(c) The following table shows the values, in thousands of dollars, of 40 randomly selected houses in the city.

34 38 95 110 137 140 155 155 169 207
209 217 274 314 314 323 343 347 349 363
369 373 388 389 397 416 448 450 483 487
488 516 571 595 600 738 762 769 863 1,084
(i) Calculate the sample midrange for the data.

The midrange is 559?

(ii) Explain why the sample midrange might be preferred to the sample mean as an estimator of the population mean.

Is the midrange preferred because the data is skewed?

7 answers

The Midrange is 559

Midrange is preferred because mean is skewed by outlier data.

Midrange = (Max + Min)/2 = (1084 + 34)/2 = 559

The midrange is preferred because the mean can be heavily affected my single outliers. It also shows us the intensity and direction of the skew in the distribution without drawing a graph.

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To calculate the sample midrange for the given data, you need to find the average (mean) of the minimum and maximum values in the data set.

(i) Calculate the sample midrange:

Step 1: Find the minimum and maximum values in the data set.
Minimum value: 34
Maximum value: 1,084

Step 2: Add the minimum and maximum values together.
34 + 1,084 = 1,118

Step 3: Divide the sum by 2.
1,118 ÷ 2 = 559

So, the sample midrange for the given data is 559.

(ii) The sample midrange might be preferred to the sample mean as an estimator of the population mean for several reasons, but one of the main reasons is related to skewed data.

When the data set has extreme values (outliers) that can greatly influence the mean, the midrange can be a better measure of central tendency. The midrange calculates the midpoint between the minimum and maximum values, which makes it less sensitive to extreme values. In this case, the sample mean might be distorted by the very high value of 1,084, while the midrange gives a more balanced estimate of the central value in the data.

Therefore, if the data is skewed or contains outliers, the sample midrange can be a more robust estimator of the population mean compared to the sample mean.