Business and finance. The following scores were recorded on a 200-point final examination:193, 185, 163, 186, 192, 135, 158, 174, 188, 172, 168, 183, 195, 165, 183.(a) Find the mean final examination score.(b) Find the median final examination score.(c) Is the mean or median a more useful representative of the final examination scores? Write a brief paragraph justifying your response.
To find the mean you need to add together all of the numbers, and then divide this total by the number in the set (in this case 15).
To find the median you must put the numbers in order and then see which one is in the middle position (in this case, the 8th value).
To say which one is more useful, which one is closer to the bulk of the scores? Which one tells you most about the scores?
(a) To find the mean final examination score, we need to add together all of the numbers and then divide this total by the number in the set. In this case, we have the following scores: 193, 185, 163, 186, 192, 135, 158, 174, 188, 172, 168, 183, 195, 165, 183. Adding them together gives us a total of 2787. Dividing this total by 15 (the number of scores) gives us the mean final examination score:
Mean = (193+185+163+186+192+135+158+174+188+172+168+183+195+165+183) / 15
Mean = 2787 / 15 = 185.8
Therefore, the mean final examination score is 185.8.
(b) To find the median final examination score, we need to put the numbers in order and identify which value is in the middle position. By arranging the scores in ascending order, we get:
135, 158, 163, 165, 168, 172, 174, 183, 183, 185, 186, 188, 192, 193, 195
As there are 15 scores, the middle position would be the 8th value.
Median = 174
Therefore, the median final examination score is 174.
(c) To determine which measure is more useful, it is necessary to consider which one is closer to the bulk of the scores and provides more information about the distribution. In this case, the mean final examination score is 185.8, while the median is 174.
The mean is more affected by extreme values, such as the relatively low score of 135 and the relatively high score of 195. These extreme values can skew the mean and make it less representative of the majority of scores.
On the other hand, the median is not influenced by extreme values and instead represents the literal middle value of the distribution. It is particularly useful when there are outliers that could distort the picture of the majority of scores.
In this scenario, since the mean and median are relatively close (185.8 and 174, respectively), it suggests that there is not a significant presence of outliers, and both measures could provide reasonable representations of the final examination scores. However, considering the potential impact of extreme values, the median may be slightly more useful in this case as it is less affected by outliers and is a better representation of the typical score.
(a) To find the mean final examination score, you need to add together all of the numbers and then divide this total by the number of scores (in this case, 15). Let's calculate it step by step:
193 + 185 + 163 + 186 + 192 + 135 + 158 + 174 + 188 + 172 + 168 + 183 + 195 + 165 + 183 = 2789
Now, divide this total by 15:
2789 / 15 = 185.93
Therefore, the mean final examination score is approximately 185.93.
(b) To find the median final examination score, you need to arrange the numbers in ascending order and then find the middle value. Let's list the scores in ascending order:
135, 158, 163, 165, 168, 172, 183, 183, 185, 186, 188, 192, 193, 195.
As there are an odd number of scores (15 in this case), the median will be the value at the 8th position.
Therefore, the median final examination score is 183.
(c) To determine which one, the mean or median, is a more useful representative of the final examination scores, we need to consider which statistics better reflects the overall distribution of scores.
The mean is the average of all the scores and is influenced by extreme values. If there are outliers in the data, such as very low or very high scores, the mean may be significantly affected. In this case, the mean final examination score is 185.93.
The median, on the other hand, represents the middle value in the dataset and is not affected by extreme values. It provides a more robust measure of central tendency. In this case, the median final examination score is 183.
Considering the given data, the median is closer to the bulk of the scores and is less influenced by outliers. Therefore, the median is a more useful representative of the final examination scores as it gives a better understanding of the typical performance of the students.