5.01 Representing Data

Histograms

A visual that shows data that appear in ranges (numerical data)

Steps to create a dot plot:

1). Draw a horizontal line using an appropriate range (or category). 2). Place a dot over the data value for each frequency in the data set. 3). Label the horizontal line, and title the graph.

The Statistical Process: There is a four-step process, called the statistical process, that is used to solve statistical problems.

1). Form a question that can be answered by data. 2). Design and implement a plan that collects appropriate data. 3). Analyze the data using graphical or numerical methods. 4). Interpret the data in relation to the original question. The statistical process is a way to collect, organize, understand, and represent data. It is important that statisticians use a standardized process so that analyses can be compared and informed decisions can be made using comparable data. Think of data as a bunch of random words in a book. Your job is to organize those words into an accurate, compelling story. That's what statisticians do.

Box Plot

A box plot is a visual that divides a list of data into four sections called quartiles. The plot is a data display that shows the minimum, lower quartile or first quartile, median, upper quartile or third quartile, and maximum of a data set. Each quartile has 25%, or one quarter, of the data points in it. A box plot can be used as part of the statistical process to analyze and interpret data.

What is Statistics?

A branch of mathematics that views numbers as data sets and teaches how to use a variety of methods to organize, interpret, and analyze these numbers to make informed decisions

Categorical Dot Plots Example

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. https://lti.flvsgl.com/flvs-cat-content/bqnasutj3mnjslcgs3shvt5epv/flvs-cat-session/temp_project_03/module05/lesson01/images/05_01_03b.png

What is Data?

Data are the numbers that give information about a given topic. Data can help someone learn more about a particular topic and make conclusions. The study of data is known as "statistics."

Presenting Data with Histograms

Data can also be presented using histograms. 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. Bar graphs and histograms are also different in that the bars in a bar graph are separate, but those in a histogram are generally touching.

Sum it Up

Dot Plots—visually show the frequency with which something occurs In general, dot plots are used for smaller data sets with smaller ranges. Histograms—used for data that appear in ranges (numerical data) Histograms show the frequency of intervals of data, which makes histograms more appropriate for larger data sets. Box Plots—There are five key pieces to a box plot: Minimum: the smallest value in the data set Lower quartile (Q1): the middle value in the lower half (or left of the median) of the data set Median: the middle value in the data set (if there is an even amount of numbers, the mean of the two middle values) Upper quartile (Q3): the middle value of the upper half (or right of the median) of the data set Maximum: the largest value in the data set

Quantitative Dot Plots Example 2

In anatomy, Percy learned that the average resting heart rate is between 60 and 100 beats per minute. He decided to record the average heart rate of people over five minutes while waiting in line at the pharmacy. His results are as follows: 68, 72, 78, 80, 69, 72, 89, 80, 78, 68, 62, 75, 76, 70, 70, 89, 80, 68, 72, 75. What can Percy conclude about the people waiting in line at the pharmacy? Percy's Dot Plot: Percy could create a table for the heart rate results, but a dot plot is a great way to visually represent the data so that analysis can be made. Analyzing Percy's Dot Plot: Percy can see from the dot plot that all 20 people that he recorded fall within the normal heart rate range. There were three people (one at 62 beats per minute and two at 89 beats per minute) that seemed to be out of the central gathering of data. https://lti.flvsgl.com/flvs-cat-content/bqnasutj3mnjslcgs3shvt5epv/flvs-cat-session/temp_project_03/module05/lesson01/images/05_01_4.gif

Forming and Designing a Plan—Step 1

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. Good statistical questions are not too broad and not too narrow. They are also measurable—which means enough data can be collected to analyze statistically. Different types of data 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 are data that fall into separate groups or distinct categories. Quantitative (numerical) data arise when the observations are counts or measurements. There are continuous data, where data fall anywhere in a range of numerical values, and discrete data, in which the measurements are counts and fall into groups of integers. Categorical: the answer will be a word that can be sorted into groups in the answer Quantitative: the answer could be a real number containing a decimal Quantitative: the answer is measurable, can vary, and may be expressed as a decimal range Categorical: the answer will fall into a distinct classification

Steps to making a Histogram

Step 1: Put the numbers in numerical order. Step 2: Determine the median. Step 3: Find Quartile 1 or lower quartile. Step 4: Find Quartile 3 or upper quartile. Step 5: Create the box plot.

Forming and Designing a Plan—Step 2

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 the data are 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 or observational studies. Experiments: An experiment is 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.

Histogram Example

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. This data should be represented with a histogram because a histogram has quantitative data that is presented in ranges. The tree heights can be broken up into intervals of 5 feet per bin in each range. Decide on possible x-axis values and y-axis values for a graph that can be constructed using these heights. Histogram; x-axis range: 60-65, 65-70, 70-75, 75-80, 80-85, 85-90 y-axis: 2, 3, 7, 12, 8, 6

Analyzing and Interpreting Data—Steps 3 and 4

The remaining two steps of the statistical process: Step 3: Analyze the data using graphical and numerical methods Step 4: Interpret the data in relation to the original question These steps are used to summarize the data and make sense of the findings. If the data aren't analyzed, interpreted, or presented well, the information can be confusing, misleading, or even useless for making informed decisions. Keep in mind, representing data is about telling a story, and pictures or graphical representations do a remarkable job of getting the point across. Let's learn the first method to represent data. Representing Data Using Dot Plots: One of the ways a set of data can be represented is by plotting dots over a number line. This is called a dot plot. In a dot plot, the number of dots over a number line tells how many times that number occurs in the data.

Lower Quartile

To find the lower quartile, take the lower half of the values. Do not include the median. Count in from both sides until you get to the middle number. This is the lower quartile, which is the median of the lower half of the data.

Median

To find the median, count in from both sides until you get to the middle value. This is the median.

Quantitative Dot Plots Example

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. Select the audio to see how to make a dot plot. 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 https://lti.flvsgl.com/flvs-cat-content/bqnasutj3mnjslcgs3shvt5epv/flvs-cat-session/temp_project_03/module05/lesson01/images/05_01_03a.png

Maximum

With the data in order, the maximum is the last value in the data set. The greatest possible amount or degree

Minimum

With the data in order, the minimum is the first number in the data set. The smallest possible amount

Upper Quartile

You follow the same steps as the lower quartile, but with the upper half of the values. Take the upper half of the values. Do not include the median. Count in from both sides until you get to the middle value. This is the upper quartile, which is the median of the upper half of the data.