Recall that the following definitions :
- mean: given a set of n data points, the mean is simply the sum of the data points divided by the total number of data points
-median: To calculate the median, you organize the data from least to greatest, the data point that has the exactly amount of data on the right and on the left, is the median. If the number of data points is even, you should add the two points that are "in the middle" and divide them by 2. Example
for the data set 5 4 2 3 1, if we organize it we get 1 2 3 4 5, so the median is 3. If we have the data set 1 2 3 4, there is no middle point, so we take 2 and 3, that are in the "middle" , we add them (2+3 =5) and divide them by 2 (2.5). So the median is 2.5
- the mode is the data point that has more observations, for example in the dataset 1 2 3 4 1 1 the mode is 1 since it appears 3 times.
a) The mean exists for every dataset. The median exists for every dataset that has more than two datapoints. In this case the mode is 63, so the three of them exist for this data set.
b) The smallest value is 31. Since we replace it by 27, the mean gets affected. Since we have the same amount of data and 27 is the smalles point, the median doesn't change. Since the occurrence of 27 is only one, the 27 remains unchanged. So only the mean changes.
c) If we remove 31, we will have less data, so the median changes. As before, the mean also changes. However, the mode remains the same.
d) If we calculate the mean of the dataset, we get that the mean is 61.5. The median is 63, so the median is greater.