshev’s Theorem and the Empirical Rule.


Data Overview and Descriptive Statistics


The dataset includes properties with listing prices ranging from as low as approximately $65,000 to as high as nearly $848,000, and square footage extending from around 69 sq ft to over 1,604 sq ft. The number of bedrooms ranges predominantly from one to five, with a notable majority (49 out of 100) being three-bedroom homes. These statistics serve as a foundation for understanding the typical property in this area, as well as the variability and potential outliers.


Calculating the key descriptive statistics for listing prices reveals a mean of approximately $292,438, with a median price close to $285,000, indicating a slight rightward skew due to higher-value outliers. The mode of $349,000 suggests that this price point is the most frequently occurring listing price. The range between minimum and maximum prices is considerable, around $782,000, emphasizing the diversity in property valuation in Cathedral City.


Similarly, for square footage, the mean is approximately 1,218 sq ft, with a median near 1,164 sq ft, and a mode of around 245 sq ft, again indicating skewness possibly due to a distribution of smaller properties interspersed with larger homes. The standard deviation for listing prices is approximately $47,659, reflecting considerable variability, a typical feature in real estate markets where property values can vary significantly based on location, size, and condition.


Distributional Analysis and Normality


The shape of the data distributions for listing prices and square footage strongly influences the choice of statistical rules. The histogram of listing prices suggests a bell-shaped curve, characteristic of a normal distribution, though some skewness is apparent. The distribution's skewness is likely positive given the presence of high-value outliers. In such cases, the Empirical Rule, which assumes normality, could provide accurate estimates of data spread within one, two, or three standard deviations of the mean.


Regarding the number of bedrooms, the frequency distribution shows a dominant cluster of three-bedroom homes (49 out of 100), with fewer two- and one-bedroom properties, and a small number of five-bedroom homes. This distribution can be considered approximately symmetric with a mode at three bedrooms, supporting the use of normal assumptions in some analyses.


Application of Chebyshev’s Theorem and Empirical Rule


Chebyshev’s Theorem states that, for any dataset regardless of distribution shape, at least \((1 - 1/k^2)\) of data points lie within \(k\) standard deviations of the mean. Conversely, the Empirical Rule applies specifically to approximately normal distributions, stating that about 68%, 95%, and 99.7% of data fall within one, two, and three standard deviations, respectively.


Since the data for listing prices and square footage display characteristics akin to a normal distribution—bell-shaped histograms with symmetry—the Empirical Rule tends to provide more precise and meaningful estimates about the data spread and outliers in these cases. For example, roughly 68% of listings should fall within one standard deviation of the mean price (\$285,000 ± \$47,659), aligning with the observed data distribution.


However, the presence of outliers and skewed data suggests that Chebyshev’s Theorem offers a more conservative estimate, applicable in all data distributions—even those far from normal. Chebyshev’s bounds are less precise but more universally valid, especially when the data’s shape is uncertain or heavily skewed.


Conclusion


In evaluating the dataset of cathedral city real estate, the Empirical Rule provides a more accurate approximation for the spread of listing prices and square footage, owing to the near-normal and symmetric nature of the data distribution. Nonetheless, Chebyshev’s Theorem remains invaluable for establishing bounds when the distribution’s shape is unknown or significantly skewed. Proper understanding of these statistical tools allows real estate analysts to interpret variability, assess outliers, and make informed market predictions effectively.


References
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