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Chapter 11: Problem 5

Grade point averages (GPA) in several colleges are on a scale of \(0-4 .\) Here are the GPAs of 45 students at a certain college. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline 1.8 & 1.9 & 1.9 & 2.0 & 2.1 & 2.1 & 2.1 & 2.2 & 2.2 & 2.3 & 2.3 & 2.4 & 2.4 & 2.4 & 2.5 \\ \hline 2.5 & 2.5 & 2.5 & 2.5 & 2.5 & 2.6 & 2.6 & 2.6 & 2.6 & 2.6 & 2.7 & 2.7 & 2.7 & 2.7 & 2.7 \\ \hline 2.8 & 2.8 & 2.8 & 2.9 & 2.9 & 2.9 & 3.0 & 3.0 & 3.0 & 3.1 & 3.1 & 3.1 & 3.2 & 3.2 & 3.4 \\ \hline \end{array}$$ Prepare a frequency histogram, a relative frequency histogram and a cumulative frequency graph. Describe the data in two to three sentences.

Short Answer

Expert verified
The GPA distribution is slightly skewed towards higher values, with a concentration in the range of 2.5 to 2.7.

Step by step solution

01

Organize Data into Frequency Table

First, we need to count how many times each GPA value appears in the data set. Here is the frequency distribution: - 1.8: 1 - 1.9: 2 - 2.0: 1 - 2.1: 3 - 2.2: 2 - 2.3: 2 - 2.4: 3 - 2.5: 5 - 2.6: 5 - 2.7: 5 - 2.8: 3 - 2.9: 3 - 3.0: 3 - 3.1: 3 - 3.2: 2 - 3.4: 1
02

Calculate Relative Frequencies

To find the relative frequency, divide each frequency by the total number of students, which is 45.- 1.8: \(\frac{1}{45} \approx 0.022\)- 1.9: \(\frac{2}{45} \approx 0.044\)- 2.0: \(\frac{1}{45} \approx 0.022\)- 2.1: \(\frac{3}{45} \approx 0.067\)- 2.2: \(\frac{2}{45} \approx 0.044\)- 2.3: \(\frac{2}{45} \approx 0.044\)- 2.4: \(\frac{3}{45} \approx 0.067\)- 2.5: \(\frac{5}{45} \approx 0.111\)- 2.6: \(\frac{5}{45} \approx 0.111\)- 2.7: \(\frac{5}{45} \approx 0.111\)- 2.8: \(\frac{3}{45} \approx 0.067\)- 2.9: \(\frac{3}{45} \approx 0.067\)- 3.0: \(\frac{3}{45} \approx 0.067\)- 3.1: \(\frac{3}{45} \approx 0.067\)- 3.2: \(\frac{2}{45} \approx 0.044\)- 3.4: \(\frac{1}{45} \approx 0.022\)
03

Create Frequency Histogram

On the horizontal axis, list the GPA values. On the vertical axis, list frequency counts from Step 1. Draw a bar above each GPA value up to its corresponding frequency.
04

Create Relative Frequency Histogram

Similar to the frequency histogram, but instead of frequency counts, use relative frequencies calculated in Step 2 on the vertical axis.
05

Construct Cumulative Frequency Table

Calculate the cumulative frequency by adding up all frequencies at or below a certain GPA level. For example, - 1.8: 1 - 1.9: 3 (1 + 2) - 2.0: 4 (3 + 1), and so on.
06

Create Cumulative Frequency Graph

Plot the cumulative frequency against GPA on a graph, with GPA on the horizontal axis and cumulative frequency on the vertical axis. Connect the points to form a step graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Histogram
A frequency histogram is a graphical representation of data that shows the frequency of each category or interval. Imagine a bar chart鈥攁 frequency histogram looks quite similar! On the x-axis, you'll have the categories or intervals, while the y-axis shows the frequency of data points in each category.

To create a frequency histogram for the given GPA data, first arrange the data into a frequency distribution table. As seen in the exercise, you count how often each GPA appears. Once done, plot this information onto the graph with bars. Each bar represents the frequency for each GPA value.

Use a consistent scale for your axes to ensure all data fits well and is understandable. The result is a clear visual representation of which GPA ranges most students fall into.
Relative Frequency
Relative frequency is all about showing the proportion of the total number of occurrences that each category represents. It is calculated by dividing the frequency of each item by the total number of observations.

For example, if the GPA 2.5 appears 5 times out of 45, the relative frequency is \ \( \frac{5}{45} \approx 0.111 \ \). This figure tells you that around 11.1% of the data is within this GPA range. It gives a sense of scale and proportion compared to just raw frequencies.

To visualize relative frequencies, you would use a relative frequency histogram. It resembles a regular frequency histogram, but the y-axis represents relative instead of absolute frequencies. This format helps to understand how common or rare each class interval is, relative to the entire dataset.
Cumulative Frequency
Cumulative frequency is a running total of frequencies through the classes or categories in a data set. It tells us how many observations fall below a certain category or value.

Calculate cumulative frequencies by adding each frequency from your frequency table to the sum of all previous frequencies. For instance, if the initial frequency for GPA 1.8 is 1, the cumulative frequency for 1.9 would be 1 + 2 = 3.

A cumulative frequency graph鈥攁lso known as an ogive鈥攊s plotted using this data. This graph shows the cumulative frequency on the y-axis against the GPA values on the x-axis. Connecting these points forms a step graph that reveals the progression of accumulated data over values.
Data Representation
Data representation is the way in which data is visually or numerically presented to convey information clearly. Effective data representation helps identify patterns, trends, and outliers easily.

In this exercise, different forms of data representation like frequency and relative frequency histograms as well as cumulative frequency graphs, serve to offer varied insights. While the frequency histogram pinpoints the most common data points, the relative frequency histogram shows their proportion. Meanwhile, the cumulative frequency graph helps identify the overall distribution and accumulation of data values.

Using these methods, one can dissect and interpret the GPA data efficiently, aiding in understanding student performance trends.

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Most popular questions from this chapter

Aptitude tests sometimes use jigsaw puzzles to test the ability of new applicants to perform precision assembly work in electronic instruments. One such company that produces the computerized parts of video and CD players gave the following results: $$\begin{array}{|c|c|} \hline \text { Time to finish the puzzle (nearest second) } & \text { Number of employees } \\ \hline 30-35 & 16 \\ \hline 35-40 & 24 \\ \hline 40-45 & 22 \\ \hline 45-50 & 26 \\ \hline 50-55 & 38 \\ \hline 55-60 & 36 \\ \hline 60-65 & 32 \\ \hline 65-70 & 18 \\ \hline \end{array}$$a) Draw a histogram of the data. b) Draw a cumulative frequency curve and estimate the median and IQR. c) Calculate the estimates of the mean and standard deviation of all such participants.

Post offices weigh the letters customers send before they decide on the amount of postage required. The table below lists the masses (in grams) of letters processed by a post office in a large city on a certain day. (Any letter heavier than \(2000 \mathrm{g}\) is considered a parcel.) Draw a histogram to illustrate the situation. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Mass } & 1-200 & 201-400 & 401-600 & 601-800 & 801-1000 & 1001-2000 \\ \hline \text { Frequency } & 3220 & 450 & 130 & 96 & 54 & 40 \\ \hline \end{array}$$

Identify the experimental units, sensible population and sample on which each of the following variables is measured. Then indicate whether the variable is quantitative or qualitative. a) Gender of a student b) Number of errors on a final exam for 10 th-grade students c) Height of a newborn child d) Eye colour for children aged less than 14 e) Amount of time it takes to travel to work f) Rating of a country's leader: excellent, good, fair, poor g) Country of origin of students at international schools

The mean score of 26 students on a 40 -point paper is \(22 .\) The mean for another group of 84 other students is \(32 .\) Find the mean of the combined group of 110 students.

State what you expect the shapes of the distributions of the following variables to be: uniform, unimodal, bimodal, symmetric, etc. Explain why. a) Number of goals shot by football players during last season. b) Weights of newborn babies in a major hospital during the course of 10 years. c) Number of countries visited by a student at an international school. d) Number of emails received by a high school student at your school per week.
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