Question

The test scores on a 100 -point test were recorded for 20 students. \(\begin{array}{llllllllll}61 & 93 & 91 & 86 & 55 & 63 & 86 & 82 & 76 & 57\end{array}\) \(\begin{array}{lllllllll}94 & 89 & 67 & 62 & 72 & 87 & 68 & 65 & 75 & 84\end{array}\) a. Use a stem and leaf plot to describe the data. b. Describe the shape and location of the scores. c. Is the shape of the distribution unusual? Can you think of any reason that the scores would have such a shape?

Short answer

Answer: The distribution is slightly skewed to the left, indicating that there's a longer tail on the lower end with few lower scores. A possible reason could be that the test was relatively easy for most students, with a few having difficulties resulting in lower scores. Another explanation could be a difference in student abilities or their preparation levels for the test.

Step-by-step solution

Organize the Scores

Arrange the given test scores in ascending order to properly represent them in a stem and leaf plot later. 2.

Create the Stem and Leaf Plot

Construct a stem and leaf plot representing the organized test scores. The "stem" will represent the tens place and the "leaf" will represent the units place of each score. 3.

Describe the Shape and Location

Using the stem and leaf plot, describe the shape of the distribution, including whether it's symmetric, skewed left, or skewed right. Additionally, analyze the location by identifying the range and central tendency measures like mean, median, and mode. 4.

Analyze the Distribution

Discuss whether the shape of the distribution is unusual or not, and if there could be any reasons for the scores to have such a shape. This might involve considering factors like test difficulty, the different abilities of the students, or any other possible explanations. Now, let's follow these steps to solve the exercise: 1. Organize the Scores: We have the following test scores sorted in ascending order: 55, 57, 61, 62, 63, 65, 67, 68, 72, 75, 76, 82, 84, 86, 86, 87, 89, 91, 93, 94 2. Create the Stem and Leaf Plot: ``` 5 | 57 6 | 123578 7 | 256 8 | 2467 9 | 134 ``` 3. Describe the Shape and Location: The distribution is slightly skewed to the left, which means that there's a longer tail on the lower end with few lower scores. Range: \(94 - 55 = 39\) Mean: \( \frac{55+57+\cdots+93+94}{20} = 75.1 \) Median: Midpoint between the 10th (\(75\)) and 11th (\(76\)) values, \(( 75 + 76 ) / 2 = 75.5\) Mode: There are 2 occurrences of the score, \(86\). 4. Analyze the Distribution: The shape of the distribution is slightly unusual, as it shows a mild negative skewness. A possible reason could be that the test was relatively easy for most of the students, with a few students having difficulties resulting in lower scores. Another explanation could be a difference in student abilities or their preparation levels for the test.

Key concepts

Data Distribution

Data distribution is the way values are spread out across a dataset. In a stem and leaf plot, each number is split into a "stem" and a "leaf." This visual tool provides a clear picture of how data points are distributed. For the test scores from 20 students, when placed in a stem and leaf plot, it's easy to see the frequency of each range of scores.
The data distribution is essential for understanding the dataset's overall landscape, including where most scores lie, the presence of outliers, and the spread of data values.

• The data ranges from 55 to 94.
• Organized, the scores provide insights into patterns and clusters.
• Grouping scores into tens (e.g., 50s, 60s, 70s) helps visualize the distribution more easily.

With such visualization, we can then better explore other statistical concepts like central tendency and skewness.

Central Tendency

Central tendency measures describe the center or typical value within a set of data. The three main measures are the mean, median, and mode, each offering different insights.

**Mean**: The mean represents the average of the scores, calculated by adding all scores and dividing by the number of observations. Here, the mean test score is 75.1, which gives an idea of the overall performance of the group.

**Median**: The median indicates the middle value when the data is ordered. It's less affected by outliers, making it a good measure for skewed distributions. For these scores, the median is 75.5.

**Mode**: The mode is the most frequently occurring value(s). In our dataset, the mode is 86, occurring twice. This helps identify the most common result and any recurring trends.
Understanding central tendency provides a snapshot of typical performance and forms the basis for further analysis of data trends.

Skewness

Skewness measures the asymmetry or deviation from the average within a distribution. This concept is important to determine how much and in which direction the data tails off.

In this case, the test scores display a slight left skew, which means there are fewer lower scores stretching the distribution to the left. Signs of left skewness appear when the mean is generally lower than the median and mode, suggesting that a few low scores pull the average down.

• Left-skewness indicates more students achieved relatively high scores compared to low scores.
• Can often point to external factors, such as students' preparation levels and test difficulty.

Recognizing skewness helps understand the potential causes behind the test scores, offering a more nuanced view than central tendency alone.

Range of Scores

The range of scores is computed as the difference between the highest and lowest values in a dataset. It provides a simple measure of data dispersion.

The range for these test scores is calculated as 94 - 55 = 39. This relatively wide range tells us that there is significant variability among the student scores.

• A high range indicates diverse performance levels among students.
• Highlights potential outliers, especially on the lower end.

While the range doesn't disclose intricate details about distribution, it is a useful starting point for grasping how widespread the data is. Looking at the range alongside other measures can assist in forming a comprehensive view of dataset characteristics.