Subjects statistics

Summary Statistics

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Summary Statistics


1. **Problem statement:** Calculate and interpret the summary statistics for two outlets based on the provided data in part (b). 2. **Arithmetic mean ($\bar{x}$):** Sum all data points for each outlet and divide by the number of data points: $$\bar{x} = \frac{\sum x_i}{n}$$ The mean represents the average value. 3. **Coefficient of Variation (CV):** This measures relative variability as a percentage: $$\text{CV} = \frac{\text{standard deviation }(s)}{\text{mean }(\bar{x})} \times 100\%$$ It shows variability relative to the mean. 4. **Mode:** The value that appears most frequently in the dataset. 5. **Median:** The middle value when all data points are sorted in ascending order. If even number of observations, average the two middle values. 6. **Geometric Mean (GM):** The nth root of the product of n values: $$\text{GM} = \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n} = \exp\left(\frac{1}{n} \sum_{i=1}^n \ln x_i\right)$$ Useful for data measured in ratios or rates. 7. **1st and 3rd Quartiles (Q1 and Q3):** Q1 is the 25th percentile and Q3 is the 75th percentile. Sort data and find values at these percentiles. 8. **Pearson’s 1st coefficient of Skewness (SK1):** Measures asymmetry: $$\text{SK1} = \frac{\bar{x} - \text{mode}}{s}$$ Positive SK1 means right-skewed, negative means left-skewed. **Interpretation:** - Mean gives central tendency. - CV shows relative spread. - Mode indicates the most common value. - Median shows middle value unaffected by outliers. - Geometric mean better for proportional data. - Quartiles describe spread and identify outliers. - Skewness tells about distribution symmetry. Final answer depends on data from part (b). Replace $x_i$, $n$, $s$, mode, and data with actual values to complete calculations.