A Random Sample Of Monthly Gasoline Bills For A Company
A random sample of monthly gasoline bills for a company's 15 salespersons is provided, and the following statistical analyses are required: calculate the mean, median, third quartile (Q3), mode, range, standard deviation, coefficient of variation (as a ratio), skewness (positively, negatively, or symmetrical), Pearson's coefficient of skewness, and the second quartile (Q2).
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Paper For Above instruction
Introduction
Statistical analysis provides valuable insights into the characteristics of a dataset. By examining measures such as the mean, median, quartiles, mode, range, standard deviation, coefficient of variation, skewness, and others, we can better understand the distribution, variability, and shape of the data. In this study, we analyze a sample of monthly gasoline bills for 15 salespersons in a company to derive these statistical measures and interpret the distributional characteristics.
Data and Methodology
The dataset comprises 15 monthly gasoline bills assigned to salespersons, though specific data points are not provided in this excerpt. For illustration, suppose the bills are as follows: (this is an example; in your actual assignment, insert the real data)
Example dataset:
$120, $135, $125, $150, $160, $130, $145, $155, $140, $165, $170, $125, $135, $140, $155
The calculations proceed as follows:
Calculations
1. Mean
The mean is calculated by summing all observations and dividing by the total number of observations (n=15):
\[
\text{Mean} = \frac{\sum x_i}{n}
\]
Using the example dataset:
\[
\text{Sum} = 120 + 135 + 125 + 150 + 160 + 130 + 145 + 155 + 140 + 165 + 170 + 125 + 135 + 140 + 155 = 2210
\]
\[
\text{Mean} = \frac{2210}{15} \approx 147.33
\]
2. Median
The median is the middle value when data is ordered:
Ordered data:
$120, $125, $125, $130, $135, $135, $140, $140, $145, $150, $155, $155, $160, $165, $170
Median position = (n + 1)/2 = 8th position
Median = $140
3. Third Quartile (Q3)
Q3 is the median of the upper half:
Upper half data: $145, $150, $155, $155, $160, $165, $170
Median of upper half (positions 12-15):
Values: $155, $160, $165, $170
Q3 = (160 + 165)/2 = 322.5/2 = $162.5
4. Mode
Most frequently occurring value:
Values: 125 appears twice, 135 appears twice, 140 appears twice, 155 appears twice; hence, multiple modes.
Modes: 125, 135, 140, 155 (multimodal distribution)
5. Range
Range = Max - Min = 170 - 120 = $50
6. Standard Deviation
Standard deviation (SD) measures dispersion:
Calculations involve:
\[
\text{Variance} = \frac{\sum(x_i - \bar{x})^2}{n-1}
\]
The deviations squared:
\[
(120 - 147.33)^2 \approx 773.33
\]
\[
(125 - 147.33)^2 \approx 495.11
\]
\[
(125 - 147.33)^2 \approx 495.11
\]
\[
(130 - 147.33)^2 \approx 296.89
\]
\[
(135 - 147.33)^2 \approx 150.89
\]
\[
(135 - 147.33)^2 \approx 150.89
\]
\[
(140 - 147.33)^2 \approx 53.11
\]
\[
(140 - 147.33)^2 \approx 53.11
\]
\[
(145 - 147.33)^2 \approx 5.44
\]
\[
(150 - 147.33)^2 \approx 7.11
\]
\[
(155 - 147.33)^2 \approx 58.43
\]
\[
(155 - 147.33)^2 \approx 58.43
\]
\[
(160 - 147.33)^2 \approx 162.78
\]
\[
(165 - 147.33)^2 \approx 311.22
\]
\[
(170 - 147.33)^2 \approx 519.56
\]
Sum of squared deviations:
\[
\approx 2735.44
\]
Variance:
\[
\frac{2735.44}{14} \approx 195.39
\]
Standard deviation:
\[
\sqrt{195.39} \approx 13.97
\]
7. Coefficient of Variation (CV)
CV = SD / Mean = 13.97 / 147.33 ≈ 0.0948
Expressed as a ratio: approximately 0.095
8. Skewness
Since the data is roughly symmetric around the median with comparable distances on both sides, the distribution appears symmetrical. Thus, the skewness indicator is:
- One word: symmetrical
9. Pearson's Coefficient of Skewness
Pearson's skewness coefficient is calculated as:
\[
\text{Sk} = \frac{3 (\bar{x} - \text{Median})}{\text{SD}}
\]
Using the example:
\[
\text{Sk} = \frac{3 (147.33 - 140)}{13.97} \approx \frac{3 \times 7.33}{13.97} \approx \frac{21.99}{13.97} \approx 1.573
\]
Since the skewness coefficient is positive but less than 2, it indicates mild positive skewness. However, in this hypothetical, the distribution is roughly symmetric, consistent with the visual assessment.
10. Second Quartile (Q2)
Q2 is the median; thus, Q2 = $140.
Conclusion
The statistical analysis of the gasoline bills suggests that the data set has an average (mean) of approximately $147.33, with a median of $140. The third quartile (Q3) is about $162.5, and the mode includes multiple values: 125, 135, 140, and 155. The range stands at $50, indicating the spread between the lowest and highest bills. The standard deviation is approximately 13.97, reflecting moderate variability, with a coefficient of variation near 0.095, indicating relative consistency in bills. The distribution appears symmetrical, as the skewness measures suggest, with a positive but mild skewness based on Pearson's coefficient.
This analysis provides insights into the expenses patterns of the salespersons and can aid in budgeting and cost control strategies.
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