Question

(a) which measures of central tendency do not exist for this data set? choose all that apply. (b) suppose that the measurement 23 (the largest measurement in the data set) were replaced by 45. which measures of central tendency would be affected by the change? choose all that apply. (c) suppose that, starting with the original data set, the smallest measurement were removed. which measures of central tendency would be changed from those of the original data set? choose all that apply. (d) which of the following best describes the distribution of the original data? choose only one.

Explanation:

(a)

The mean is calculated by summing all values and dividing by the number of values. The median is the middle - value when data is ordered. The mode is the most frequently occurring value. For a data set (even if it's a histogram - based data set where we can assume values from intervals), all these measures can be calculated.

(b)

The formula for the mean is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). If we change a value (in this case, increase the largest value from \(x = 23\) to \(x = 45\)), the sum \(\sum_{i=1}^{n}x_{i}\) changes, so the mean \(\bar{x}\) changes. The median depends on the position of the middle - value. Since we are only changing the largest value, the order of the middle - value (when data is ordered) does not change. The mode is the most frequently occurring value, and changing the largest value does not affect the frequency of the most common value.

(c)

The original data set has some values. Removing the smallest value:

• The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), both \(\sum_{i = 1}^{n}x_{i}\) and \(n\) change. But if the data set had \(n\) values (say \(n\) is odd, \(n=2k + 1\) and after removing one value \(n'=2k\)), the median calculation (average of the \(k\) - th and \((k + 1)\) - th values in the ordered new data set) may not change if the relative position of the middle - values (excluding the smallest) is the same as in the original ordered data set. The mode is based on frequency, and removing the smallest value (assuming it was not the mode) does not change the mode.

(d)

In a positively skewed distribution, the tail is on the right - hand side. Looking at the histogram (assuming the intervals: for example, if we assume the first interval (0 - 5) has some non - zero frequency, then intervals (5 - 10), (10 - 15), (15 - 20) have higher frequencies, and the last interval (20 - 25) has a non - zero but relatively lower frequency compared to the middle intervals but the tail is on the right.

Answer:

(a) All of these measures exist
(b) Mean
(c) None of these measures
(d) Positively skewed