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# Applied Math 40S March 17, 2008

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Working with grouped data, 1-variable statistics, and the standard deviation.

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### Applied Math 40S March 17, 2008

1. 1. Working With Grouped Data Group Photo by premasagar
2. 2. Working with Grouped Data A frequency distribution table shows the number of elements of data (frequency) at each measure. Sometimes the measures need to be grouped, especially if the measures are continuous. Example: The table below is a frequency distribution table that shows the heights of 100 Senior 4 students. The students are grouped into suitable height groups in 7 cm. intervals. height interval interval mean # of students 153.5 to 160.5 157 5 160.5 to 167.5 164 16 167.5 to 174.5 171 43 174.5 to 181.5 178 27 181.5 to 188.5 185 9 Total 100 Determine the mean, median, mode, range, and standard deviation for the student heights. mean = 172.33 median = 171.00 mode = 171.00 range = 28 standard deviation = 6.837
3. 3. Working with Grouped Data A probability distribution table shows the percent of elements of data (probability) of each measure. Sometimes the measures need to be grouped, especially if the measures are continuous. Example: The table below is a frequency distribution table that shows the heights of 100 Senior 4 students. The students are grouped into suitable height groups in 7 cm. intervals. height interval interval mean % of students 153.5 to 160.5 157 0.05 160.5 to 167.5 164 0.16 167.5 to 174.5 171 0.43 174.5 to 181.5 178 0.27 181.5 to 188.5 185 0.09 Total 1
4. 4. Grouped Data and Histograms A histogram is a bar graph that shows equal intervals of a measured or counted quantity on the horizontal axis, and the frequencies associated with these intervals on the vertical axis. Drawing a histogram is useful because it shows the distribution of the heights of the students. A histogram is known as a Frequency Distribution Graph when the data is obtained from a frequency distribution. A histogram is known as a Probability Distribution Graph when the data is obtained from a probability distribution. Learn more about constructing a Histogram. Click: Contents > 2. Seeing Data > 2.3 Histogram
5. 5. Let's apply what we've learned ... mark interval mark # of students The frequency distribution table at 29 to 37 33 1 right shows the midterm marks of 85 38 to 46 42 4 Grade 12 math students. The first 47 to 55 51 12 column shows the mark interval, the 56 to 64 60 18 second column the average mark 65 to 73 69 24 within each mark interval, and the 74 to 82 78 16 third column the number of students 83 to 91 87 7 at each mark. 92 to 100 96 3 Total 85 (a) Calculate the mean and std. dev. to two decimal places. (b) Calculate the number of students that have marks within one std. dev. of the mean. (c) What percent of students have marks within one std. dev. of the mean?
6. 6. Let's apply what we've learned ... mark interval mark # of students The frequency distribution table at 29 to 37 33 1 right shows the midterm marks of 85 38 to 46 42 4 Grade 12 math students. The first 47 to 55 51 12 column shows the mark interval, the 56 to 64 60 18 second column the average mark 65 to 73 69 24 within each mark interval, and the 74 to 82 78 16 third column the number of students 83 to 91 87 7 at each mark. 92 to 100 96 3 Total 85 (b) Calculate the number of students that have marks within one std. dev. of the mean. (c) What percent of students have marks within one std. dev. of the mean?
7. 7. An experiment was performed to determine the approximate mass of a penny. Three hundred pennies were weighed, and the weights were recorded in the frequency distribution table shown below. Mass (grams) 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 Frequency 2 4 34 71 94 74 17 4 Determine the mean, median, mode, range, and standard deviation of the data. Create a histogram that shows the frequencies of different masses of this set of pennies.
8. 8. HOMEWORK Four hundred people were surveyed No. of Videos Returned No. of Persons to find how many videos they had 1 28 rented during the last month. 2 102 Determine the mean and median of the 3 160 frequency distribution shown below, 4 70 and draw a probability distribution 5 25 histogram. Also, determine the mode 6 13 by inspecting the frequency 7 0 distribution and the histogram. 8 2
9. 9. HOMEWORK The table shows the weights weight interval mean interval # of infants (in pounds) of 125 newborn 3.5 to 4.5 4 4 infants. The first column 4.5 to 5.5 5 11 shows the weight interval, 5.5 to 6.5 6 19 the second column the 6.5 to 7.5 7 33 average weight within each 7.5 to 8.5 8 29 weight interval, and the third 8.5 to 9.5 9 17 column the number of 9.5 to 10.5 10 8 newborn infants at each 10.5 to 11.5 11 4 weight. Total 125 (a) Calculate the mean weight and standard deviation. (b) Calculate the weight of an infant at one standard deviation below the mean weight, and one standard deviation above the mean. (c) Determine the number of infants whose weights are within one standard deviation of the mean weight. (d) What percent of the infants have weights that are within one standard deviation of the mean weight?
10. 10. HOMEWORK The table shows the lengths in millimetres of 52 arrowheads. 16 16 17 17 18 18 18 18 19 20 20 21 21 21 22 22 22 23 23 23 24 24 25 25 25 26 26 26 26 27 27 27 27 27 28 28 28 28 29 30 30 30 30 30 30 31 33 33 34 35 39 40 (a) Calculate the mean length and the standard deviation. (b) Determine the lengths of arrowheads one standard deviation below and one standard deviation above the mean. (c) How many arrowheads are within one standard deviation of the mean? (d) What percent of the arrowheads are within one standard deviation of the mean length?