Question

A survey of major universities asked what percentage of incoming freshmen usually graduate "on time" in 4 years. Use the summary statistics given to answer the questions that follow. $$ \begin{array}{l|c} & \% \text { on Time } \\ \hline \text { Count } & 48 \\ \text { Mean } & 68.35 \\ \text { Median } & 69.90 \\ \text { StdDev } & 10.20 \\ \text { Min } & 43.20 \\ \text { Max } & 87.40 \\ \text { Range } & 44.20 \\ \text { 25th \%tile } & 59.15 \\ \text { 75th \%tile } & 74.75 \end{array} $$ a) Would you describe this distribution as symmetric or skewed? Explain. b) Are there any outliers? Explain. c) Create a boxplot of these data. d) Write a few sentences about the graduation rates.

Short answer

Left-skewed distribution; no outliers; depicts consistent graduation rates.

Step-by-step solution

Analyze Symmetry or Skewness

To describe the distribution as symmetric or skewed, we compare the mean and median. When the mean is less than the median, the distribution is typically left-skewed. When the mean is greater than the median, it is typically right-skewed. In this case, the mean is 68.35 and the median is 69.90, which suggests the distribution is left-skewed.

Identify Possible Outliers

Check for outliers using the 1.5*IQR rule (Interquartile Range). First, multiply the IQR (Interquartile Range) by 1.5.\( \text{IQR} = \text{75th percentile} - \text{25th percentile} = 74.75 - 59.15 = 15.6 \)Calculate 1.5*IQR: \( 1.5 \times \text{IQR} = 1.5 \times 15.6 = 23.4 \)Calculate the bounds:- Lower Bound = 25th percentile - 1.5*IQR = 59.15 - 23.4 = 35.75- Upper Bound = 75th percentile + 1.5*IQR = 74.75 + 23.4 = 98.15Since the minimum is 43.20 and maximum is 87.40, there are no data points outside these bounds, hence no outliers.

Create a Boxplot

A boxplot consists of a rectangular box extending from the 25th percentile (59.15) to the 75th percentile (74.75), with a vertical line at the median (69.90) inside the box. "Whiskers" extend from the box to the minimum (43.20) and maximum (87.40) values. Sketch these on a number line to visualize the data distribution.

Interpret Graduation Data

The graduation rates for the universities surveyed have an average rate of 68.35% with only slight skewness towards lower values, suggesting most rates align closely to the mean. With no outliers, the data reflects consistent education performance among these universities.

Key concepts

Distribution Symmetry

In statistics, understanding the symmetry of a distribution is crucial for interpreting data correctly. The concepts of skewness and symmetry help us know if the data tends to lean towards lower or higher values. When we describe a distribution, we often consider the relationship between the mean and the median.
- If the mean is less than the median, the data is typically left-skewed, meaning there is a tail on the left side of the distribution. - If the mean is greater than the median, it indicates right-skewness, suggesting a longer tail on the right.
In our situation, with a mean of 68.35 and a median of 69.90, a conclusion of left-skewness can be drawn. This implies that the data perhaps have some lower value graduation rates pulling the mean down below the median. Understanding this concept allows us to anticipate how data points are spread, providing insights into their possible impact on real-world education scenarios.

Outliers

Outliers are data points that differ significantly from the rest of the dataset. Detecting outliers is important because they can distort the overall analysis. The Interquartile Range (IQR) method is a common technique for spotting outliers. It involves calculating the range between the first quartile (25th percentile) and the third quartile (75th percentile).
Steps to identify outliers: - Compute the IQR: 75th percentile - 25th percentile. In our case, it is 74.75 - 59.15 = 15.6. - Multiply the IQR by 1.5 to get 23.4. - Determine the "fences" beyond which data might be outliers: - Lower bound: 25th percentile - 1.5 * IQR = 35.75 - Upper bound: 75th percentile + 1.5 * IQR = 98.15
Since the actual data range (from 43.20 to 87.40) fits within these calculations, we identify no outliers in the graduation rate data. Recognizing the absence of outliers allows for a more straightforward interpretation of the data without the need to account for anomalies.

Boxplot

Boxplots are visual tools used in statistics to depict the distribution of numerical data sets through their quartiles. They provide a straightforward way to interpret data's spread, position, and overall variability.
Creating a boxplot involves: - Drawing a box from the first quartile (25th percentile) to the third quartile (75th percentile), here from 59.15 to 74.75. - Placing a line inside this box at the median, which is 69.90. - Extending "whiskers" from the edges of the box to the minimum (43.20) and maximum (87.40) values of the data set. - Outliers, if any, would be plotted as individual points, but as we've determined, there are none.
The boxplot is advantageous for identifying the central tendency and variability. It also easily shows data symmetry or skewness. The visualization of a boxplot, with its central box highlighting key percentiles, is invaluable for quickly understanding the characteristics of the graduation rates of the surveyed universities.

Graduation Rates

Graduation rates are a key statistic reflecting the effectiveness and efficiency of educational institutions. They represent the percentage of students who graduate within a specified time, often four years for on-time completion. Understanding graduation rates helps stakeholders evaluate how well universities uphold academic and student support standards.
In the data provided, the average graduation rate is 68.35% with a median of 69.90%. This indicates that over half of the surveyed universities have rates around this mark, suggesting a consistent level of performance. - The range, from 43.20 to 87.40%, shows variances amongst institutions, with the IQR spotlighting the central half of data points from 59.15 to 74.75%. - This range suggests that while some schools might be struggling, most maintain a moderate to high performance.
Analyzing graduation rates allows education policymakers to identify successful institutions and address those in need of improvement. It also aids prospective students in making informed decisions by understanding which universities boast high success rates.