Chapter 2: Frequency Chapter 2: Frequency Distributions and GraphsDistributions and Graphs

Organizing DataOrganizing Data• Raw dataRaw data are data in original form.

• A frequency distributionfrequency distribution is the organization of raw data in table form, using classes and frequencies.

• The frequencyfrequency is the number of values in a specific class of the distribution.

• The categorical frequencycategorical frequency distribution is used for data that can be placed in specific categories, such as nominal or ordinal data.

Birth Month - DataBirth Month - DataSeptember May July October July

August May May March September

April August June July February

January July January November April

July March October December July

October January December May March

March February November April August

July October February April June

August August November July September

December August July September October

August December April September September

June November November July July

December November May August August

December January September May

Grouped Frequency Grouped Frequency DistributionDistribution

• When the rangerange of the data is large, the data must be grouped into classes that are more than one unit in width.

• The lower class limitlower class limit is the smallest data value that can be included in the class.

• The upper class limitupper class limit is the largest data value that can be included in the class.

• The class boundariesclass boundaries are used to separate the classes so that there are no gaps in the frequency distribution.

Grouped Frequency Grouped Frequency DistributionDistribution

• The class limits should have the decimal place value as the data, but the class boundaries should have one additional place value and end in a 5.

• The class width is found by subtracting the lower ( or upper) class limit of one class from the lower ( or upper) class limit of the other class. The class width can also be found by subtracting the lower boundary from the upper boundary.

• The class width cannot be found by subtracting the limits of a single class.

Rules for Constructing aRules for Constructing aFrequency DistributionFrequency Distribution

• There should be between 5 and 20 classes.• The class width should be an odd number. This

ensures that the midpoint of each class has the same place value as the limits.

– The class midpoint is obtained by adding the lower and upper boundaries and dividing by 2, or adding the lower and upper limits and dividing by 2.

• The classes must be mutually exclusive.• The classes must be continuous. Do not omit

classes with a frequency of zero unless they occur at the beginning or ending of the distribution.

• The classes must be exhaustive.• The classes must be equal in width.

How to Construct a How to Construct a Frequency DistributionFrequency Distribution

• To determine the classes– Find the highest and lowest values

– Find the range: Range=R = highest – lowest

– Width = Range/number of classes (rounded up)

– Select a starting point for the first class limit. This can be the smallest data value or any convenient number less than the smallest data value.

Example: Class Age DataExample: Class Age Data

• Construct a grouped frequency distribution using 7 classes.

23 19 19 22 20 18 32 27

21 22 22 19 25 20 26 19

19 20 21 22 26 21 18 27

19 27 20 20 23 39 30 21

19 20 49 20 34 24 30 24

22 17 20 19 19 28 27 19

27 24 21 21 20 19 34

19 20 22 25 22 20 20

19 20 24 24 26 20 26

23

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Class Class

Classes Boundaries Tally Frequency Frequency

• Cumulative frequenciesCumulative frequencies are used to show how many data values are accumulated up to and including a specific class.

• When the range of data values is relatively small a frequency distribution can be constructed using single data values for each class. This type of distribution is called an ungrouped ungrouped frequency distributionfrequency distribution.

Example: BUN CountExample: BUN Count

• Example: The blood urea nitrogen (BUN) count of 20 randomly selected patients is given here in milligrams per deciliter. Construct an ungrouped frequency distribution for the data.

17 18 13 14

12 17 11 20

13 18 19 17

14 16 17 12

16 15 19 22

Reasons For Constructing a Reasons For Constructing a Frequency DistributionFrequency Distribution

• To organize the data in a meaningful, intelligible way.

• To enable the reader to determine the nature and shape of the distribution

• To facilitate computational procedures for measures of average and spread

• To enable the researcher to draw charts and graphs for the presentation of data.

• To enable the reader to make comparisons among different data sets.

2.3 Histograms, Frequency 2.3 Histograms, Frequency Polygons, and OgivesPolygons, and Ogives

• The histogramhistogram is a graph that displays the data by using contiguous vertical bars (unless the frequency of a class is 0) of various heights to represent the frequencies of the classes.

• Step 1. Draw and label the x and y axes.• Step 2. Represent the frequency on the y axis

and the class boundaries on the x-axis.• Step 3. Using the frequencies as the heights,

draw vertical bars for each class.

Frequency PolygonFrequency Polygon

• The frequency polygonfrequency polygon is a graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequencies are represented by the heights of the points. Distribution of 69 Student Ages

0

5

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17 22 27 32 37 42 47

Age

Fre

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ency

OgiveOgive

• The ogiveogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution

• Step 1. Find the cumulative frequency for each class.

• Step 2. Draw the x and y axes. Label the x-axis with the class boundaries.

• Step 3. Plot the cumulative frequency at each upper class boundary.

OgiveOgive

Cumulative Frequency Distribution of Ages of Students

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14.5 19.5 24.5 29.5 34.5 39.5 44.5 49.5

Age

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Relative FrequencyRelative Frequency

• Relative frequency graphs use proportions instead of actual numbers as y-axis values.

• To convert a frequency into a proportion or relative frequency, divide the frequency for each class by the total of frequencies.

• The sum of the relative frequencies will always be one.

• The shape of the graph is the same as those that use frequencies.

Relative FrequencyRelative FrequencyRelative Cumulative

Rel. Cumul.• Classes Frequency Frequency Frequency

Frequency

Distribution ShapesDistribution Shapes

2.4 Other Types of 2.4 Other Types of GraphsGraphs

• A Pareto chartPareto chart is used to represent a frequency distribution for a categorical variable, and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest.

Distribution of Births by Month

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Birth Month

Freq

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Series1

Time Series GraphTime Series Graph

• A time series graph represents data that occur over a specific time period.

Example: Time Series Example: Time Series GraphGraph

• Draw a time series graph to represent the data for the number of airline departures (in millions) for the given years.

Year 1994 1995 1996 1997 1998 1999 2000

No. of Departures

7.5 8.1 8.2 8.2 8.3 8.6 9.0

Airline Departures from 1994 - 2000

6.5

7

7.5

8

8.5

9

9.5

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Year

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s(i

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Pie GraphPie Graph

• A pie graphpie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.

Example: Pie GraphExample: Pie GraphBirth Month Frequency Percentage

DegreesJanuary 4February 3March 4April 5May 6June 3July 11August 9September 7October 5November 6December 6

Stem-and-Leaf PlotStem-and-Leaf Plot

• A stem and leaf plotstem and leaf plot is a data plot that uses part of the data values as the stem and part of the data values as the leaf to form groups or classes.

Example: Stem & Leaf Example: Stem & Leaf PlotPlot

• The following data represents the number of grams of fat in breakfast meals offered at McDonald’s. Construct a stem-and-leaf-plot.

12 23 28 2 31 37 15 23 38

31 16 11 8 17 20 34 8