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. $$ \begin{array}{llllllllll} 61 & 93 & 91 & 86 & 55 & 63 & 86 & 82 & 76 & 57 \\ 94 & 89 & 67 & 62 & 72 & 87 & 68 & 65 & 75 & 84 \end{array} $$ Describe the shape and location of the scores.

Short answer

Answer: The shape of the test scores is roughly bell-shaped, with the highest concentration of scores in the middle intervals (60-67, 68-75, and 84-91). The location of the scores is mostly in the range of 60 to 91.

Step-by-step solution

1. List the given scores in ascending order.

Here are the test scores listed in ascending order: $$ \begin{array}{llllllllll} 55 & 57 & 61 & 62 & 63 & 65 & 67 & 68 & 72 & 75 \\ 76 & 82 & 84 & 86 & 86 & 87 & 89 & 91 & 93 & 94 \end{array} $$

2. Determine the class intervals

Based on the given width of 8 and starting point of 52, we will have the following class intervals: 1. 52-59 2. 60-67 3. 68-75 4. 76-83 5. 84-91 6. 92-99

3. Assign the scores to their respective class intervals

Now, let's group the scores within the intervals: 1. 52-59: 55, 57 2. 60-67: 61, 62, 63, 65, 67 3. 68-75: 68, 72, 75 4. 76-83: 76, 82 5. 84-91: 84, 86, 86, 87, 89, 91 6. 92-99: 93, 94

4. Calculate the frequencies for each class interval

Next, count the number of scores in each class interval (frequencies): 1. 52-59: 2 2. 60-67: 5 3. 68-75: 3 4. 76-83: 2 5. 84-91: 6 6. 92-99: 2

5. Calculate the relative frequencies for each class interval

Now that we have the frequencies, we can find the relative frequencies by dividing the frequency by the total number of data points (20 students): 1. 52-59: 2/20 = 0.10 2. 60-67: 5/20 = 0.25 3. 68-75: 3/20 = 0.15 4. 76-83: 2/20 = 0.10 5. 84-91: 6/20 = 0.30 6. 92-99: 2/20 = 0.10

6. Analyze the shape and location of the scores

Based on the relative frequency distribution, we can see that the scores have a somewhat symmetric shape, with the highest relative frequency (0.30) in the 84-91 interval. The test scores are mostly concentrated in the middle intervals (60-67, 68-75, and 84-91). We can say that the shape of the scores is roughly bell-shaped. The location of the scores is mostly in the range of 60 to 91.

Key concepts

Class Intervals

When analyzing statistical data, it's important to group data points into manageable categories known as class intervals. These intervals help us summarize and interpret the data easily. They divide a set of observations into ranges, and each range encompasses the values that fall within it. In our exercise, the class intervals have a width of 8, starting at 52. This creates uniform ranges such as 52-59, 60-67, and so on.

Setting up appropriate class intervals is crucial as it influences the level of detail you'll receive from the data analysis. If the intervals are too wide, you may miss out on important variations in the data; too narrow, and the summary may become overly complex. In educational contexts, like evaluating test scores, class intervals are used to determine how the scores are distributed and to identify clusters where students fall within a particular performance range.

Statistical Data Analysis

Statistical data analysis involves collecting, presenting, and interpreting data in order to make informed decisions. It's a crucial step in a variety of fields including education, business, and healthcare. In the exercise, we've collected test scores and our goal is to analyze the distribution of these scores across specified class intervals.

By constructing a relative frequency distribution, we can gain insights into the performance of a group of students on a test. Analysis can be descriptive, where we summarize data in a meaningful way using tools like frequency distributions, or inferential, where we make predictions or generalizations about a population based on our sample data. In a classroom setting, this form of analysis can help educators identify areas where students are excelling or struggling.

Frequency Calculation

The term frequency calculation refers to counting how often each value occurs within a dataset. It is a foundational step in creating frequency distributions, which can be used in both descriptive and inferential statistics. In our exercise, we counted the number of test scores that fell into each class interval, known as the frequency of that interval. For example, between 84-91, six test scores fall within this range, making its frequency 6.

To convert these frequencies into relative terms, we divide each frequency by the total number of observations. Doing so gives us a clearer idea of how large a part of the whole each class interval represents, helping to compare different datasets or different classes within the same dataset. Relative frequencies can also be expressed as percentages, providing a straightforward interpretation of data distribution.

Descriptive Statistics

Descriptive statistics is a branch of statistics that deals with the summarization and description of data. It presents data in a meaningful way, without drawing conclusions or inferences about the population from which the sample was drawn. This includes measures of central tendency (mean, median, mode), measures of variability (range, variance, standard deviation), and frequency distributions, like the one created in our exercise.

By calculating the relative frequencies of each class interval of test scores, we can describe the distribution of scores. From our analysis, we see that the data is somewhat symmetric and most scores are clustered around the middle intervals. This summary helps determine if the scores are spread out evenly or if there are any outliers. Descriptive statistics is a vital tool for teachers, researchers, and students as it forms the basis for further statistical analysis and decision-making.