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155S2.12 Frequency Distributions
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Chapter 2Summarizing and Graphing Data
MAT 155Dr. Claude Moore
Cape Fear Community College
21 Review and Preview22 Frequency Distributions23 Histograms24 Statistical Graphics25 Critical Thinking: Bad Graphs
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1. Center: A representative or average value that indicates where the middle of the data set is located.2. Variation: A measure of the amount that the data values vary.3. Distribution: The nature or shape of the spread of data over the range of values (such as bellshaped, uniform, or skewed).
Preview
4. Outliers: Sample values that lie very far away from the vast majority of other sample values.5. Time: Changing characteristics of the data over time.
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Key Concept When working with large data sets, it is often helpful to organize and summarize data by constructing a table called a frequency distribution, defined later. Because computer software and calculators can generate frequency distributions, the details of constructing them are not as important as what they tell us about data sets. It helps us understand the nature of the distribution of a data set.
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• Frequency Distribution• (or Frequency Table)
shows how a data set is partitioned among all of several categories (or classes) by listing all of the categories along with the number of data values in each of the categories.
Definition
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Pulse Rates of Females and MalesOriginal Data
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Frequency Distribution Pulse Rates of Females
The frequency for a particular class is the number of original values that fall into that class.
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Lower
are the smallest numbers that can actually belong to different classes
Lower Class Limits22 Frequency Distributions
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Upper
Upper Class Limits are the largest numbers that can actually belong to different classes
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ClassBoundaries
59.569.579.589.599.5109.5119.5129.5
are the numbers used to separate classes, but without the gaps created by class limits
Class Boundaries
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Class Midpointsare the values in the middle of the classes and can be found by adding the lower class limit to the upper class limit and dividing the sum by two.
ClassMidpoints
64.574.584.594.5104.5114.5124.5
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Class Widthis the difference between two consecutive lower class limits or two consecutive lower class boundaries.
Class Width
101010101010
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1. Large data sets can be summarized.2. We can analyze the nature of data.3. We have a basis for constructing important graphs.
Reasons for Constructing Frequency Distributions
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Suggested technique to find class width:2k Rule
k = suggested number of classesn = sample size or total number of data values
Find the smallest whole number, k, such that 2k > n.
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3. Starting point: Choose the minimum data value or a convenient value below it as the first lower class limit.Using the first lower class limit and class width, proceed to list the other lower class limits.5. List the lower class limits in a vertical column and proceed to enter the upper class limits.6. Take each individual data value and put a tally mark in the appropriate class. Add the tally marks to get the frequency.
Constructing A Frequency Distribution1. Determine the number of classes (should be between 5 and 20).2. Calculate the class width (round up).
class width =(maximum value) – (minimum value)
number of classes
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Relative Frequency Distribution
relative frequency =class frequency
sum of all frequencies
includes the same class limits as a frequency distribution, but the frequency of a class is replaced with a relative frequencies (a proportion) or a percentage frequency ( a percent).
percentage frequencyclass frequency
sum of all frequencies=
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Relative Frequency Distribution
* (12/40)(100%) = 30%
*
Total Frequency = 40
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Cumulative Frequency Distribution
Cum
ulative Frequencies
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Frequency Tables22 Frequency Distributions
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Critical Thinking Interpreting Frequency Distributions
In later chapters, there will be frequent reference to data with a normal distribution. One key characteristic of a normal distribution is that it has a “bell” shape.The frequencies start low, then increase to one or two high frequencies, then decrease to a low frequency.
The distribution is approximately symmetric, with frequencies preceding the maximum being roughly a mirror image of those that follow the maximum.
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GapsGapsThe presence of gaps can show that we have data from two or more different populations.
However, the converse is not true, because data from different populations do not necessarily result in gaps.
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RecapIn this Section we have discussed• Important characteristics of data• Frequency distributions• Procedures for constructing frequency distributions• Relative frequency distributions• Cumulative frequency distributions
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22 Frequency Distributions59/6. In Exercises 5– 8, identify the class width, class midpoints, and class boundaries for the given frequency distribution. The frequency distributions are based on data from Appendix B.
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22 Frequency Distributions60/8. In Exercises 5– 8, identify the class width, class midpoints, and class boundaries for the given frequency distribution. The frequency distributions are based on data from Appendix B.
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22 Frequency Distributions61/19. Nicotine in Nonfiltered Cigarettes Refer to Data Set 4 in Appendix B and use the 25 nicotine amounts (in mg) listed for the nonfiltered kingsize cigarettes. Construct a frequency distribution. Begin with a lower class limit of 1.0 mg, and use a class width of 0.20 mg.
Use Statdisk to create a Histogram per the instructions in Exercise 19. Turn labels on to get the frequency for each bar (class). Construct the frequency distribution using the information in the histogram.
Class Frequency1.0 1.2 141.2 1.4 41.4 1.6 31.6 1.8 31.8 2.0 1
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22 Frequency Distributions61/20. Nicotine in Filtered Cigarettes Refer to Data Set 4 in Appendix B and use the 25 nicotine amounts (in mg) listed for the filtered and nonmenthol cigarettes. Construct a frequency distribution. Begin with a lower class limit of 0.2 mg, and use a class width of 0.20 mg. Compare the frequency distribution to the result from Exercise 19.
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