Question

send data to calculator (a) which measures of central tendency do not exist for this data set? choose all that apply. mean median mode all of these measures exist (b) suppose that the measurement 35 (the largest measurement in the data set) were replaced by 70. which measures of central tendency would be affected by the change? choose all that apply. mean median mode none of these measures (c) suppose that, starting with the original data set, the largest measurement were removed. which measures of central tendency would be changed from those of the original data set? choose all that apply. mean median mode none of these measures (d) which of the following best describes the distribution of the original data? choose only one. negatively skewed positively skewed roughly symmetrical

Explanation:

Step1: Recall definitions

Mean, median and mode are well - defined for any non - empty data set. Mean is the sum of all values divided by the number of values, median is the middle value when data is ordered, and mode is the most frequently occurring value.

Step2: Analyze part (a)

Since the data set is non - empty, all of mean, median and mode exist.

Step3: Analyze part (b)

The mean is calculated as $\sum_{i = 1}^{n}x_{i}/n$. Changing a single value (from 35 to 70) will change the sum $\sum_{i = 1}^{n}x_{i}$, so the mean will be affected. The median depends on the order of the data. If the number of data points is odd, it is the middle value; if even, the average of the two middle values. Changing the largest value may not change the middle value(s), so the median may not be affected. The mode is the most frequent value, and changing the largest value does not affect the frequency of the most common value, so the mode is not affected.

Step4: Analyze part (c)

Removing the largest value will change the sum for the mean calculation, so the mean will change. If the number of data points changes from odd to even or vice - versa, the median may change. The mode is based on frequency and removing a single non - modal value does not change the mode.

Step5: Analyze part (d)

The tail of the distribution extends to the right (towards larger positive values like 35 and 40), so the distribution is positively skewed.

Answer:

(a) All of these measures exist
(b) Mean
(c) Mean, Median
(d) Positively skewed