Answer :
Alright, let’s break down the solution step by step.
### Step-by-Step Solution:
#### i. Variance and Standard Deviation
First, we need to calculate the variance and standard deviation of the given data set:
[tex]\[ 176, 105, 133, 140, 305, 215, 207, 210, 173, 156, 78, 96 \][/tex]
Using the given data, the variance is:
[tex]\[ 3974.33 \][/tex]
The standard deviation (which is the square root of the variance) is:
[tex]\[ 63.04 \][/tex]
#### ii. The Quartiles [tex]\( Q_1, Q_2, Q_3 \)[/tex]
To find the quartiles, we order the data set and then find the appropriate percentiles.
The sorted data is:
[tex]\[ 78, 96, 105, 133, 140, 156, 173, 176, 207, 210, 215, 305 \][/tex]
- [tex]\( Q_1 \)[/tex] (the first quartile) is the 25th percentile, which is:
[tex]\[ 126.0 \][/tex]
- [tex]\( Q_2 \)[/tex] (the second quartile or median) is the 50th percentile, which is:
[tex]\[ 164.5 \][/tex]
- [tex]\( Q_3 \)[/tex] (the third quartile) is the 75th percentile, which is:
[tex]\[ 207.75 \][/tex]
#### iii. Bowley's Coefficient of Skewness
Bowley's coefficient of skewness (also known as the Bowley skewness or quartile skewness) is given by the formula:
[tex]\[ \text{Skewness} = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1} \][/tex]
Plugging in the values:
[tex]\[ Q_1 = 126.0 \][/tex]
[tex]\[ Q_2 = 164.5 \][/tex]
[tex]\[ Q_3 = 207.75 \][/tex]
The skewness is:
[tex]\[ \frac{207.75 + 126.0 - 2 \cdot 164.5}{207.75 - 126.0} \approx 0.058 \][/tex]
### Interpretation:
- Variance and Standard Deviation provide measures of spread or dispersion of the data. A standard deviation of [tex]\(63.04\)[/tex] indicates a relatively moderate spread around the mean price of the land plots.
- Quartiles provide a way of understanding the distribution of the data. For instance, we can infer that 25% of the surveyed land plots are priced below [tex]\(126.0\)[/tex] USD, 50% below [tex]\(164.5\)[/tex] USD, and 75% below [tex]\(207.75\)[/tex] USD.
- The skewness measure of approximately [tex]\(0.058\)[/tex] suggests that the distribution of prices is nearly symmetrical. A Bowley's coefficient closer to 0 indicates little to no skewness, with values approaching -1 or 1 indicating significant negative or positive skewness, respectively. Hence, in this case, the price distribution is nearly symmetrical with no significant skew.
### Step-by-Step Solution:
#### i. Variance and Standard Deviation
First, we need to calculate the variance and standard deviation of the given data set:
[tex]\[ 176, 105, 133, 140, 305, 215, 207, 210, 173, 156, 78, 96 \][/tex]
Using the given data, the variance is:
[tex]\[ 3974.33 \][/tex]
The standard deviation (which is the square root of the variance) is:
[tex]\[ 63.04 \][/tex]
#### ii. The Quartiles [tex]\( Q_1, Q_2, Q_3 \)[/tex]
To find the quartiles, we order the data set and then find the appropriate percentiles.
The sorted data is:
[tex]\[ 78, 96, 105, 133, 140, 156, 173, 176, 207, 210, 215, 305 \][/tex]
- [tex]\( Q_1 \)[/tex] (the first quartile) is the 25th percentile, which is:
[tex]\[ 126.0 \][/tex]
- [tex]\( Q_2 \)[/tex] (the second quartile or median) is the 50th percentile, which is:
[tex]\[ 164.5 \][/tex]
- [tex]\( Q_3 \)[/tex] (the third quartile) is the 75th percentile, which is:
[tex]\[ 207.75 \][/tex]
#### iii. Bowley's Coefficient of Skewness
Bowley's coefficient of skewness (also known as the Bowley skewness or quartile skewness) is given by the formula:
[tex]\[ \text{Skewness} = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1} \][/tex]
Plugging in the values:
[tex]\[ Q_1 = 126.0 \][/tex]
[tex]\[ Q_2 = 164.5 \][/tex]
[tex]\[ Q_3 = 207.75 \][/tex]
The skewness is:
[tex]\[ \frac{207.75 + 126.0 - 2 \cdot 164.5}{207.75 - 126.0} \approx 0.058 \][/tex]
### Interpretation:
- Variance and Standard Deviation provide measures of spread or dispersion of the data. A standard deviation of [tex]\(63.04\)[/tex] indicates a relatively moderate spread around the mean price of the land plots.
- Quartiles provide a way of understanding the distribution of the data. For instance, we can infer that 25% of the surveyed land plots are priced below [tex]\(126.0\)[/tex] USD, 50% below [tex]\(164.5\)[/tex] USD, and 75% below [tex]\(207.75\)[/tex] USD.
- The skewness measure of approximately [tex]\(0.058\)[/tex] suggests that the distribution of prices is nearly symmetrical. A Bowley's coefficient closer to 0 indicates little to no skewness, with values approaching -1 or 1 indicating significant negative or positive skewness, respectively. Hence, in this case, the price distribution is nearly symmetrical with no significant skew.
