STK110 Chapter 2: Tabular and Graphical Methods Lecture 1 of 2 ritakeller.com mathspig.wordpress.com
Frequency distribution Example Data from a sample of 50 soft drink purchases
Frequency Distribution of soft drink purchases Soft Drink Count Frequency Coke Classic Diet Coke Dr. Pepper Pepsi-Cola Sprite Total
Frequency table
Other Measures
Frequency table Soft drink Coke Classic Frequency 19 Diet Coke 8 Dr. Pepper 5 Pepsi-Cola 13 Sprite 5 Relative frequency 19/50=0.38 8/50=0.16 5/50=0.10 13/50=0.26 5/50=0.10 Percent frequency 38 16 10 26 10 Total 50 1 100
Bar Graph of soft drink purchases Bar Graph of soft drink purchases Frequency 20 18 16 14 12 10 8 6 4 2 0 Coke Classic Diet Coke Dr. Pepper Pepsi-Cola Sprite Soft Drink o x-axis (horizontal axis): The labels that are used for the classes (categories) of data. o y-axis (vertical axis): The frequency, relative frequency, or percent frequency.
The sector of the pie chart labelled Coke Classic consists of 0.38 360 136.8.
Example The time in days required to complete year-end audits for a sample of 20 clients of Sanderson and Clifford, a small public accounting firm. Year-end audit times (in days) 12 14 19 18 15 15 18 17 20 27 22 23 22 21 33 28 14 18 16 13 Take note that the sample size is 20 (n = 20).
Constructing a frequency distribution Number of classes: The number of classes is usually between 5 and 20 classes. For a larger number of data items, a larger number of classes is required. That is, for a large dataset (n 30): 15 20 classes. For a smaller number of data items, as few as 5 or 6 classes may be used to summarize the data. That is, for a small dataset (n < 30): 5 or 6 classes. For our example, since n = 20 (< 30) we will use 5 classes. Take note: You could have worked with 6 classes.
Approximate class width: Approx. class width = Note: Always round up: 4.2~5
Frequency distribution for the audit time data: Audit Time (in days) Frequency 10-14 4 15-19 8 20-24 5 25-29 2 30-34 1 Total 20
Class limits: The lower class limit identifies the smallest possible data value assigned to the class. The upper class limit identifies the largest possible data value assigned to the class. For example, for the 1 st class: Lower class limit = 10 and Upper class limit = 14.
Class midpoints: By adding the lower class limit and the upper class limit and dividing the total by 2, we obtain the class midpoints. 10 14 12 2
Midpoints Audit Time (in days) Frequency Class Midpoint 10-14 4 12 15-19 8 20-24 5 25-29 2 30-34 1 Total 20
Starting point of the 1 st class: Why does the 1 st class start at 10 when the smallest value in the data set is 12? The starting point of the 1 st class could be taken to be the smallest value in the data set or a value just before the smallest value. For this dataset, the starting value of the 1 st class could have been taken to be 10, 11 or 12. For example, if you started at 12, then the frequency distribution would look as follows:
Audit Time (in days) Frequency 12-16 7 17-21 7 22-26 3 27-31 2 32-36 1 Total 20
Relative and percent frequency distributions for the audit time data: Audit Time (in days) Frequency Relative Frequency Percent Frequency 10-14 4 15-19 8 20-24 5 0.25 25 25-29 2 0.10 10 30-34 1 0.05 5 Total 20 1 100
Histogram x-axis (horizontal axis): The variable of interest is placed here y-axis (vertical axis): Frequency, relative frequency, or percent frequency Histogram for the Audit Time Data Frequency 9 8 7 6 5 4 3 2 1 0 10-14 15-19 20-24 25-29 30-34 Audit Time (days)
Symmetric
Moderately Skewed Left
Moderately Skewed Right
Recall that the frequency, relative frequency and percent frequency distributions were given by: Audit time (in days) Frequency Relative frequency Percent frequency 10-14 4 0.20 20 15-19 8 0.40 40 20-24 5 0.25 25 25-29 2 0.10 10 30-34 1 0.05 5 Total 20 1 100
Table A: Cumulative distributions: Audit time (in days) 14 19 24 29 34 Cumulative frequency Cumulative relative frequency Cumulative percent frequency
Audit time (in days) Cumulative frequency Cumulative relative frequency Cumulative percent frequency 14 4 0.2 20 19 12 0.6 60 24 17 0.85 85 29 19 0.95 95 34 20 1.00 100
Questions 1. What % of the audits has an audit-time of at most 24 days? 2. What proportion of the audits has an audit-time of more than 19 days? 3. How many audits have an audit-time of more than 24 days?
How many of the audit-times took more than 19 days AND were less than or equal to 29 days? Audit time (in days) Cumulative frequency Cumulative relative frequency Cumulative percent frequency 14 4 0.2 20 19 12 0.6 60 24 17 0.85 85 29 19 0.95 95 34 20 1.00 100
How many of the audit-times took more than 19 days AND were less than or equal to 29 days?
Audit time (in days) Cumulative percent frequency 14 20 19 60 24 85 29 95 34 100 If this is the only information available, How many audits have a time of above 24 days?
Graph of the cumulative frequency distribution called the OGIVE Ogive for the audit-time data 20 Cumulative Frequency 18 16 14 12 10 8 6 4 2 0 0 5 10 Audit 15Times (in 20 days) 25 30 35 40
Questions on Ogive How many audits took at most 22 days? What percentage of audits had an audit time of more than 22 days? How many audits had an audit time between 16 and 22 days? What is the median value? The upper 25% of values is above days.
The Stem-and-Leaf Display 61 112 73 126 82 92 115 108 86 97 94 106 75 102 118 Construct a Stem-and-Leaf display [Leaf unit = 1] Take note: If the leaf unit is not given, then the leaf unit is equal to one.
Example 1565 1852 1644 1766 1888 1912 1954 2044 1812 1790 1679 2008 1852 1967 1733 Leaf unit = 10
Stem and Leaf Advantages of the Stem-and-Leaf display: Summarizes data quickly. Shows rank order of data. The user can see the shape of the data.
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