A Providence Hospital experiment involves two different waiting line configurations for patients arriving Friday missions the waiting times in seconds are recorded with a single line configuration that feeds four stations and another configuration with individual lines at the four stations find the coefficient of variation for each of the two samples and then compare the variation the single line is 389 398 402 409 425 437 445 460 460 460 the individual lines is 254 324 348 373 402 462 462 511 559 599 find the coefficient of variation of the single line data set and the individual line data set

To find the coefficient of variation (CV) for each dataset, we need to calculate the mean and standard deviation for both the single line configuration and the individual line configuration. The coefficient of variation is then computed using the formula:

\[
\text{CV} = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100\%
\]

### 1. Single Line Configuration Data
**Data:**
389, 398, 402, 409, 425, 437, 445, 460, 460, 460

**Calculations:**

- **Mean**:
\[
\text{Mean} = \frac{\sum \text{data}}{n} = \frac{389 + 398 + 402 + 409 + 425 + 437 + 445 + 460 + 460 + 460}{10} = \frac{4425}{10} = 442.5
\]

- **Standard Deviation**:
First, calculate the variance:
\[
\text{Variance} = \frac{\sum (x - \text{Mean})^2}{n}
\]
Calculating each squared deviation:
- (389 - 442.5)² = 2840.25
- (398 - 442.5)² = 1980.25
- (402 - 442.5)² = 1640.25
- (409 - 442.5)² = 1106.25
- (425 - 442.5)² = 306.25
- (437 - 442.5)² = 30.25
- (445 - 442.5)² = 6.25
- (460 - 442.5)² = 306.25
- (460 - 442.5)² = 306.25
- (460 - 442.5)² = 306.25

Summing these squared deviations:
\[
\text{Sum} = 2840.25 + 1980.25 + 1640.25 + 1106.25 + 306.25 + 30.25 + 6.25 + 306.25 + 306.25 + 306.25 = 6203.5
\]

Variance:
\[
\text{Variance} = \frac{6203.5}{10} = 620.35
\]

Standard deviation:
\[
\text{Standard Deviation} = \sqrt{620.35} \approx 24.9
\]

- **Coefficient of Variation**:
\[
\text{CV} = \left( \frac{24.9}{442.5} \right) \times 100 \approx 5.63\%
\]

### 2. Individual Line Configuration Data
**Data:**
254, 324, 348, 373, 402, 462, 462, 511, 559, 599

**Calculations:**

- **Mean**:
\[
\text{Mean} = \frac{\sum \text{data}}{n} = \frac{254 + 324 + 348 + 373 + 402 + 462 + 462 + 511 + 559 + 599}{10} = \frac{3900}{10} = 390
\]

- **Standard Deviation**:
Calculating each squared deviation:
- (254 - 390)² = 18496
- (324 - 390)² = 4356
- (348 - 390)² = 1764
- (373 - 390)² = 289
- (402 - 390)² = 144
- (462 - 390)² = 5184
- (462 - 390)² = 5184
- (511 - 390)² = 14641
- (559 - 390)² = 28561
- (599 - 390)² = 43681

Summing these squared deviations:
\[
\text{Sum} = 18496 + 4356 + 1764 + 289 + 144 + 5184 + 5184 + 14641 + 28561 + 43681 = 110455
\]

Variance:
\[
\text{Variance} = \frac{110455}{10} = 11045.5
\]

Standard deviation:
\[
\text{Standard Deviation} = \sqrt{11045.5} \approx 105.08
\]

- **Coefficient of Variation**:
\[
\text{CV} = \left( \frac{105.08}{390} \right) \times 100 \approx 26.94\%
\]

### Summary of Results
- **Single Line Configuration CV**: 5.63%
- **Individual Line Configuration CV**: 26.94%

### Comparison
The individual line configuration has a higher coefficient of variation (26.94%) compared to the single line configuration (5.63%), indicating that the individual lines exhibit greater relative variability in waiting times.

If you have any questions or need further details, feel free to ask! Here are some related questions to consider:

1. How might the choice of waiting line configuration affect patient satisfaction?
2. What other factors could influence waiting times in a hospital setting?
3. How could you improve the efficiency of the waiting line system?
4. What statistical methods can be used to analyze waiting times further?
5. Are there any other metrics you would consider for assessing waiting line performance? 

**Tip**: Always ensure that your data is clean and represents the population adequately for reliable statistical analysis.

Explore how to calculate the coefficient of variation for waiting times in different line configurations at a hospital. The single line configuration yielded a CV of 5.63%, while individual lines showed a CV of 26.94%, indicating greater variability in the latter setup.

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