Frequency Distributions
Imagine that a large class of students is given a quiz. We examine the results in the form of tables and graphs.
Practice 1
In our hypothetical class, 130 students take a 10-question quiz. The results are described to us in long-winded verbal form. Here's what we're told: "Nobody missed all the questions (that is, got a score of 0 correct); 4 people got 1 correct answer; 7 people got 2 correct answers; 10 people got 3 correct answers; 15 people got 4 correct answers; 24 people got 5 correct answers; 22 people got 6 correct answers; 24 people got 7 correct answers; 15 people got 8 correct answers; 7 people got 9 correct answers; and 2 people wrote perfect papers (that is, got 10 correct answers)."
Portray these results in the form of a table, showing the test scores in ascending order from top to bottom in the left-hand column, and the absolute frequencies for each score in the right-hand column.
Solution 1
Table 8-1 shows the results of the quiz in tabular form. Note that in this depiction, the lowest score is at the top, and the highest score is at the bottom. The table is arranged this way because that's how we are asked to do it.
Table 8-1 Table for Practice 1. The lowest score is at the top and the highest score is at the bottom.
Practice 2
How else can the data from Practice 1 be arranged in a table?
Solution 2
The quiz results can be portrayed in a table upside-down relative to Fig. 8-1, that is, with the highest score at the top and the lowest score at the bottom (Table 8-2), and it shows us the information just as well.
Fig. 8-1. Illustration for Practice 1-3.
Table 8-2 Table for Practice 2. This is the same data as that shown in Table 8-1, but with the highest score at the top and the lowest score at the bottom.
The table can also be arranged with the columns and rows interchanged, so it has 2 rows and 11 columns (not counting the column with the headers). This can be done in either of two ways: the lowest score at the left and the highest score at the right (Table 8-3A), or the highest score at the left and the lowest score at the right (Table 8-3B).
Table 8-3A Table for Practice 2. This is the same data as that shown in Table 8-1, but with the data arranged horizontally. The lowest score is at the left and the highest score is at the right.
Table 8-3B Another table for Practice 2. This is the same data as that shown in Table 8-1, but with the data arranged horizontally. The highest score is at the left and the lowest score is at the right.
Practice 3
Render the data from Practice 1 in the form of a vertical bar graph, showing the lowest score at the left and the highest score at the right. Do not put numbers for the absolute frequency values at the tops of the bars.
Solution 3
Figure 8-1 shows the results of the quiz as a vertical bar graph, without absolute frequency values shown at the tops of the bars. The advantage of showing the numbers, if there's room to do so, is the fact that it eliminates the need for the observer having to guess at the values. In this graph, it would be a "tight squeeze" to show the numbers, and the result would look crowded and might even cause confusion in reading the graph.
Fig. 8-1. Illustration for Practice 1-3.
Practice 4
Render the data from Practice 1 in the form of a horizontal bar graph, showing the highest score at the top and the lowest score at the bottom. Include the absolute frequency values at the right-hand ends of the bars.
Solution 4
Figure 8-2 shows the results of the quiz as a horizontal bar graph, with the absolute frequency values indicated at the right-hand ends of the bars. In this graph, it is not quite so messy to show the numbers, and they provide useful information.
Fig. 8-2. Illustration for Solution 4.
Practice 5
Render the data from Practice 1 in the form of a point-to-point graph, showing the lowest score at the left and the highest score at the right on the horizontal scale, and showing the absolute frequency referenced to the vertical scale with the lowest values at the bottom and the highest values at the top.
Solution 5
Figure 8-3 is an example of such a graph. Data values are shown by the points. The straight lines create an impression of the general shape of the distribution, but are not part of the graph itself.
Fig. 8-3. Illustration for Solution 5.
Practice 6
Portray the results of our hypothetical quiz in the form of a table similar to Table 8-1, with the lowest score at the top and the highest score at the bottom.
Table 8-1 Table for Practice 6. The lowest score is at the top and the highest score is at the bottom.
But in addition to the absolute frequency values, include a column showing cumulative absolute frequencies in ascending order from top to bottom.
Solution 6
See Table 8-4. Note that the values in the third column, which shows the cumulative absolute frequency, always increase as we go down the table. That is, each number is greater than the one above it. In addition, the largest value is equal to the total number of elements in the statistical group, in this case 130, the number of students in the class.
Table 8-4 Table for Practice 6. Note that the cumulative absolute frequency values constantly increase as you read down the table.
Practice 7
Render the data from Practice 1 in the form of a dual point-to-point graph, showing the lowest score at the left and the highest score at the right on the horizontal scale. Show the absolute frequency values referenced to a vertical scale at the left-hand side of the graph. Show the cumulative absolute frequency values as a dashed line, referenced to a vertical scale at the right-hand side of the graph.
Solution 7
See Fig. 8-4. To further help in differentiating between the graph of the absolute frequency and the graph of the cumulative absolute frequency, open circles are used to indicate the points corresponding to the cumulative absolute frequency values. The solid black dots and the solid black line are referenced to the left-hand scale; the open circles and the dashed line are referenced to the right-hand scale.
Fig. 8-4. Illustration for Solution 7.
Practice 8
What is the mean score for the quiz results in the class we have been discussing in the past few paragraphs?
Solution 8
Multiply each score by its absolute frequency, obtaining a set of products. Add up the products and divide the result by the number of papers in the class, in this case 130. Table 8-5 shows the products, along with the cumulative sums. Making a table and then double-checking the results can be helpful in situations like this, because errors are easy to make. (If a mistake occurs, it propagates through the rest of the calculation and gets multiplied, worsening the inaccuracy of the final result.) The population mean is 731/130, or approximately 5.623. It can be symbolized μ.
Table 8-5 Table for Practice 8. The lowest score is at the top and the highest score is at the bottom. The number at the lower right is divided by the number of elements in the population (in this case 130) to obtain the mean, which turns out to be approximately 5.623.
Practice 9
What is the median score for the quiz results in the class we have been discussing in the past few paragraphs?
Solution 9
Recall the definition of the median. If the number of elements in a distribution is even, then the median is the value such that half the elements are greater than or equal to it, and half the elements are less than or equal to it. If the number of elements is odd, then the median is the value such that the number of elements greater than or equal to it is the same as the number of elements less than or equal to it. In this case, the "elements" are the test results for each individual in the class. Table 8-6 shows how the median is determined. When the scores of all 130 individual papers are tallied up so they are in order, the scores of the 65th and 66th papers – the two in the middle – are found to be 6 correct. Thus, the median score is 6, because half the students scored 6 or above, and the other half scored 6 or below.
Table 8-6 Table for Practice 9. The median can be determined by tabulating the cumulative absolute frequencies.
Practice 10
What is the mode score for the quiz results in the class we have been discussing in the past few paragraphs?
Solution 10
The mode is the score that occurs most often. In this situation there are two such scores, so this distribution is bimodal. The mode scores are 5 and 7.
Practice 11
Show the mean, median, and modes as vertical dashed lines in a point-to-point graph similar to the plot of Fig. 8-3.
Fig. 8-3. Illustration for Practice 11.
Solution 11
See Fig. 8-5. The mean, median, and modes are labeled, and are all referenced to the horizontal scale.
Fig. 8-5. Illustration for Solution 11.
More practice problems for these concepts can be found at:
Statistics Practical Problems Practice Test
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