1. **Problem statement:** We need to find five numbers for each of the following conditions: i) mode < median < mean ii) mean < mode < median iii) mode < mean < median Then justify why one of these is not possible. 2. **Recall definitions:** - Mode: the most frequent number in the set. - Median: the middle number when the set is ordered. - Mean: the average of all numbers. 3. **Construct example for (i) mode < median < mean:** Let's try numbers: 1, 2, 2, 5, 10 - Mode = 2 (appears twice) - Median = 2 (middle number) - Mean = (1+2+2+5+10)/5 = 20/5 = 4 Here mode = 2, median = 2, mean = 4, so mode = median < mean, not mode < median. Try: 1, 3, 3, 4, 10 - Mode = 3 - Median = 3 - Mean = (1+3+3+4+10)/5 = 21/5 = 4.2 Again mode = median < mean. Try: 1, 2, 3, 3, 10 - Mode = 3 - Median = 3 - Mean = (1+2+3+3+10)/5 = 19/5 = 3.8 Mode = median < mean. To get mode < median, mode must be less than the middle number. Try: 1, 1, 4, 5, 6 - Mode = 1 - Median = 4 - Mean = (1+1+4+5+6)/5 = 17/5 = 3.4 Here mode = 1 < median = 4, but mean = 3.4 < median, so mode < mean < median, not mode < median < mean. Try: 1, 1, 5, 6, 10 - Mode = 1 - Median = 5 - Mean = (1+1+5+6+10)/5 = 23/5 = 4.6 So mode = 1 < mean = 4.6 < median = 5, which is mode < mean < median, not mode < median < mean. Try: 1, 2, 2, 6, 10 - Mode = 2 - Median = 2 - Mean = (1+2+2+6+10)/5 = 21/5 = 4.2 Mode = median < mean. Try: 1, 2, 3, 4, 10 - Mode: no repeats, no mode So no mode. Try: 1, 2, 2, 3, 10 - Mode = 2 - Median = 2 - Mean = (1+2+2+3+10)/5 = 18/5 = 3.6 Mode = median < mean. Try: 1, 2, 3, 3, 10 - Mode = 3 - Median = 3 - Mean = 19/5 = 3.8 Mode = median < mean. Try: 1, 1, 3, 4, 10 - Mode = 1 - Median = 3 - Mean = 19/5 = 3.8 Mode = 1 < median = 3 < mean = 3.8 This satisfies mode < median < mean. 4. **Construct example for (ii) mean < mode < median:** Try: 1, 5, 5, 6, 10 - Mode = 5 - Median = 5 - Mean = (1+5+5+6+10)/5 = 27/5 = 5.4 Mean = 5.4 > mode = 5, so mean < mode is false. Try: 1, 1, 5, 6, 10 - Mode = 1 - Median = 5 - Mean = 23/5 = 4.6 Mean = 4.6 > mode = 1, so mean < mode false. Try: 1, 2, 2, 6, 10 - Mode = 2 - Median = 2 - Mean = 21/5 = 4.2 Mean = 4.2 > mode = 2, mean < mode false. Try: 1, 1, 2, 3, 10 - Mode = 1 - Median = 2 - Mean = 17/5 = 3.4 Mean = 3.4 > mode = 1, mean < mode false. Try: 1, 1, 1, 3, 10 - Mode = 1 - Median = 1 - Mean = 16/5 = 3.2 Mean = 3.2 > mode = 1, mean < mode false. Try: 1, 2, 3, 4, 5 - No mode. So no example found where mean < mode < median. 5. **Construct example for (iii) mode < mean < median:** Try: 1, 1, 5, 6, 10 - Mode = 1 - Mean = 23/5 = 4.6 - Median = 5 Mode = 1 < mean = 4.6 < median = 5, satisfies condition. 6. **Justification why (ii) mean < mode < median is not possible:** The mode is the most frequent value, so it must be a value in the data set. The mean is the average, which is influenced by all values. For mean < mode to hold, the average must be less than the most frequent value. But if the mode is less than the median, and the mean is less than the mode, the mean is less than the median. However, the median is the middle value, so the mean being less than the mode and mode less than median contradicts the ordering of values in the data set. Hence, mean < mode < median is not possible for a set of five numbers. **Final answers:** - (i) Example: 1, 1, 3, 4, 10 - (ii) Not possible - (iii) Example: 1, 1, 5, 6, 10