GS ECO 302 CH 2 Describing Data: Frequency Tables, Frequency Distributions, and Graphic Presentation

Frequency Distribution: Step 1. A useful recipe to determine the number of classes (k) is the

"2 to the k rule." This guide suggests you select the smallest number (k) for the number of classes such that 2k (in words, 2 raised to the power of k) is greater than the number of observations (n). Example, there were 180 vehicles sold. So n =180. If we try k 7, which means we would use 7 classes, 2^7= 128, which is less than 180. Hence, 7 is too few classes. If we let k= 8, then 2^8= 256, which is greater than 180. So the recommended number of classes is 8.

raw data or ungrouped data

unorganized information

pie chart Pie charts and bar charts serve much the same function. What are the criteria for selecting one over the other? In most cases, pie charts are the most informative when the goal is to compare the relative difference in the percentage of observations for each of the nominal scale variables.

A chart that shows the proportion or percentage that each class represents of the total number of frequencies.

bar chart (relative frequency bar chart) A graph that shows qualitative classes on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are proportional to the heights of the bars. The variable location is of nominal scale, so the order of the locations on the horizontal axis does not matter. Listing this variable alphabetically or by some type of geographical arrangement might also be appropriate. Bar charts are preferred when the goal is to compare the number of observations in each category.

A chart that represents the count (or percentage) of each category in a categorical variable as a bar, allowing easy visual comparisons across categories. The most common graphic form to present a qualitative variable The vertical axis shows the frequency or fraction of each of the possible outcomes. A distinguishing feature of a bar chart is there is distance or a gap between the bars. That is, because the variable of interest is qualitative, the bars are not adjacent to each other. Thus, a bar chart graphically describes a frequency table using a series of uniformly wide rectangles, where the height of each rectangle is the class frequency.

Class Interval (width)

A group of values that is used to analyze the distribution of data. To determine the class interval, subtract the lower limit of the class from the lower limit of the next class. The class interval of the Applewood data is $400, which we find by subtracting the lower limit of the first class, $200, from the lower limit of the next class; that is, $600. ($600 - $200 = $400.) You can also determine the class interval by finding the difference between consecutive midpoints. The midpoint of the first class is $400 and the midpoint of the second class is $800. The difference is $400.

frequency table

A grouping of qualitative data into mutually exclusive classes showing the number of observations in each class.

class midpoint (Class Lower Limit + Class Upper Limit) / 2

A point that divides a class into two equal parts. This is the average of the upper and lower class limits. The midpoint is halfway between the lower limits of two consecutive classes. It is computed by adding the lower limits of consecutive classes and dividing the result by 2. Referring to Table 2-7, the lower class limit of the first class is $200 and the next class limit is $600. The class midpoint is $400, found by ($600 + $200) / 2. (Class Lower Limit + Class Upper Limit) / 2 The midpoint of $400 best represents, or is typical of, the profits of the vehicles in that class.

relative frequency

A ratio that compares the frequency of each category to the total. captures the relationship between a class total and the total number of observations. To convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of observations. CLASS % OF TOTAL POPULATION

cumulative frequency distribution

A tabular summary of quantitative data showing the number of data values that are less than or equal to the upper class limit of each class.

Cumulative Frequency Polygon OGIVE CURVE A graph of all quantities of a numerical variable. Measurements are sorted from smallest to largest. Displays cumulative frequency for continuous variables

To plot a cumulative frequency distribution • scale the upper limit of each class along the X-axis • and the corresponding cumulative frequencies along the Y-axis. • To provide additional information, you can label the vertical axis on the left in units and the vertical axis on the right in percent. To begin, the first plot is at X = 200 and Y = 0. None of the vehicles sold for a profit of less than $200.

Frequency Distribution: Step 3: Set the individual class limits. State clear class limits so you can put each observation into only one category. For example, classes such as $1,300-$1,400 $1,400-$1,500 should not be used,

Classes stated as $1,300-$1,400 $1,500-$1,600 are frequently used, but may also be confusing without the additional common convention of rounding all data at or above $1,450 up to the second class and data below $1,450 down to the first class. In this text, we will generally use the format $1,300 up to $1,400 $1,400 up to $1,500 and so on. With this format it is clear that $1,399 goes into the first class and $1,400 in the second.

frequency distribution A grouping of data into mutually exclusive classes showing the number of observations in each class.

How do we develop a frequency distribution? The first step is to tally the data into a table that shows the classes and the number of observations in each class. Step 1: Decide on the number of classes. The goal is to use just enough groupings or classes to reveal the shape of the set of observations. 2 to the K rule Step 2: Determine the class interval or class width. Generally the class interval or class width is the same for all classes. I >= (H-L) / K Step 3: Set the individual class limits. State clear class limits so you can put each observation into only one category. Step 4: Tally the vehicle profit into the classes. Step 5: Count the number of items in each class.

Learning Objectives When you have completed this chapter, you will be able to:

LO1 Make a frequency table for a set of data. LO2 Organize data into a bar chart. LO3 Present a set of data in a pie chart. LO4 Create a frequency distribution for a data set. LO5 Understand a relative frequency distribution. LO6 Present data from a frequency distribution in a histogram or frequency polygon. LO7 Construct and interpret a cumulative frequency distribution.

Frequency Distribution: Step 3: Set the individual class limits. (continued) Because we round the class interval up to get a convenient class size, we cover a larger than necessary range. For example, using 8 classes with a width of $400 in the Applewood Auto Group example results in a range of 8($400) =$3,200.

The actual range is $2,998, found by ($3,292-$294). Comparing that value to $3,200, we have an excess of $202. Because we need to cover only the distance (H-L), it is natural to put approximately equal amounts of the excess in each of the two tails. Note: The excess was $200, so start the first class at $200; use a class (interval) of $400 and create 8 classes (from previous formulas).

Histogram • A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. • The class frequencies are represented by the heights of the bars, and the bars are drawn adjacent to each other. the histogram has the advantage of depicting each class as a rectangle, with the height of the rectangular bar representing the number in each class.

a frequency distribution based on quantitative data is similar to the bar chart showing the distribution of qualitative data. However, there is one important difference based on the nature of the data. Quantitative data are usually measured using scales that are continuous, not discrete. Therefore, the horizontal axis represents all possible values, and the bars are drawn adjacent to each other to show the continuous nature of the data.

frequency polygon also shows the shape of a distribution and is similar to a histogram. It consists of • line segments connecting the points formed by the intersections of the class midpoints and the class frequencies. Note in Chart 2-5 that, to complete the frequency polygon, midpoints of $0 and $3,600 are added to the X-axis to "anchor" the polygon at zero frequencies. These two values, $0 and $3,600, were derived by subtracting the class interval of $400 from the lowest midpoint ($400) and by adding $400 to the highest midpoint ($3,200) in the frequency distribution. (where possible) The frequency polygon, in turn, has an advantage over the histogram. It allows us to compare directly two or more frequency distributions.

graph of a frequency distribution that shows the number of instances of obtained scores, usually with the data points connect by straight lines The midpoint of each class is scaled on the X-axis and the class frequencies on the Y-axis. Recall that the class midpoint is the value at the center of a class and represents the typical values in that class. The X and the Y values of this point are called the coordinates.

Frequency Distribution: Step 2. class interval or class width The classes all taken together must cover at least the distance from the lowest value in the data up to the highest value.

i >= (H-L) / k where i is the class interval, H is the highest observed value, L is the lowest observed value, and k is the number of classes. Example For the Applewood Auto Group, the lowest value is $294 and the highest value is $3,292. If we need 8 classes, the interval should be: i = ($3,292-$294)/8 = $374.75 In practice, this interval size is usually rounded up to some convenient number, such as a multiple of 10 or 100. The value of $400 is a reasonable choice.

descriptive statistics

statistics that summarize the data collected in a study organize data to show the general pattern of the data and where values tend to concentrate and to expose extreme or unusual data values

class frequency

the number of observations in the data set falling into a particular mutually exclusive (distinctive) class COUNT

In frequency distributions, equal class intervals are preferred. However,

unequal class intervals may be necessary in certain situations to avoid a large number of empty, or almost empty, classes. Such is the case in Table 2-6. The Internal Revenue Service used unequal-sized class intervals to report the adjusted gross income on individual tax returns. Had they used an equal-sized interval of, say, $1,000, more than 1,000 classes would have been required to describe all the incomes.

The frequency polygon, in turn, has an advantage over the histogram. It allows us to compare directly two or more frequency distributions.

• The typical vehicle profit is larger at Fowler Motors—about $2,000 for Applewood and about $2,400 for Fowler. • There is less dispersion in the profits at Fowler Motors than at Applewood. The lower limit of the first class for Applewood is $0 and the upper limit is $3,600. For Fowler Motors, the lower limit is $800 and the upper limit is the same: $3,600.