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CH CHAAPPTTEERR 2 2
Frequency Distributions and Graphs
Ob Objjeeccttiivveess
- Organize data using frequency distributions.
- Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives.
- Represent data using Pareto charts, time series graphs, and pie graphs.
- Draw and interpret a stem and leaf plot.
- Draw and Interpret a scatter plot for a set of paired data.
This chapter will show how to organize data and then construct appropriate graphs to represent the data in a concise, easy-to-understand form.
Se Seccttiioonn 2 (^2) - - 11 OOrrggaanniizziinngg DDaattaa
Ba Bassiicc VVooccaabbuullaarryy
- When data are collected in original form, they are called raw data.
- A frequency distribution is the organization of raw data in table form, using classes and frequencies.
- The two most common distributions are categorical frequency distribution and the grouped frequency distribution.
Fr Freeqquueennccyy DDiissttrriibbuuttiioonnss Categorical Frequency Distributions count how many times each distinct category has occurred and summarize the results in a table format
Example 1: Letter grades for Math 227 Spring 2005: C A B C D F B B A C C F C B D A C C C F C C A A C
a) Construct a frequency distribution for the categorical data.
b) What percentage of the students pass the class with the grade C or better?
Fr Freeqquueennccyy DDiissttrriibbuuttiioonnss
Gr Groouuppeedd FFrreeqquueennccyy DDiissttrriibbuuttiioonnss--When the range of the data is large, the data must be grouped into classes that are more than one unit in width
Ba Bassiicc VVooccaabbuullaarryy The lower class limit represents the smallest value that can be included in the class.
The upper class limit represents the largest value that can be included in the class.
The class boundaries are used to separate the classes so that there are no gaps in the frequency distribution.
Cl Claassss BBoouunnddaarriieess SSiiggnniiffiiccaanntt FFiigguurreess
- Rule of Thumb: Class limits should have the same decimal place value as the data, but the class boundaries have one additional place value and end in a 5.
e.g. data were whole numbers lower class boundary = lower class limit – 0. upper class boundary = upper class limit + 0.
e.g. data were one decimal place lower class boundary = lower class limit – 0. upper class boundary = upper class limit +0.
Cl Claassss MMiiddppooiinnttss The class midpoint (mark) is found by adding the lower and upper boundaries (or limits) and dividing by 2.
Cl Claassss WWiiddtthh The class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the the lower (or upper) class limit of the next class.
Frequency Distributions An ungrouped frequency distribution is used for numerical data and when the range of data is small.
Example: The number of incoming telephone calls per day over the first 25 days of business:
4, 4, 1, 10, 12, 6, 4, 6, 9, 12, 12, 1, 1, 1, 12, 10, 4, 6, 4, 8, 8, 9, 8, 4, 1
Construct an ungrouped frequency distribution
T Tyyppeess ooff FFrreeqquueennccyy DDiissttrriibbuuttiioonnss (^) ((ssuummmmaarryy))
A ccaatteeggoorriiccaall ffrreeqquueennccyy ddiissttrriibbuuttiioonn is used when the data is nominal.
- A grgroouuppeedd ffrreeqquueennccyy ddiissttrriibbuuttiioonn is used when the range is large and classes of several units in width are needed.
- An ununggrroouuppeedd ffrreeqquueennccyy ddiissttrriibbuuttiioonn is used for numerical data and when the range of data is small.
Wh Whyy CCoonnssttrruucctt FFrreeqquueennccyy DDiissttrriibbuuttiioonnss??
- To organize the data in a meaningful, intelligible way.
- To enable the reader to make comparisons among different data sets.
- To facilitate computational procedures for measures of average and spread.
- To enable the reader to determine the nature or shape of the distribution.
- To enable the researcher to draw charts and graphs for the presentation of data.
Se Seccttiioonn 2 (^2) - - 22 HHiissttooggrraamm,, FFrreqequueennccyy (^) PPoollyyggoonnss,, OOggiivveess
This section will show how to organize data and then construct appropriate graphs to represent the data in a concise, easy-to-understand form.
Th Thee RRoollee ooff GGrraapphhss
- The purpose of graphs in statistics is to convey the data to the viewer in pictorial form.
- Graphs are useful in getting the audience’s attention in a publication or a presentation.
Th Thrreeee MMoosstt CCoommmmoonn GGrraapphhss
- The histogram displays the data by using vertical bars of various heights to represent the frequencies.
- The frequency polygon displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes.
- The cumulative frequency polygon or ogive represents the cumulative frequencies for the classes in a frequency distribution.
d) Construct an ogive. (x-axis: class boundaries; y-axis: cumulative frequency)
e) Construct a (i) relative frequency histogram, (ii) relative frequency polygon, and (iii) relative cumulative frequency ogive.
Di Dissttrriibbuuttiioonn sshhaappeess
Se Seccttiioonn 2 (^2) - - 33 OOtthheerr TTyyppeess ooff GGrraapphhss
A Pareto chart is used to represent a frequency distribution for categorical variable, and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest. (x-axis: categorical variables; y-axis: frequencies, which are arranged in order from highest to lowest)
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.
Example 1: Grade received for Math 227 C A B B D C C C C B B A F F a) Construct a pareto chart.
b) Construct a pie chart.
Example 2 : Use the data in example 1 to construct a double stem and leaf plot. e.g. split each stem into two parts, with leaves 0 – 4 on one part and 5 – 9 on the other.
A stem-and-leaf plot portrays the shape of a distribution and maintains the original data values. It is also useful for spotting outliers. Outliers are data values that are extremely large or extremely small in comparison to the norm.
22 - - 44 PaPaiirreedd DDaattaa aanndd SSccaatttteerr PPlloottss
- Many times researchers are interested in determining if a relationship between two variables exist.
- To do this, the researcher collects data consisting of two measures that are paired with another.
- The variable first mentioned is called the independent variable ; the second variable is the dependent variable.
Scatter Plot – is a graph of order pairs of data values that is used to determine if a relationship exists between two variables.
An Anaallyyzziinngg tthhee SSccaatttteerr PPlloott
- A popossiittiivvee lliinneeaarr rreellaattiioonnsshhiipp exists when the points fall approximately in an ascending straight line and both the x and y values increase at the same time.
- A neneggaattiivvee lliinneeaarr rreellaattiioonnsshhiipp exists when the points fall approximately in a straight line descending from left to right.
- A nononnlliinneeaarr rreellaattiioonnsshhiipp exists when the points fall along a curve.
- (^) NNoo rreellaattiioonnsshhiipp exists when there is no discernable pattern of the points.
Examples of Scatter Plots and Relationships
Example 1: A researcher wishes to determine if there is a relationship between the number of days an employee missed a year and the person’s age. Draw a scatter plot and comment on the nature of the relationship.
Age (x) 22 30 25 35 65 50 27 53 42 58
Days missed (y) 0 4 1 2 14 7 3 8 6 4
S Sumummmaarryy ooff GGrraapphhss aanndd UUsseess
• HHiissttooggrraammss ,, frfreeqquueennccyy ppoollyyggoonnss ,, and ogogiivveess are used when the data are contained in
a grouped frequency distribution.
• PPaarreettoo cchhaarrttss are used to show frequencies for nominal variables.
• TTiimmee sseerriieess ggrraapphhss are used to show a pattern or trend that occurs over time.
• PPiiee ggrraapphhss are used to show the relationship between the parts and the whole.
• When data are collected in pairs, the relationship, if one exists, can be determined by
looking at a ssccaatttteerr pplloott.
Co Conncclluussiioonnss
• Data can be organized in some meaningful way using frequency distributions. Once
the frequency distribution is constructed, the representation of the data by graphs is a simple task.
