1. **Problem statement:** We have 18 headache pain ratings: 15, 16, 16, 16, 19, 27, 28, 30, 31, 32, 37, 44, 45, 53, 63, 71, 85, 90. We analyze measures of central tendency (mean, median, mode) for these data and answer questions (a) to (d). --- 2. **Definitions and rules:** - Mean is the average: $$\text{Mean} = \frac{\sum x_i}{n}$$ - Median is the middle value when data are sorted. - Mode is the most frequent value(s). If multiple values share the highest frequency, mode is multimodal (more than one mode). --- 3. **Calculate mean:** $$\sum x_i = 15 + 16 + 16 + 16 + 19 + 27 + 28 + 30 + 31 + 32 + 37 + 44 + 45 + 53 + 63 + 71 + 85 + 90 = 718$$ Number of data points $n=18$ $$\text{Mean} = \frac{718}{18} \approx 39.89$$ --- 4. **Calculate median:** Sorted data (already sorted): 15, 16, 16, 16, 19, 27, 28, 30, 31, 32, 37, 44, 45, 53, 63, 71, 85, 90 Since $n=18$ (even), median is average of 9th and 10th values: 9th value = 31, 10th value = 32 $$\text{Median} = \frac{31 + 32}{2} = 31.5$$ --- 5. **Calculate mode:** Value 16 appears 3 times, more than any other value. Mode = 16 (single mode) --- 6. **Answer (a):** - Mean: single value - Median: single value - Mode: single value (16) **Which measures take more than one value?** None. --- 7. **Answer (b):** Replace 15 by 2. - Mean changes because sum changes: new sum = 718 - 15 + 2 = 705 New mean = $$\frac{705}{18} \approx 39.17$$ (changed) - Median: data sorted with 2 instead of 15 shifts smallest value but middle values (9th and 10th) remain 31 and 32, so median unchanged. - Mode: 16 still appears 3 times, no change. Affected measures: Mean only. --- 8. **Answer (c):** Remove largest measurement 90. - New sum = 718 - 90 = 628 - New $n=17$ - New mean = $$\frac{628}{17} \approx 36.94$$ (changed) - Median: now 17 values, median is 9th value in sorted data without 90. Sorted without 90: 15,16,16,16,19,27,28,30,31,32,37,44,45,53,63,71,85 9th value = 31 (same as before median average) Median changes from 31.5 to 31 (changed) - Mode: still 16 appears 3 times, no change. Affected measures: Mean and Median. --- 9. **Answer (d):** Mean $\approx 39.89$, Median = 31.5 Mean > Median, which is typical when data are right-skewed (long tail to higher values). --- **Final answers:** (a) None of these measures take more than one value. (b) Mean is affected. (c) Mean and Median are affected. (d) Mean is greater.