A garage wishes to analyse the costs of servicing its range of cars. The costs of servicing cars over the past month are shown.
250 200 375 394 415 405 340 365 417 400
336 245 280 442 422 460 300 405 417 435
193 275 355 225 318 375 390 149 390 468
390 345 95 355 525 575 400 413 312 570
350 411 148 395 380 175 125 198 370 433
Required
(a) How might this data have been collected and recorded?
(b) By choosing suitable class intervals, calculate the mean servicing cost.
(c) What other measures of dispersion could have been used to analyse the above data? Discuss the relative merits of each method.
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(a) The data could have been collected and recorded through various methods. One possible method is by manually recording the servicing costs for each car at the end of each day or after each servicing job. The person responsible for recording the data would write down the cost of each service in a logbook or spreadsheet. Another method could involve using an electronic system to automatically track and record the servicing costs as the work is being done. This could be done through a computer-based system, where the servicing costs are logged into a database or software program.
(b) To calculate the mean servicing cost, we need to add up all the costs and divide by the number of data points. In this case, we have 50 data points:
250 + 200 + 375 + 394 + 415 + 405 + 340 + 365 + 417 + 400 +
336 + 245 + 280 + 442 + 422 + 460 + 300 + 405 + 417 + 435 +
193 + 275 + 355 + 225 + 318 + 375 + 390 + 149 + 390 + 468 +
390 + 345 + 95 + 355 + 525 + 575 + 400 + 413 + 312 + 570 +
350 + 411 + 148 + 395 + 380 + 175 + 125 + 198 + 370 + 433 = 20185
Mean servicing cost = 20185 / 50 = 403.7
Therefore, the mean servicing cost is 403.7.
(c) Other measures of dispersion that could be used to analyze the above data include the range, variance, and standard deviation.
- Range: The range is the difference between the largest and smallest values in the data set. It provides a simple measure of the spread of the data.
- Variance: The variance calculates the average of the squared differences between each data point and the mean. It measures how much the data varies or deviates from the mean.
- Standard Deviation: The standard deviation is the square root of the variance. It measures the dispersion or spread of the data, similar to the variance but in the original units of the data.
The relative merits of each method depend on the context and purpose of the analysis. The range is simple to calculate and easy to interpret, but it can be heavily influenced by outliers. The variance and standard deviation provide more robust measures of dispersion, giving weight to all data points, but they require more calculations and might not be as intuitive for interpretation. Ultimately, the choice of measure depends on the specific goals and requirements of the analysis.