Answer to Question #297606 in Statistics and Probability for kaye

Question

Raphael’s grades in his statistics classes are as follows: Quizzes: 62, 88, 82 Laboratories: 89, 96 Examinations: 87, 99 a. In this class, quizzes count once, laboratories count twice as much as a quiz, and examinations count three times as much as a quiz. Determine the mode, mean and median. b. If Raphael’s quiz grade of 62 was removed from the data, briefly describe the anticipated effect on the statistics you calculated in part (a). 

Answer 1

Solution:

With the weightage associated with the different grade types, we can restate the values as

(a):

Quizzes: 62, 88, 82

Laboratories: 89, 89, 96, 96

Examinations: 87, 87, 87, 99, 99, 99

Thus, overall we have a set of 13 values. The mean, median and mode can now, therefore, we determined for the combined data set

{62, 88, 82, 89, 89, 96, 96, 87, 87, 87, 99, 99, 99}

The mode is the data or set of values that repeat most often, i.e. have highest frequency. By virtue of their weight, the grades obtained in Examnations have the highest frequency. Thus, we have multi modal data, with modes as 87 and 99

The mean is the average value, and is found as

Mean = Sum of values ÷ No of values = (62+88+82+89+89+96+96+87+87+87+99+99+99) ÷ 13 = 89.23 (approx)

The median is the central or middle value. For n = 13 values, the 7th value, when data is arranged in ascending order, is the median. The arranged values are

62, 82, 87, 87, 87, 88, 89, 89, 96, 96, 99, 99, 99

The 7th value is 89, and hence it is the median.

(b):

If 62 is removed from the data set, then mode shall not change since the frequency of data that repeats the most often has not changed. The median might change, but even if it does so the change cannot be too drastic since we are looking at the middle value. If we evaluate in this case, we shall only have 12 values. So median shall be the average of 6th and 7th values, viz. still 89. So it does not change at all. The mean, on the other hand, shall change the most. 62 being the least score, and seemingly an outlier (i.e. too far away from other values) shall increase the mean significantly. This happens since mean puts an equal weightage to all the values in a data set.