Algebra Statistics - Representing Data

Presenting Data—Bar Graphs and Histograms

Data can also be presented using histograms and bar graphs. Bar graphs and histograms look very similar, but they are used for different types of data. A bar graph is typically used to represent categorical data where histograms are typically used to present quantitative data. Remember that even though histograms and bar graphs look similar, they are fundamentally different in the type of information they display. They are also different in that the bars in a bar graph are separate, but those in a histogram are generally touching.

Step 2: Design and implement a plan that collects appropriate data

In the Statistical Process, the plan must be designed so that the type of variables and the way in which the data is collected are appropriate to the question and the desired outcome. Depending on the type of data (categorical or numerical), one of the following is used: Experiments An experiment can be defined as when a researcher deliberately imposes some treatment on individuals and observes their responses. Observational Studies Observational studies are when a researcher observes individuals and measures variables of interest, but does not attempt to influence the responses. Surveys are an important type of observational study and are used quite often in statistical analysis. It is important to remember that if an experiment or study is only to be administered to a sample of the population, those subjects should be chosen at random to avoid any prejudice or bias in the outcomes. For example, it would be a poor choice of subjects to survey the hockey team and ask "Which of the following is your favorite sport: football, basketball, baseball, hockey, or soccer?" It would be a better idea and yield more accurate statistical data to survey a variety of 9th grade students—both athletes and non-athletes.

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In this lesson, you learned how to represent data with different graphs. In particular, you learned how to make dot plots, bar graphs, and histograms. Dot plots visually show the frequency with which something occurs. Bar graphs are used for data that appear in categories, and histograms are used for data that appear in ranges (numerical data). You have also learned about the Statistical Process. The four steps of the Statistical Process are: Form a question that can be answered by data. Design and implement a plan that collects appropriate data. Analyze the data using graphical or numerical methods. Interpret the data in relation to the original question. Dot plots and histograms are two ways you can display data by the frequency of values. In general, dot plots are used for smaller data sets with smaller ranges. Histograms show the frequency of intervals of data, which makes histograms more appropriate for larger data sets. To create a dot plot: Draw a horizontal line using an appropriate range (or category). Place a dot over the data value for each frequency in the data set. Label the horizontal line and title the graph. Example Frequency table and dot plot showing frequency of quiz scores. The score of 50 has 4 dots, 60 has 2 dots, 70 has 7 dots, 80 has 4 dots, 90 has 2 dots, and 100 has 1 dot. Creating a histogram is much like creating a bar graph. Here are the steps you must follow: Group the data values into appropriate bin intervals and determine the frequency. Draw and label the horizontal and vertical axes. Draw bars to a certain height based on the frequency of each bin. Do not put spaces between the bars. Remember to title the graph. Example Frequency table and histogram showing frequency of fish in tanks. Bin 1 to 5 has a frequency of 5, 6 to 10 has a frequency of 2, 11 to 15 has a frequency of 3, and 16 to 20 has a frequency of 1.

Categorical Dot Plots

Sometimes the data collected are categorical data. So, instead of numbering the horizontal line, you would write the categories. For example, let's say a survey was conducted to determine which color is the most popular. Compare the frequency table on the left with its dot plot on the right. The horizontal line is labeled with the various colors. Because there are four students who said blue is their favorite color, there are four dots above blue. The same happens for the other colors. Two dots for green and red, five for purple, and two for black. Dot plots give you a nice visual of the frequency of each data value in a data set. Of course, do not forget your label and title. Print Frequency table and dot plot for categorical data blue, green, red, purple, and black. The frequencies and number of dots are four, two, two, five, and two, respectively. Analyzing the Dot Plot Dot plots are useful for comparing the frequency of one category to the others. In the example above, it is apparent that purple is the most common favorite color in the class. Even though the same information can be shown in a table, some information can be quite large, making it difficult to see things in a table that you can easily get from a visual, like a dot plot.

Step 1: Form a question that can be answered by data

Statistical questions specify populations and measurements of interest. The question should also anticipate answers based on data that vary, but anticipate fixed answers—no randomness is involved. The answers to statistical questions will address the variation in data, use probability statements, and apply only to the population specified. Statistical and Non-Statistical Questions Non-Statistical Question Statistical Question Reasoning Does the pesticide Raid harm butterflies? What is the mortality rate in Colias Eruytheme one hour after treatment with Raid? The non-statistical question is too broad and not measurable enough to produce statistical data. How has the Common Core State Standards changed our school? Since the implementation of the Common Core State Standards, what is the change in the mean EOC score for Algebra 1 students? How tall is Sheila? How tall are the students in my Algebra 1 class? The non-statistical question is too specific and does not allow for enough data to be collected to produce statistical data. How many words are there in this sentence? How many words are there in each sentence of Chapter 12 of The Scarlett Letter? There are different types of data which can be used in statistical analysis. Two of these types are categorical and quantitative (numerical) data—the type of data will depend on the wording of the question asked in step 1. Categorical Data Categorical data is data that falls into separate groups or distinct categories. A survey was conducted to find out 9th graders' favorite sport. The following question was asked, "What is your favorite sport—football, basketball, baseball, hockey, or soccer?" The data could fit into the following categories: football, basketball, baseball, hockey, and soccer. The data would represent the five categories and how each 9th grader answered the survey. The data generated could be represented in a table. Sport Football Basketball Baseball Hockey Soccer # of People 25 22 17 6 19 Each category contains a count that represents the specific number of students who responded with that sport. Quantitative Data Quantitative (numerical) arise when the observations are counts or measurements. There is continuous data where data fall anywhere in a range of numerical values and discrete data if the measurements are counts and fall into groups of integers. If a survey of 27 basketball players was conducted to find out their free-throw percentages, it would be numerical or quantitative. This means the data points could be arranged from the best free-throw shooter to the worst shooter. In order to make sense of the data, the range can be broken up into segments or bins containing similar free-throw percentages. Percentage 45-55 56-65 66-75 76-85 86-95 People 4 6 7 8 2 The players' average percentages range from 45% to 95% and all the players fall within that range. Take a look at some number sets to determine if they are quantitative or categorical in nature.

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The following is a list of cherry tree heights (in feet) that were found in town: 61, 72, 84, 88, 77, 67, 76, 79, 63, 79, 69, 70, 86, 78, 82, 82, 80, 73, 87, 90, 73, 76, 79, 71, 75, 79, 76, 83, 84, 87, 72, 81, 89, 74, 77, 78, 81, 84. Histogram; x-axis range: 60-65, 65-70, 70-75, 75-80, 80-85, 85-90 y-axis: 2, 3, 7, 12, 8, 6

Quantitative Dot Plots

Wesley's class is doing an activity about data collection. The students are collecting data to answer the question: On average, how many dandelions can be found inside the area of one hula hoop? All the groups threw their hoop, and their data are 9, 10, 11, 12, 9, 10, 13, 10, 8, 10, and 11. Take Wesley's class data and list it in order to identify the least and greatest values: 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 13 This list shows that you should draw a number line ranging from eight to thirteen. Next, place a dot above the number line for each time that number occurs in the data set. Eight occurs once, so place one dot above eight. Nine occurs twice, so place two dots above nine. Ten occurs four times, so place four dots above ten, and so on. Because there are eleven values in the data set, there will be eleven dots on the graph. Last, label the horizontal line and give it a title. The data points represent the frequency of dandelions. Print Take Wesley's class data and list it in order to identify the least and greatest values: 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 13 A dot plot of class activity. Horizontal line drawn from 8 to 13 with integers labeled. There is one dot above 8, 12, and 13, there are two dots above 9 and 11, and there are four dots above 10.