Question

The test scores on a l00-point test were recorded for 20 students. Construct a relative frequency distribution for the data, using 6 classes of width \(8,\) and starting at \(52 .\) Then answer the questions. $$ \begin{array}{llllllllll} 61 & 93 & 91 & 86 & 55 & 63 & 86 & 82 & 76 & 57 \\ 94 & 89 & 67 & 62 & 72 & 87 & 68 & 65 & 75 & 84 \end{array} $$ Is the shape of the distribution unusual? Can you think of any reason that the scores would have such a shape?

Short answer

The shape of the distribution is somewhat irregular, with higher frequencies in the middle classes. It is not entirely symmetric, and it is not a perfect bell curve. One possible reason for this pattern could be that the test scores come from different groups of students with varying levels of preparation or ability. Some students might have access to better resources and support, resulting in a higher concentration of scores in higher-score classes, while others might face challenges that lead to lower test scores and a concentration of scores in the lower-score classes.

Step-by-step solution

Sort the scores in ascending order

First, we should arrange the given scores in ascending order: 55, 57, 61, 62, 63, 65, 67, 68, 72, 75, 76, 82, 84, 86, 86, 87, 89, 91, 93, 94

Define the class intervals

Since we need to create 6 classes of width 8 starting at 52, the class intervals are as follows: - Class 1: 52–59 - Class 2: 60–67 - Class 3: 68–75 - Class 4: 76–83 - Class 5: 84–91 - Class 6: 92–99

Calculate the frequency for each class

Count the number of scores that fall into each class interval: - Class 1: 3 (55, 57, 61) - Class 2: 5 (62, 63, 65, 67, 68) - Class 3: 4 (72, 75, 76, 82) - Class 4: 3 (84, 86, 86) - Class 5: 3 (87, 89, 91) - Class 6: 2 (93, 94)

Calculate the relative frequencies

Divide each class frequency by the total number of scores (20) to get the relative frequency: - Class 1: 3/20 = 0.15 - Class 2: 5/20 = 0.25 - Class 3: 4/20 = 0.2 - Class 4: 3/20 = 0.15 - Class 5: 3/20 = 0.15 - Class 6: 2/20 = 0.1

Create the relative frequency distribution table

Now we can create the relative frequency distribution table: | Class Interval | Frequency | Relative Frequency | |---|---|---| | 52–59 | 3 | 0.15 | | 60–67 | 5 | 0.25 | | 68–75 | 4 | 0.2 | | 76–83 | 3 | 0.15 | | 84–91 | 3 | 0.15 | | 92–99 | 2 | 0.1 |

Analyze the shape of the distribution

Looking at the relative frequency distribution, we can see a somewhat irregular pattern, with higher frequencies in the middle classes. The distribution is not entirely symmetric, and it is not a perfect bell curve. One possible reason for this shape could be that these test scores come from different groups of students with varying levels of preparation or ability. For instance, some students might have access to better resources and support, resulting in a higher concentration of scores in higher-score classes. Conversely, some students might face challenges that lead to lower test scores, resulting in a concentration of scores in the lower-score classes.

Key concepts

Statistical Data Analysis

Statistical data analysis is a crucial component in understanding and interpreting numerical data. It encompasses a variety of techniques and methods used to summarize, examine, and draw conclusions from data sets. Such analysis can inform decision-making processes, highlight trends, and even predict future outcomes based on historical data.

When analyzing test scores for students like in the provided exercise, statistical data analysis allows us to observe patterns and irregularities in the data. It can also offer insights into student performance and the effectiveness of teaching methods. By constructing a relative frequency distribution, we transform raw data into a format that is easier to understand and interpret.

Class Intervals

In the context of statistics, class intervals are an essential component when organizing data into a frequency distribution table. A class interval, or bin, is a range of values within which a set of data points is grouped. The width of the interval is the difference between the upper and lower boundaries of the bin.

Choosing the right class intervals can significantly impact the presentation and analysis of data. If intervals are too wide, you might miss important patterns; if too narrow, the distribution might be too fragmented to discern any trends. In the exercise, the choice of six class intervals of width 8 is a strategic one, aimed at providing a clear yet comprehensive overview of the test score data.

Frequency Distribution Table

A frequency distribution table is a visual representation that displays how frequently each category, or class interval, occurs in the dataset. It typically contains columns for class intervals, absolute frequency (the count of data points in each interval), and relative frequency, which is the proportion of data points in each interval relative to the total number of observations.

The relative frequency is calculated by dividing the absolute frequency by the total number of data points, as demonstrated in Step 4 of the exercise. Creating a frequency distribution table, like the one for the test scores, not only helps in visualizing data but also in identifying the shape of the distribution—whether it is symmetric, skewed, or has any gaps or outliers, which are significant for further statistical analysis.