A variable [Typically called a “random” variable since we do not know it’s value until we observe it] is some characteristic of a population or sample.
E.g. student grades, weight of a potato, # heads in 10 flips of a coin, etc.
Typically denoted with a capital letter: X, Y, Z…
The values of the variable are the range of possible values for a variable.
E.g. student marks (0..100)
Data are the observed values of a random variable.
Continuous Data – Data can be any real number within a given range. Normally measurement data [weights, Age, Prices, etc]
Discrete Data – Data can only be very specific values which we can list. Normally count data [# of firecrackers in a package of 100 that fail to pop, # of accidents on the UTA campus each week, etc]
Nominal Data [ has no natural order to the values].
E.g. responses to questions about marital status: Single = 1, Married = 2, Divorced = 3, Widowed = 4
Arithmetic operations don’t make any sense (e.g. does Widowed ÷ 2 = Married?!)
Ordinal Data [values have a natural order ] :
E.g. College course rating system: poor = 1, fair = 2, good = 3, very good = 4, excellent = 5
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Graphical & Tabular Techniques for Nominal Data…
The only allowable calculation on nominal data is to count the frequency of each value of the variable.
We can summarize the data in a table that presents the categories and their counts called a frequency distribution.
A relative frequency distribution lists the categories and the proportion with which each occurs.
Since Nominal data has no order, if we arrange the outcomes from the most frequently occurring to the least frequently occurring, we call this a “pareto chart”
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Nominal Data (Frequency) 2. Bar Charts are often used to display frequencies… Is there a better way to order these? Would Bar Chart look different if we plotted “relative frequency” rather than “frequency”?
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Nominal Data (Relative Frequency) 2. Pie Charts show relative frequencies…
A sample of 30 employees from large companies was selected, and these employees were asked how stressful their jobs were. The responses of these employees are recorded next where very represents very stressful, somewhat means somewhat stressful, and none stands for not stressful at all.
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Example 2.2 Construct a frequency distribution table for these data. Some what None Somewhat Very Very None Very Somewhat Somewhat Very Somewhat Somewhat Very Somewhat None Very None Somewhat Somewhat Very Somewhat Somewhat Very None Somewhat Very very somewhat None Somewhat
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Solution 2.2 Table 2.2 Frequency Distribution of Stress on Job Stress on Job Tally Frequency ( f ) Very Somewhat None |||| |||| |||| |||| |||| |||| | 10 14 6 Sum = 30
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Relative Frequency and Percentage Distributions
Calculating Relative Frequency of a Category
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Relative Frequency and Percentage Distributions cont.
A graph made of bars whose heights represent the frequencies of respective categories is called a bar graph .
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Figure 2.2 Bar graph for the frequency distribution of Table 2.3
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Graphical Presentation of Qualitative Data cont.
Definition
A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories is called a pie chart .
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Table 2.4 Calculating Angle Sizes for the Pie Chart Stress on Job Relative Frequency Angle Size Very Somewhat None .333 .467 .200 360(.333) = 119.88 360(.467) = 168.12 360(.200) = 72.00 Sum = 1.00 Sum = 360
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Figure 2.4 Pie chart for the percentage distribution of Table 2.5.
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Frequency Distributions Table 2.7 Weekly Earnings of 100 Employees of a Company Variable Third class Lower limit of the sixth class Upper limit of the sixth class Frequency of the third class Frequency column Weekly Earnings (dollars) Number of Employees f 401 to 600 601 to 800 801 to 1000 1001 to 1200 1201 to 1400 1401 to 1600 9 22 39 15 9 6
A frequency distribution for quantitative data lists all the classes and the number of values that belong to each class. Data presented in the form of a frequency distribution are called grouped data .
STEP 4. Write the classes or categories starting with the lowest score. Stop when the class already includes the highest score.
Add the class width to the starting point to get the second lower class limit. Add the class width to the second lower class limit to get the third, and so on. List the lower class limits in a vertical column and enter the upper class limits, which can be easily identified at this stage.
Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week. Here are the ages of 50 men randomly selected from that population. Construct a frequency distribution table.
Table 2.9 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season. Construct a frequency distribution table.
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Table 2.9 Home Runs Hit by Major League Baseball Teams During the 2002 Season Team Home Runs Team Home Runs Anaheim Arizona Atlanta Baltimore Boston Chicago Cubs Chicago White Sox Cincinnati Cleveland Colorado Detroit Florida Houston Kansas City Los Angeles 152 165 164 165 177 200 217 169 192 152 124 146 167 140 155 Milwaukee Minnesota Montreal New York Mets New York Yankees Oakland Philadelphia Pittsburgh St. Louis San Diego San Francisco Seattle Tampa Bay Texas Toronto 139 167 162 160 223 205 165 142 175 136 198 152 133 230 187
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Solution 2-3 Now we round this approximate width to a convenient number – say, 22.
The lower limit of the first class can be taken as 124 or any number less than 124. Suppose we take 124 as the lower limit of the first class. Then our classes will be
124 – 145, 146 – 167, 168 – 189, 190 – 211,
and 212 - 233
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Table 2.10 Frequency Distribution for the Data of Table 2.9 Total Home Runs Tally f 124 – 145 146 – 167 168 – 189 190 – 211 212 - 233 |||| | |||| |||| ||| |||| |||| ||| 6 13 4 4 3 ∑ f = 30
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Relative Frequency and Percentage Distributions
Calculate the relative frequencies and percentages for Table 2.10
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Solution 2-4 Table 2.11 Relative Frequency and Percentage Distributions for Table 2.10 Total Home Runs Class Boundaries Relative Frequency Percentage 124 – 145 146 – 167 168 – 189 190 – 211 212 - 233 123.5 to less than 145.5 145.5 to less than 167.5 167.5 to less than 189.5 189.5 to less than 211.5 211.5 to less than 233.5 .200 .433 .133 .133 .100 20.0 43.3 13.3 13.3 10.0 Sum = .999 Sum = 99.9%
A histogram is a graph in which classes are marked on the horizontal axis and the frequencies, relative frequencies, or percentages are marked on the vertical axis. The frequencies, relative frequencies, or percentages are represented by the heights of the bars. In a histogram, the bars are drawn adjacent to each other.
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Figure 2.3 Frequency histogram for Table 2.10. 124 - 145 146 - 167 168 - 189 190 - 211 212 - 233 Total home runs 15 12 9 6 3 0 Frequency
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Figure 2.4 Relative frequency histogram for Table 2.10. 124 - 145 146 - 167 168 - 189 190 - 211 212 - 233 Total home runs .50 .40 .30 .20 .10 0 Relative Frequency
The following data give the average travel time from home to work (in minutes) for 50 states. The data are based on a sample survey of 700,000 households conducted by the Census Bureau (USA TODAY, August 6, 2001).
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Example 2-5 Construct a frequency distribution table. Calculate the relative frequencies and percentages for all classes. 22.4 19.7 21.6 15.4 21.1 18.2 27.0 21.9 22.1 25.4 23.7 21.7 23.2 19.6 24.9 19.8 17.6 16.0 21.4 25.5 26.7 17.7 16.1 23.8 20.1 23.4 22.5 22.3 21.9 17.1 23.5 23.7 24.4 21.9 22.5 21.2 28.7 15.6 24.3 29.2 19.9 22.7 26.7 26.1 31.2 23.6 24.2 22.7 22.6 20.8
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Solution 2-5 Table 2.12 Frequency, Relative Frequency, and Percentage Distributions of Average Travel Time to Work Class Boundaries f Relative Frequency Percentage 15 to less than 18 18 to less than 21 21 to less than 24 24 to less than 27 27 to less than 30 30 to less than 33 7 7 23 9 3 1 .14 .14 .46 .18 .06 .02 14 14 46 18 6 2 Σ f = 50 Sum = 1.00 Sum = 100%
The administration in a large city wanted to know the distribution of vehicles owned by households in that city. A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned:
5 1 1 2 0 1 1 2 1 1
1 3 3 0 2 5 1 2 3 4
2 1 2 2 1 2 2 1 1 1
4 2 1 1 2 1 1 4 1 3
Construct a frequency distribution table for these data, and draw a bar graph.
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Solution 2-6 Table 2.13 Frequency Distribution of Vehicles Owned Vehicles Owned Number of Households ( f ) 0 1 2 3 4 5 2 18 11 4 3 2 Σ f = 40
In a stem-and-leaf display of quantitative data, each value is divided into two portions – a stem and a leaf. The leaves for each stem are shown separately in a display.
To construct a stem-and-leaf display for these scores, we split each score into two parts. The first part contains the first digit, which is called the stem. The second part contains the second digit, which is called the leaf.
After we have listed the stems, we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line.
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