Methods for Describing Quantitative Data Histograms Frequency Distribution Methods for Describing Quantitative Data

2-2 Frequency Distributions A frequency distribution is the organizing of raw data in table form, using classes and frequencies. The following slide shows an example of a frequency distribution.

2-2 Blood Type Frequency Distribution - Example

2-2 Ungrouped Frequency Distributions 2-9 Ungrouped frequency distributions - can be used for data that can be enumerated and when the range of values in the data set is not large. Examples - number of miles your instructors have to travel from home to campus, number of girls in a 4-child family etc.

2-2 Grouped Frequency Distributions Grouped frequency distributions - can be used when the range of values in the data set is very large. The data must be grouped into classes that are more than one unit in width. Examples - the life of boat batteries in hours.

2-2 Lifetimes of Boat Batteries - Example Class Class Frequency Cumulative limits Boundaries frequency 24 - 30 23.5 - 37.5 4 4 38 - 51 37.5 - 51.5 14 18 52 - 65 51.5 - 65.5 7 25

2-2 Terms Associated with a Grouped Frequency Distribution Class limits represent the smallest and largest data values that can be included in a class. In the lifetimes of boat batteries example, the values 24 and 30 of the first class are the class limits. The lower class limit is 24 and the upper class limit is 30.

2-2 Terms Associated with a Grouped Frequency Distribution 2-14 The class boundaries are used to separate the classes so that there are no gaps in the frequency distribution.

2-2 Terms Associated with a Grouped Frequency Distribution The class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class minus the lower (or upper) class limit of the previous class.

There should be between 5 and 20 classes. 2-2 Guidelines for Constructing a Frequency Distribution There should be between 5 and 20 classes. The class width should be an odd number. The classes must be mutually exclusive.

The classes must be continuous. The classes must be exhaustive. 2-2 Guidelines for Constructing a Frequency Distribution The classes must be continuous. The classes must be exhaustive. The class must be equal in width.

Find the upper class limits. Find the boundaries. 2-2 Procedure for Constructing a Grouped Frequency Distribution Select a starting point (usually the lowest value); add the width to get the lower limits. Find the upper class limits. Find the boundaries. Tally the data, find the frequencies and find the cumulative frequency.

2-2 Grouped Frequency Distribution - Example In a survey of 20 patients who smoked, the following data were obtained. Each value represents the number of cigarettes the patient smoked per day. Construct a frequency distribution using six classes. (The data is given on the next slide.)

2-2 Grouped Frequency Distribution - Example

2-22 Step 1: Find the highest and lowest values: H = 22 and L = 5. 2-2 Grouped Frequency Distribution - Example 2-22 Step 1: Find the highest and lowest values: H = 22 and L = 5. Step 2: Find the range: R = H – L = 22 – 5 = 17. Step 3: Select the number of classes desired. In this case it is equal to 6.

2-2 Grouped Frequency Distribution - Example Step 4: Find the class width by dividing the range by the number of classes. Width = 17/6 = 2.83. This value is rounded up to 3.

2-2 Grouped Frequency Distribution - Example Step 5: Select a starting point for the lowest class limit. For convenience, this value is chosen to be 5, the smallest data value. The lower class limits will be 5, 8, 11, 14, 17 and 20.

Table 1. EPA Mileage Rating on 100 Cars* 36.3 41.0 36.9 37.1 44.9 36.8 30.0 37.2 42.1 36.7 32.7 37.3 41.2 36.6 32.9 36.5 33.2 37.4 37.5 33.6 40.5 37.6 33.9 40.2 36.4 37.7 40.0 34.2 36.2 37.9 36.0 35.9 38.2 38.3 35.7 35.6 35.1 38.5 39.0 35.5 34.8 38.6 39.4 35.3 34.4 38.8 39.7 32.5 36.1 38.4 39.3 31.8 33.1 37.0 38.7 35.8 40.7 37.8 34.5 40.3 40.1 38.0 35.2 39.5 39.9 33.8 39.8 34.0 35.0 38.1

Mean SAT I Math Scores by State (2003) Data Source: The College Board, New York, NY 515 501 491 518 543 576 578 613 514 502 504 519 510 549 562 588 522 498 516 540 552 559 496 525 591 594 474 512 527 517 560 582 554 521 500 541 597 506 532 553 596 583 551

Measurement Class (BIN) Measurement Classes, Frequencies, and Relative Frequencies for the SAT Math Scores Measurement Class (BIN) Frequency Relative Frequency  500 5 .098 500 – 510 9 .176 510 – 520 8 .157 520 – 530 4 .078 530 – 540 3 .059 540 – 550 550 – 560 560 – 570 1 .020 570 – 580 2 .039 580 – 590 590 – 600 Greater than 600 Totals 51 1.00

Assignment

2-3 Visualizing Data The three most commonly used graphs in research are: The histogram. The frequency polygon. The cumulative frequency graph, or ogive (pronounced o-jive).

Histogram Example (SAT Math Scores)

Histograms Histograms condense the data by grouping similar data values into the same class (bin) in the graph. Summarize Data Sort Min /Max Range (Max – Min) Mean (Average values) Group Plot

Frequency Versus Relative Frequency

2-2 Number of Miles Traveled - Example

2-2 Lifetimes of Boat Batteries - Example Class Class Frequency Cumulative limits Boundaries frequency 24 - 30 23.5 - 37.5 4 4 38 - 51 37.5 - 51.5 14 18 52 - 65 51.5 - 65.5 7 25

2-3 Visualizing Data A frequency polygon is a graph that displays the data by using lines that connect points plotted for frequencies at the midpoint of classes. The frequencies represent the heights of the midpoints.

Example of a Frequency Polygon 6 5 y c n 4 e u q 3 e r F 2 1 2 5 8 1 1 1 4 1 7 2 2 3 2 6 N u m b e r o f C i g a r e t t e s S m o k e d p e r D a y

2-3 Visualizing Data A cumulative frequency graph or ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.

Example of an Ogive Ogive

2-3 Other Types of Graphs 2-41 Pie graph - A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.

2-3 Other Types of Graphs - Pie Graph Robbery (29, 12.1%) Pie Chart of the Number of Crimes Investigated by Law Enforcement Officers In U.S. National Parks During 1995 Rape (34, 14.2%) Homicide (13, 5.4%) Assaults (164, 68.3%)

Tabulation of data Stub Headings Sub head Sub Head Table Number and Title ( Head or Prefatory Note ) Stub Headings Caption Sub head Sub Head Column Head Column Head Total (rows) Stub Entries Total (Columns) Body or Field Foot Note Source Note