Sove the following application.
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) If the mean or median a more useful representative of the final examination scores? write a brief paragraph justifying your respones.
Do you know how to find the mean and median?
The arithmetic mean, also called the average, of a series of quantities is obtained by finding the sum of the quantities and dividing it by the number of quantities. In the series 1, 3, 5, 18, 19, 20, 25, the mean or average is 13—in other words, 91 divided by 7.
The median of a series is that point which so divides it that half the quantities are on one side, half on the other. In the above series, the median is 18.
The median often better expresses the common-run, since it is not, as is the mean, affected by an excessively high or low figure. In the series 1, 3, 4, 7, 55, the median of 4 is a truer expression of the common-run than is the mean of 14.
To solve this problem, you need to follow these steps:
(a) Find the mean final examination score:
1. Add up all the scores: 193 + 185 + 163 + 186 + 192 + 135 + 158 + 174 + 188 + 172 + 168 + 183 + 195 + 165 + 183 = 2,813.
2. Divide the sum by the number of scores (15 in this case): 2,813 / 15 = 187.53.
3. The mean final examination score is 187.53.
(b) Find the median final examination score:
1. Arrange the scores in ascending order: 135, 158, 163, 165, 168, 172, 174, 183, 183, 185, 186, 188, 192, 193, 195.
2. Since there are an odd number of scores (15), the median is the middle value. In this case, the middle value is the 8th number from the sorted list: 183.
3. The median final examination score is 183.
(c) Justifying the usefulness of mean or median as a representative measure:
The mean and median are both measures of central tendency, but they may reflect different aspects of the data. In this case, the mean final examination score is 187.53, and the median score is 183.
If there are no extreme outliers or skewed distributions, the mean tends to provide a more balanced representation of the data. However, if the data contains extreme values or is skewed, the median may be a more useful measure.
In this scenario, since the scores are relatively close to one another and there are no apparent outliers, both the mean and median are fairly similar. Thus, it could be argued that either measure would provide a representative assessment of the final examination scores. The choice between mean and median would ultimately depend on the specific context and the researcher's goals.