Question

Here are the summary statistics for Verbal SAT scores for a high school graduating class: $$ \begin{array}{l|c|c|c|c|c|c|c|c} & n & \text { Mean } & \text { Median } & \text { SD } & \text { Min } & \text { Max } & \text { Q1 } & \text { Q3 } \\ \hline \text { Male } & 80 & 590 & 600 & 97.2 & 310 & 800 & 515 & 650 \\ \text { Female } & 82 & 602 & 625 & 102.0 & 360 & 770 & 530 & 680 \end{array} $$ a) Create parallel boxplots comparing the scores of boys and girls as best you can from the information given. b) Write a brief report on these results. Be sure to discuss the shape, center, and spread of the scores.

Short answer

Females have a slightly higher median and larger IQR than males. Scores for males are slightly right-skewed; females are more symmetric.

Step-by-step solution

Understanding the Data

We are given summary statistics for Verbal SAT scores for male and female students, including the number of students \(n\), mean, median, standard deviation (SD), minimum, maximum, first quartile (Q1), and third quartile (Q3) values for both groups.

Drawing Parallel Boxplots

To construct the boxplots for both male and female scores, we plot the minimum, Q1, median, Q3, and maximum values for each group on the same number line: - For males: Min = 310, Q1 = 515, Median = 600, Q3 = 650, Max = 800. - For females: Min = 360, Q1 = 530, Median = 625, Q3 = 680, Max = 770. Each boxplot will have a box from Q1 to Q3, with a line at the median. Whiskers will extend from the minimum to the Q1 and from the Q3 to the maximum. Parallel the plots for easy comparison.

Analyzing the Shape, Center, and Spread

The shape of the distribution can be inferred from the symmetry around the median within the box: - Male scores are slightly skewed right (median closer to Q1). - Female scores appear more symmetric (median roughly in the center). The center of the distributions can be compared using the median: - Male median = 600, Female median = 625. The spread can be compared using the range and interquartile range (IQR): - Male range = 800 - 310 = 490, IQR = 650 - 515 = 135. - Female range = 770 - 360 = 410, IQR = 680 - 530 = 150.

Writing the Report

The boxplots illustrate that the female scores have a slightly higher center, as indicated by the median of 625 compared to 600 for males. While the spread for females is slightly narrower than for males in terms of range, the interquartile range (IQR) is larger for females (150) compared to males (135), suggesting more variability within the central half of the data for females. Both distributions have similar maximum values, but males have a lower minimum score, contributing to a wider overall spread.

Key concepts

Summary Statistics

Summary statistics provide a comprehensive snapshot of a data set, and they are essential for understanding the key features of Verbal SAT scores. Summary statistics include several components:

• Mean: This is the average score. It's calculated by summing all the scores and dividing by the number of students. The mean gives a general idea of the overall performance level.
• Median: The middle score when all scores are ordered from least to greatest. It is a more resistant measure than the mean as it is not influenced by outliers.
• Standard Deviation (SD): Represents how much the scores vary from the mean. A high SD indicates a wide spread of scores, while a low SD suggests scores are close to the mean.
• Minimum and Maximum: They define the range of the data, showing the lowest and highest scores achieved.
• Quartiles (Q1 and Q3): Q1 is the 25th percentile and Q3 is the 75th percentile, helping to define the spread and identify potential outliers.

In understanding Verbal SAT scores, these statistics help assess overall trends, central tendency, and variability among the student scores.

Verbal SAT Scores

Verbal SAT scores are a critical component of college admissions tests. They assess a student's reading and language skills, which are essential for academic success. The scores typically range from 200 to 800, with higher scores indicating better performance. When analyzing these scores, as in the case of the given data:

• Comparative Analysis: Male students had a median score of 600, while female students had a median score of 625. This suggests that female students, on average, performed slightly better in Verbal SAT scores.
• Distribution Insights: The scores also show a significant range for both genders. Female scores range from 360 to 770, and male scores range from 310 to 800. Each of these figures provides insight into the spread and possible outliers within the data.

Understanding Verbal SAT scores through these statistics allows schools and educators to identify trends and areas in need of improvement, and assists in making informed decisions about educational strategies and curriculum adjustments.

Distribution Analysis

Distribution analysis is crucial for interpreting how data spreads across a range of values, providing insights into the frequency of different scores. When analyzing Verbal SAT scores with boxplots:

• Shape of Distribution: The arrangement of scores can indicate whether a distribution is symmetric, skewed left (scores concentrated towards higher values), or skewed right (scores concentrated towards lower values). For males, the skew is slightly right, suggesting a concentration of scores lower than the median. In contrast, female scores appear more symmetric.
• Center of Distribution: The median score provides a powerful indication of the typical score. In this case, the female median is higher by 25 points, indicating a superior central tendency compared to males.
• Spread of Distribution: By assessing the range and interquartile range (IQR), we can understand variability. The female IQR is larger than the male IQR, indicating more variability in scores among the middle 50% of female students.

Analyzing score distributions in this manner allows for more nuanced insights into performance, enabling educators to target interventions and tailor support to improve student outcomes effectively.