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## Elementary Statistics

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**Elementary Statistics**Q: What is data? Q: What does the data look like? Q: What conclusions can we draw from the data? Q: Where is the middle of the data? Q: Why is the spread of the data important? Q: Can we model the data? Q: How do we know if we have a good model? Q: Is our data affected by other variables?**Definitions**Individuals : Objects described by a set of data. Individuals may be people, but they may also be animals or things. Variable : Any characteristic of an individual. A variable can take on different values for different individuals. Categorical and Quantitative Variables Categorical variable : Places an individual into one or several categories. Quantitative variable : Takes numerical values for which arithmetic operations make sense. Distribution : Tells what values the data takes and how often it takes these values.**Homework**1, 2, 4, 6**Exploring Data**Two Basic Strategies : 1) Begin by examining each variable by itself. Then move on to study the relationships among variables. 2) Begin with a graph or graphs. Then add numerical summaries of specific aspects of data. Different types of graphs : Bar graph, Pie chart, Stemplot, back-to-back Stemplot, Histogram, Time plot**Bar Graphs**Grade A B C D Other Count Bar graph - A graph which displays the data using heights of bars to represent the counts of the variables. Example : Consider the following grade distribution : 6 12 15 9 3 How could we display the data using a bar graph ?**Bar Graphs**15 12 9 Grade A B C D Other 6 Count 6 12 15 9 3 3 A B C D F**Pie Charts**Pie Chart : 1) A chart which represents the data using percentages. 2) Break up a circle (pie) into the respected percentages.**Pie Charts**Percent Grade A B C D Other Count 6 12 15 9 3 13 27 33 20 7 B A C D F**Homework**13, 14, 16**Stemplot**How to make a Stemplot : 1) Separate each observation into a stem consisting of all but the final (rightmost) digit, and a leaf, the final digit. Stems may have as many digits as needed, but each leaf contains only a single digit. 2) Write the stems in a vertical column with the smallest at the top, and draw a vertical line at the right of this column. 3) Write each leaf in a row to the right of the stem, in increasing order out from the stem.**Stemplot**4 4 5 5 Steps 1 and 2 : 6 6 Step 3 : 7 7 8 8 9 9 10 10 Example: Here are the grades Max achieved while in school his first two years. Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100, 58, 76, 83, 88, 72, 66 5 8 6 2 7 7 6 2 8 3 3 6 2 3 8 1 0 4 1 0**Stemplot**4 4 5 5 5 8 Steps 1 and 2 : 6 6 6 Step 3 : 7 7 2 2 6 7 7 8 8 2 3 3 3 6 8 8 9 9 0 1 1 4 10 10 0 Example: Here are the grades Max achieved while in school his first two years. Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100, 58, 76, 83, 88, 72, 66**Back-To-Back Stemplot**• This is a stemplot which allows you to see and compare the • distribution of two related data sets Example : Here are the grades Lulu received during her first two years at college : Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55, 73, 95, 87, 76, 93, 82, 66, 75 • To make a Back-To-Back Stemplot, you make the stem, and the • stems going off to the right and the left. You want the smaller • values closer to the stem.**Back-To-Back Stemplot**4 5 5 8 6 6 7 2 2 6 7 7 8 2 3 3 3 6 8 8 9 0 1 1 4 10 0 Lulu’s Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55, 73, 95, 87, 76, 93, 82, 66, 75 Max’s Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100, 58, 76, 83, 88, 72, 66 5 6 6 5 6 3 1 8 8 7 2 7 2 6 4 3 5 3 0 2**Back-To-Back Stemplot**4 5 5 5 8 6 6 6 6 8 8 7 6 5 3 1 7 2 2 6 7 7 7 6 4 2 2 8 2 3 3 3 6 8 8 5 3 3 2 0 9 0 1 1 4 10 0 Lulu’s Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55, 73, 95, 87, 76, 93, 82, 66, 75 Max’s Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100, 58, 76, 83, 88, 72, 66**Splitting Stems**7 8 9 • If you have a large data set (leaves), then sometimes a stemplot • will not work very well. For instance, if you have a large amount • of leaves, and only a few stems, you might want to split the stems. Example : Consider the following test scores : 71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84, 85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97 Normally we would set up the stems as follows :**Splitting Stems**7 8 9 • If you have a large data set (leaves), then sometimes a stemplot • will not work very well. For instance, if you have a large amount • of leaves, and only a few stems, you might want to split the stems. Example : Consider the following test scores : 71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84, 85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97 Normally we would set up the stems as follows : 1, 1, 2, 4, 5, 5, 6, 7, 9 0, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 8, 9 0, 0, 0, 1, 3, 5, 6 , 7**Splitting Stems**7 7 This stem gets scores 70 - 74 8 8 This stem gets scores 75 - 79 9 9 • If you have a large data set (leaves), then sometimes a stemplot • will not work very well. For instance, if you have a large amount • of leaves, and only a few stems, you might want to split the stems. Example : Consider the following test scores : 71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84, 85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97 However, we could set up the stems as follows : 1 1 2 4 5 5 5 6 7 9 0 1 1 2 3 3 3 3 4 5 5 8 9 0 0 0 1 3 5 6 7**Rounding Stems**29.1 29.0 5.7 5.6 5.5 Q: What if we have a lot of stems, but not a lot of leaves? A: One might want to join the stems into larger stems by rounding. Example: Consider the following charges for filling a car with gas : 9.73 10.12 8.72 6.53 12.89 15.67 5.50 16.97 11.38 10.77 7.77 9.00 10.50 8.00 17.12 13.00 21.00 18.11 9.99 25.12 22.57 15.00 23.00 29.11 What would this stem look like ?**Rounding Stems**2 1 0 Q: What if we have a lot of stems, but not a lot of leaves? A: One might want to join the stems into larger stems by rounding. Example: Consider the following charges for filling a car with gas : 9.73 10.12 8.72 6.53 12.89 15.67 5.50 16.97 11.38 10.77 7.77 9.00 10.50 8.00 17.12 13.00 21.00 18.11 9.99 25.12 22.57 15.00 23.00 29.11 We could round the stems to be $10 stems : 1 5 2 3 9 0 2 5 6 1 0 0 7 3 8 5 9 8 6 5 7 9 8 9**Homework**20, 22, 23, 26**Histograms**A histogram breaks the range of variables up into (equal) intervals, and displays only the count or percent of the observations which fall into the particular intervals. Notes: • You can choose the intervals (usually equal) • Slower to construct than stemplots • Histograms do not display the individual observations • In case a score falls on an interval point, you must decide in • advance which interval in which the point will go.**Histograms**Steps to drawing a histogram : 1) Divide the range into classes of equal width. 2) Count the number of observations in each class. These are called frequencies. 3) Draw the histogram.**Histograms**Grade Amount Percent 90 - 100 8 20 80 - 90 10 25 70 - 80 10 25 60 - 70 8 20 50 - 60 4 10 Frequency Table 10 10 8 8 4 Example : Suppose the final breakdown in grades looks like this : 50 60 70 80 90 100**Histograms**Grade Amount Percent 90 - 100 8 20 80 - 90 10 25 70 - 80 10 25 60 - 70 8 20 50 - 60 4 10 25% 25% 20% 20% 10% Example : Suppose the final breakdown in grades looks like this : 50 60 70 80 90 100**Homework**31, 32**Time Plot**Variable Time A Time Plot is a graph with two axis. One axis represents time ,and the other axis represents the variable being measured.**Time Plot**89 90 91 92 93 94 95 96 97 98 Year HR 33 39 22 42 9 9 39 52 58 70 Example : The following are homerun totals for a certain baseball player the last 10 years : Construct a timeplot for this data set.**Time Plot**89 90 91 92 93 94 95 96 97 98 Year HR 33 39 22 42 9 9 39 52 58 70 Home Run Year**Time Plot**89 90 91 92 93 94 95 96 97 98 Year 70 HR 33 39 22 42 9 9 39 52 58 70 60 50 40 30 20 10 89 90 91 92 93 94 95 96 97 98**Homework**35, 36