Question

A sample of 25 freshmen at a major university completed a survey that measured their degree of racial prejudice (the higher the score, the greater the prejudice). a. Compute the median and mean scores for these data. \(\begin{array}{lllll} 10 & 43 & 30 & 30 & 45 \\ 40 & 12 & 40 & 42 & 35 \\ 45 & 25 & 10 & 33 & 50 \\ 42 & 32 & 38 & 11 & 47 \\ 22 & 26 & 37 & 38 & 10 \end{array}\) b. These same 25 students completed the same survey during their senior year. Compute the median and mean for this second set of scores, and compare them to the earlier set. What happened? \(\begin{array}{lllll} 10 & 45 & 35 & 27 & 50 \\ 35 & 10 & 50 & 40 & 30 \\ 40 & 10 & 10 & 37 & 10 \\ 40 & 15 & 30 & 20 & 43 \\ 23 & 25 & 30 & 40 & 10 \end{array}\)

Short answer

Freshman Year: Median = 35, Mean = 34.72.Senior Year: Median = 30, Mean = 31.8.Prejudice scores decreased over time.

Step-by-step solution

Organize the Data for Freshman Year

Arrange the data in ascending order. The given scores are: [10, 43, 30, 30, 45, 40, 12, 40, 42, 35, 45, 25, 10, 33, 50, 42, 32, 38, 11, 47, 22, 26, 37, 38, 10].When arranged: [10, 10, 10, 11, 12, 22, 25, 26, 30, 30, 32, 33, 35, 37, 38, 38, 40, 40, 42, 42, 43, 45, 45, 47, 50].

Compute the Median for Freshman Year

Since there are 25 scores, the median is the 13th value in the ordered list.The 13th value is 35. Therefore, median = 35.

Compute the Mean for Freshman Year

Sum all the scores and divide by the number of scores.Sum = 10 + 10 + 10 + 11 + 12 + 22 + 25 + 26 + 30 + 30 + 32 + 33 + 35 + 37 + 38 + 38 + 40 + 40 + 42 + 42 + 43 + 45 + 45 + 47 + 50 = 868.Mean = 868 / 25 = 34.72.

Organize the Data for Senior Year

Arrange the data in ascending order. The given scores are: [10, 45, 35, 27, 50, 35, 10, 50, 40, 30, 40, 10, 10, 37, 10, 40, 15, 30, 20, 43, 23, 25, 30, 40, 10].When arranged: [10, 10, 10, 10, 10, 10, 15, 20, 23, 25, 27, 30, 30, 30, 30, 35, 35, 37, 40, 40, 40, 43, 45, 50, 50].

Compute the Median for Senior Year

Since there are 25 scores, the median is the 13th value in the ordered list.The 13th value is 30. Therefore, median = 30.

Compute the Mean for Senior Year

Sum all the scores and divide by the number of scores.Sum = 10 + 10 + 10 + 10 + 10 + 10 + 15 + 20 + 23 + 25 + 27 + 30 + 30 + 30 + 30 + 35 + 35 + 37 + 40 + 40 + 40 + 43 + 45 + 50 + 50 = 795.Mean = 795 / 25 = 31.8.

Compare the Results

For Freshman Year: Median = 35, Mean = 34.72.For Senior Year: Median = 30, Mean = 31.8.The median and mean scores have decreased from the freshman year to the senior year, indicating a decrease in racial prejudice over time.

Key concepts

racial prejudice measurement

Racial prejudice measurement is an important aspect in social research as it helps in understanding changes in social attitudes over time. In this exercise, students were surveyed at the beginning and end of their university experience to measure their level of racial prejudice. The scores reflect their attitudes, with higher scores indicating greater prejudice.
This measurement is crucial as it provides data on how education and exposure to diverse environments might influence racial attitudes.
By comparing these scores, researchers can assess the impact university life has on reducing or increasing racial prejudice among students.
This type of data can inform policies and programs aimed at fostering more inclusive environments.

data organization

Organizing data is a fundamental step in any statistical analysis. In this exercise, the first task was to arrange the scores in ascending order.
Here's why this is important:

• It allows for the easy identification of the median score, as the median is the middle value in an organized list.
• It helps in visualizing the distribution of the data, which can indicate patterns such as clustering or outliers.

Proper data organization makes subsequent calculations straightforward and accurate.
For instance, when the scores from the freshman year were sorted, it became clear which value was the median and what the overall range of scores was.
Without organizing data first, any statistical analysis would be chaotic and prone to errors.

comparison of means and medians

Comparing means and medians provides a deeper understanding of data distribution.
Both are measures of central tendency, but they offer different insights:

• **Median**: The middle number when data is ordered. It is not affected by extreme values (outliers).
• **Mean**: The average value of the data. It can be skewed by outliers and extreme values.

In the exercise, the freshman year had a median of 35 and a mean of 34.72, whereas the senior year's median was 30 and mean was 31.8. These results showed a decrease in both median and mean scores, indicating a general decline in racial prejudice over the college years.
This comparison helps in understanding whether a few extreme scores are influencing the average or if the change is consistent across the whole dataset.

statistical analysis in social research

Statistical analysis is a powerful tool in social research, allowing researchers to draw meaningful insights from data.
Key elements include:

• **Descriptive Statistics**: Summarizes data through means, medians, and ranges to provide a clear overview of the dataset.
• **Inferential Statistics**: Allows researchers to make predictions or inferences about a population based on a sample.

In this exercise, the statistical analysis of racial prejudice scores highlighted changes over time, demonstrating how exposure to a university environment might influence students' attitudes.
These insights are valuable for developing targeted interventions aimed at reducing prejudice and promoting inclusivity.
Such analyses can inform policy decisions and help shape educational programs to create a more equitable society.