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# Applied 40S March 24, 2009

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Working with grouped data. Frequency and probability histograms.

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### Applied 40S March 24, 2009

1. 1. Looking For Patterns in Grouped Data Script Ohio by ﬂickr user Junior Sam
2. 2. Looking For Patterns in Grouped Data Script Ohio by ﬂickr user Junior Sam
3. 3. Measures of Dispersion (Variability) determine how quot;spread outquot; or variedquot; a set of data is. Standard Deviation (σ): What's the difference between quot;σquot; and quot;squot;? The symbol for standard deviation of a population or large sample is quot;σquot; (sometimes written as quot;σx quot;), and the symbol for standard deviation of a sample is 's'. A large sample is deﬁned as a sample with 30 or more data items. In this course, we will use only quot;σquot; (sigma), which represents the standard deviation of the population. Let's take a look at a visual explanation of why we calculate the standard deviation this way ... http://www.seeingstatistics.com/
4. 4. 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
5. 5. 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
6. 6. 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
7. 7. 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 http://www.seeingstatistics.com/
8. 8. Let's apply what we've learned ... mark interval mark # of students The frequency distribution table 29 to 37 33 1 at right shows the midterm marks 38 to 46 42 4 of 85 Grade 12 math students. 47 to 55 51 12 The ﬁrst column shows the mark 56 to 64 60 18 65 to 73 69 24 interval, the second column the 74 to 82 78 16 average mark within each mark 83 to 91 87 7 interval, and the third column the 92 to 100 96 3 number of students at each mark. Total 85 This is the (a) Calculate the mean and std. dev. to two decimal places. correct way Mean = 66.99 σ = 13.27 to do this. (b) Calculate the number of students that have marks within one std. dev. of the mean. 58 or 70 (c) What percent of students have marks within one std. dev. of the mean? 68% or 82%
9. 9. HOMEWORK 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.
10. 10. HOMEWORK No. of Videos Four hundred people were No. of Persons Returned surveyed to ﬁnd how many videos 1 28 they had rented during the last 2 102 month. Determine the mean and 3 160 median of the frequency 4 70 distribution shown below, and draw 5 25 6 13 a probability distribution 7 0 histogram. Also, determine the 8 2 mode by inspecting the frequency distribution and the histogram.
11. 11. HOMEWORK weight interval mean interval # of infants The table shows the 3.5 to 4.5 4 4 weights (in pounds) of 125 4.5 to 5.5 5 11 newborn infants. The ﬁrst 5.5 to 6.5 6 19 column shows the weight 6.5 to 7.5 7 33 7.5 to 8.5 8 29 interval, the second column 8.5 to 9.5 9 17 the average weight within 9.5 to 10.5 10 8 each weight interval, and 10.5 to 11.5 11 4 the third column the TOTAL 125 number of newborn infants at each weight. (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?