The problem describes a dataset of the areas (in $m^2$) of nine apartments in a building: 118, 70, 36, 84, 94, 144, 60, 48, 78. We need to answer several questions about this dataset, including identifying the population and statistical unit, the nature of the variable, graphical representation, calculating arithmetic, harmonic, geometric, and quadratic means, finding the median, range, mean absolute deviation (about the mean and median), standard deviation, and coefficient of variation.

1. Problem Description

The problem describes a dataset of the areas (in $m^2$) of nine apartments in a building: 118, 70, 36, 84, 94, 144, 60, 48,

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8. We need to answer several questions about this dataset, including identifying the population and statistical unit, the nature of the variable, graphical representation, calculating arithmetic, harmonic, geometric, and quadratic means, finding the median, range, mean absolute deviation (about the mean and median), standard deviation, and coefficient of variation.

2. Solution Steps

1. Population and Statistical Unit:

The population studied is the set of the nine apartments in the residence.

The statistical unit is one apartment.

2. Observed Variable and its Nature:

The observed variable is the area of the apartment (in $m^2$).

The nature of the variable is quantitative continuous.

3. Graphical Representation:

Since the prompt requires the calculation to be performed but does not specify the tools to create graphical representations, this question will be skipped here.

4. Calculating Means:

First, let's sort the data: 36, 48, 60, 70, 78, 84, 94, 118, 144

n = 9

*Arithmetic Mean (X)*:

$X = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{36+48+60+70+78+84+94+118+144}{9} = \frac{732}{9} = 81.33$

*Harmonic Mean (H)*:

$H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} = \frac{9}{\frac{1}{36}+\frac{1}{48}+\frac{1}{60}+\frac{1}{70}+\frac{1}{78}+\frac{1}{84}+\frac{1}{94}+\frac{1}{118}+\frac{1}{144}} = \frac{9}{0.0278+0.0208+0.0167+0.0143+0.0128+0.0119+0.0106+0.0085+0.0069} = \frac{9}{0.1299} = 69.28$

*Geometric Mean (G)*:

$G = \sqrt[n]{\prod_{i=1}^{n} x_i} = \sqrt[9]{36 \cdot 48 \cdot 60 \cdot 70 \cdot 78 \cdot 84 \cdot 94 \cdot 118 \cdot 144} = \sqrt[9]{6.515e+14} = 75.14$

*Quadratic Mean (Q)*:

$Q = \sqrt{\frac{\sum_{i=1}^{n} x_i^2}{n}} = \sqrt{\frac{36^2+48^2+60^2+70^2+78^2+84^2+94^2+118^2+144^2}{9}} = \sqrt{\frac{70264}{9}} = \sqrt{7807.11} = 88.36$

The order is: $H < G < X < Q$, $69.28 < 75.14 < 81.33 < 88.36$

5. Median:

The sorted data is 36, 48, 60, 70, 78, 84, 94, 118,

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4. Since n = 9 (odd), the median is the middle value.

Median =

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6. Dispersion Characteristics:

*Range*:

Range = Maximum value - Minimum value = 144 - 36 = 108

*Mean Absolute Deviation (MAD) about the Mean*:

$MAD_{mean} = \frac{\sum_{i=1}^{n} |x_i - X|}{n} = \frac{|36-81.33|+|48-81.33|+|60-81.33|+|70-81.33|+|78-81.33|+|84-81.33|+|94-81.33|+|118-81.33|+|144-81.33|}{9} = \frac{45.33+33.33+21.33+11.33+3.33+2.67+12.67+36.67+62.67}{9} = \frac{229.33}{9} = 25.48$

*Mean Absolute Deviation (MAD) about the Median*:

$MAD_{median} = \frac{\sum_{i=1}^{n} |x_i - Median|}{n} = \frac{|36-78|+|48-78|+|60-78|+|70-78|+|78-78|+|84-78|+|94-78|+|118-78|+|144-78|}{9} = \frac{42+30+18+8+0+6+16+40+66}{9} = \frac{226}{9} = 25.11$

*Standard Deviation (s)*:

$s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - X)^2}{n-1}} = \sqrt{\frac{(36-81.33)^2+(48-81.33)^2+(60-81.33)^2+(70-81.33)^2+(78-81.33)^2+(84-81.33)^2+(94-81.33)^2+(118-81.33)^2+(144-81.33)^2}{9-1}} = \sqrt{\frac{2054.78+1110.78+454.98+128.38+11.09+7.13+160.53+1344.53+3935.69}{8}} = \sqrt{\frac{9207.9}{8}} = \sqrt{1150.99} = 33.93$

*Coefficient of Variation (CV)*:

$CV = \frac{s}{X} = \frac{33.93}{81.33} = 0.4172 = 41.72\%$

3. Final Answer

1. Population: The nine apartments in the residence. Statistical unit: One apartment.

2. Observed variable: Area of the apartment (in $m^2$). Nature: Quantitative continuous.

3. Skipped (Graphical representation).

4. Arithmetic Mean (X) = 81.33, Harmonic Mean (H) = 69.28, Geometric Mean (G) = 75.14, Quadratic Mean (Q) = 88.

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6. $H < G < X < Q$.

5. Median =

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6. Range = 108, MAD (about mean) = 25.48, MAD (about median) = 25.11, Standard Deviation = 33.93, Coefficient of Variation = 41.72%.