Question

define and distinguish among mean, median, and mode

choose the correct description of the mean below

a. the mean is the sum of all the values divided by the number of values. it can be strongly affected by outliers
b. the mean is the most common value in a data set. it is not affected by outliers
c. the mean is the sum of all the values divided by the number of values. it is not affected by outliers
d. the mean is the middle value in a data set. it is not affected by outliers
e. the mean is the most common value in a data set. it can be strongly affected by outliers
f. the mean is the middle value in a data set. it can be strongly affected by outliers

choose the correct description of the median below

a. the median is the most common value in a data set. it is not affected by outliers
b. the median is the most common value in a data set. it can be strongly affected by outliers
c. the median is the sum of all the values divided by the number of values. it is not affected by outliers
d. the median is the middle value in a data set. it can be strongly affected by outliers
e. the median is the middle value in a data set. it is not affected by outliers
f. the median is the sum of all the values divided by the number of values. it can be strongly affected by outliers

Explanation:

• Mean: The mean is calculated as the sum of all values divided by the number of values. Since it involves every data point in its calculation, extreme values (outliers) can significantly pull the mean up or down. For example, in a dataset {1, 2, 3, 100}, the mean is \((1 + 2+3 + 100)\div4=26.5\), and the outlier 100 has a strong effect.
• Median: The median is the middle - value when the data is ordered. For an odd - numbered dataset \(\{a_1,a_2,\cdots,a_{2n + 1}\}\) sorted in ascending order, the median is \(a_{n+1}\); for an even - numbered dataset \(\{a_1,a_2,\cdots,a_{2n}\}\), it is \(\frac{a_{n}+a_{n + 1}}{2}\). Outliers have little impact because only the position of the middle value(s) matters. For instance, in the dataset \(\{1,2,3,100\}\), the median is \(\frac{2 + 3}{2}=2.5\), and the outlier 100 does not affect it much.

Answer:

• For the mean: A. The mean is the sum of all the values divided by the number of values. It can be strongly affected by outliers.
• For the median: E. The median is the middle value in a data set. It is not affected by outliers.