Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In an examination given to a class of 20 students, the following test scores were obtained: $$\begin{array}{lllllllllr}40 & 45 & 50 & 50 & 55 & 60 & 60 & 75 & 75 & 80 \\ 80 & 85 & 85 & 85 & 85 & 90 & 90 & 95 & 95 & 100\end{array}$$ a. Find the mean (or average) score, the mode, and the median score. b. Which of these three measures of central tendency do you think is the least representative of the set of scores?

Step-by-Step Solution

Answer

The mean score is 79.5, the mode is 85, and the median is 80. The least representative measure of central tendency is the mode (85) since it appears only four times and does not account for the other scores in the dataset.

Step 1: Calculate the Mean (Average) Score

To find the mean score, we add up all the scores and then divide by the total number of scores (20 in this case). Mean = \(\frac{Sum \: of \: scores}{Number \: of \: scores}\)

Step 2: Calculate the Sum of Scores

Here's the sum of all the scores: Sum = 40 + 45 + 50 + 50 + 55 + 60 + 60 + 75 + 75 + 80 + 80 + 85 + 85 + 85 + 85 + 90 + 90 + 95 + 95 + 100 Sum = 1590

Step 3: Calculate the Mean

Now, we calculate the mean by dividing the sum by the total number of scores: Mean = \(\frac{1590}{20}\) Mean = 79.5

Step 4: Find the Mode

The mode is the number that appears most frequently in the dataset. In this case, 85 appears four times, which is the highest frequency, so: Mode = 85

Step 5: Find the Median

To find the median, first, we order the scores in ascending order (which they already are), and then we find the middle value. Since there are 20 scores (an even number of scores), we take the mean of the two middle values (10th and 11th scores): Median = \(\frac{80+80}{2}\) Median = 80

Step 6: Analyze the Measures of Central Tendency

We have calculated the mean (79.5), mode (85), and median (80). Now, we need to discuss which of these measures is the least representative of the set of scores. The mode (85) appears only four times out of the 20 scores, so it may not represent the scores' overall distribution well. The median (80) is the middle value, and it sits well within the range of values, so it could generally represent the scores. The mean (79.5) takes into account every score, but some extreme values such as low scores (40 and 45) and high scores (95 and 100) may not represent the overall group well.

Step 7: Conclusion

Considering the mean, mode, and median, the least representative measure of central tendency for this dataset is the mode (85) because it appears only a few times and does not account for the other scores in the dataset.

Key Concepts

Mean

Understanding the concept of the "Mean" is crucial for interpreting data sets. The mean, often referred to as the average, is calculated by adding up all the values in a data set and then dividing by the number of values. It offers insights into the overall level of all the individual values combined.

For example, in our examination data, the mean score is computed by totaling all the scores and dividing by how many scores there are. By doing so, you arrive at a mean score of 79.5.

• This mean takes every score into account.
• Extremes, such as the lowest and highest scores, can skew this average.

The mean provides a middle ground, but it's essential to be cautious about how outliers (very high or very low scores) might impact the representation of a group.

Median

The "Median" is a measure of central tendency that identifies the middle point of a data set. It's found by listing numbers in order and locating the midpoint. If the data set has an even number of observations, the median is the mean of the two central numbers.

In the exam scores given, the scores are already sorted. With 20 scores, the median is the average of the 10th and 11th scores in this ordered list. Both are 80, making the median 80.

• The median is not affected by extremely high or low values.
• It provides a central point that splits the data into two equal halves.

Hence, the median is a reliable indicator of central tendency, especially when dealing with skewed distributions.

Mode

The "Mode" is the value that appears most frequently in a data set. It reflects the most common point or frequency within data, making it straightforward but unique among measures of central tendency.

In the situation of these exam scores, the number 85 appears more often than any other score (four times), making it the mode.

• Mode is excellent for understanding the most frequent data point.
• It may not always reflect the center of a data set if it occurs infrequently.

Thus, while the mode can show popular scores, it's less effective if the most common score doesn't form a significant part of the data set, as seen in the example where it isn't very spread out.