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ASU MAT 142 - Statistics

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MAT 142 College Mathematics Module STStatisticsTerri Miller revised December 13, 20101. Population, Sample, and Data1.1. Basic Terms.A population is the set of all objects under study, a sample is any subset of a popultion,and a data point is an element of a set of data.Example 1. Population, sample, data pointPopulation: all ASU studentsSample: 1000 radomly selected ASU studentsdata: 10, 15, 13, 25, 22, 53, 47data point: 13data point: 53The frequency is the number of times a particular data point occurs in the set of data. Afrequency distribution is a table that list each data point and its frequency. The relativefrequency is the frequency of a data point expressed as a percentage of the total numberof data points.Example 2. Frequency, relative frequency, frequency distributiondata: 1, 3, 6, 4, 5, 6, 3, 4, 6, 3, 6frequency of the data point 1 is 1frequency of the data point 6 is 4the relative frequency of the data point 6 is (4/11) × 100% ≈ 36.35%the frequency distribution for this set of data is: (where x is a data point and f is thefrequency for that point)x f1 13 34 42 25 1Data is often described as ungrouped or grouped. Ungrouped data is data given as indi-vidual data points. Grouped data is data given in intervals.Example 3. Ungrouped data without a frequency distribution.1, 3, 6, 4, 5, 6, 3, 4, 6, 3, 62 StatisticsExample 4. Ungrouped data with a frequency distribution.Number oftelevision sets Frequency0 21 132 183 04 105 2Total 45Example 5. Grouped data.Exam score Frequency90-99 780-89 570-79 1560-69 450-59 540-49 030-39 1Total 371.2. Organizing Data.Given a set of data, it is helpful to organize it. This is usually done by creating a frequencydistribution.Example 6. Ungrouped data.Given the following set of data, we would like to create a frequency distribution.1 5 7 8 23 7 2 8 7To do this we will count up the data by making a tally (a tick mark in the tally column foreach occurance of the data point. As before, we will designate the data points by x.x tally1 |2 ||3 |45 |67 |||8 ||MAT 142 - Module ST 3Now we add a column for the frequency, this will simply be the number of tick marks foreach data point. We will also total the number of data points. As we have done previously,we will represent the frequency with f.x tally f1 | 12 || 23 | 14 05 | 16 07 ||| 38 || 2Total 10This is a frequency distribution for the data given. We could also include a column for therelative frequency as part of the frequency distribution. We will use rel f to indicate therelative frequency.x tally f rel f1 | 1110∗ 100% = 10%2 || 2210∗ 100% = 20%3 | 1 10%4 0 0%5 | 1 10%6 0 0%7 ||| 3310∗ 100% = 30%8 || 2 20%Totals 10 100%For our next example, we will use the data to create groups (or categories) for the data andthen make a frequency distribution.Example 7. Grouped Data.Given the following set of data, we want to organize the data into groups. We have decidedthat we want to have 5 intervals.26 18 21 34 1838 22 27 22 3025 25 38 29 2024 28 32 33 18Since we want to group the data, we will need to find out the size of each interval. To dothis we must first identify the highest and the lowest data point. In our data the highestdata point is 38 and the lowest is 18. Since we want 5 intervals, we make the computionhighest − lowestnumber of intervals=38 − 185=205= 44 StatisticsSince we need to include all points, we always take the next highest integer from that whichwas computed to get the length of our interval. Since we computed 4, the length of ourintervals will be 5. Now we set up the first intervallowest ≤ x < lowest + 5 which results in 18 ≤ x < 23.Our next interval is obtained by adding 5 to each end of the first one:18 + 5 ≤ x < 23 + 5 which results in 23 ≤ x < 28.We continue in this manner to get all of our intervals:18 ≤ x < 2323 ≤ x < 2828 ≤ x < 3333 ≤ x < 3838 ≤ x < 43.Now we are ready to tally the data and make the frequency distribution. Be careful to makesure that a data point that is the same number as the end of the interval is placed in thecorrect interval. This means that the data point 33 is counted in the interval 33 ≤ x < 38and NOT in the interval 28 ≤ x < 33.x tally f rel f18 ≤ x < 23 ||||||| 7720∗ 100% = 35%23 ≤ x < 28 ||||| 5520∗ 100% = 25%28 ≤ x < 33 |||| 4420∗ 100% = 20%33 ≤ x < 38 || 2220∗ 100% = 10%38 ≤ x < 43 || 2 10%Totals 20 100%1.3. Histogram.Now that we have the data organized, we want a way to display the data. One such displayis a histogram which is a bar chart that shows how the data are distrubuted among eachdata point (ungrouped) or in each interval (grouped)Example 8. Histogram for ungrouped data.Given the following frequency distribution:Number oftelevision sets Frequency0 21 132 183 04 105 2The histogram would look as follows:MAT 142 - Module ST 5! " # $ % &'"$&(!'!"!$!&!("'Example 9. Histogram example for grouped data. We will use the data from example 7.The histogram would look as follows.2. Measures of Central Tendency2.1. Mode.The mode is the data point which occurs most frequently. It is possible to have more thanone mode, if there are two modes the data is said to be bimodal. It is also possible fora set of data to not have any mode, this situation occurs if the number of modes gets tobe “too large”. It it not really possible to define “too large” but one should exercise goodjudgement. A resaonable, though very generous, rule of thumb is that if the number of datapoints accounted for in the list of modes is half or more of the data points, then there is nomode.6 StatisticsNote: if the data is given as a list of data points, it is often easiest to find the mode bycreaating a frequency distribution. This is certainly the most organized method for findingit. In our examples we will use frequency distributions.Example 10. A data set with a single mode.Consider the data from example 8:Number oftelevision sets Frequency0 21 132 183 04 105 2You can see from the table that the data point which occurs most frequently is 2 as it has afrequency of 18. So the mode is 2.Example 11. A data set with two modes.Consider the data:Number ofhours of television Frequency0 10.5 41 81.5 92 132.5 103 113.5 134 54.5 3You can see from the table that the data points 2 and 3.5 both occur with the highestfrequency of 13. So the modes are 2 and 3.5.Example 12. A data set with no mode.Consider the data:MAT 142 - Module ST 7Age Frequency18 1219 520 321 922 123 824 1225 1226 527 3Total 71You can see from the table that the data


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ASU MAT 142 - Statistics

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