2 4 measures of center

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2 4 measures of center

  1. 1. Describing, Exploring, and Comparing Data <ul><li>Summarizing a set of data with a… </li></ul><ul><ul><li>Tables </li></ul></ul><ul><ul><li>Pictures </li></ul></ul><ul><ul><li>A small set of numbers that describes the data’s </li></ul></ul><ul><ul><ul><li>Center </li></ul></ul></ul><ul><ul><ul><li>Variation </li></ul></ul></ul><ul><ul><ul><li>Distribution </li></ul></ul></ul><ul><ul><ul><li>Outliers </li></ul></ul></ul><ul><li>Comparing summaries </li></ul><ul><ul><li>Several different groups </li></ul></ul><ul><ul><li>Changes over time within the same group </li></ul></ul>
  2. 2. Summarizing data: Measures of Center <ul><li>Or measures of central tendency </li></ul><ul><li>Mean </li></ul><ul><ul><li>AKA: arithmetic mean, average </li></ul></ul><ul><ul><li>The ‘definitive’ single value for characterizing a set of data. </li></ul></ul><ul><li>Median </li></ul><ul><ul><li>Middle value </li></ul></ul><ul><li>Mode </li></ul><ul><ul><li>Most common value(s) </li></ul></ul><ul><li>Midrange </li></ul>
  3. 3. Mean <ul><li>Sum of the values divided by the number of values </li></ul><ul><li>Sample mean: </li></ul><ul><li>Population mean: </li></ul>
  4. 4. Mean: For example 1300 990 990 SAT 1300 1290 1190 1100 1050 1500 1200
  5. 5. Weighted Mean <ul><li>The values in a list vary in importance aside from their actual number </li></ul><ul><ul><li>E.g., in a list of SAT scores, the seniors’ scores might be more important that the juniors </li></ul></ul><ul><li>We quantify the importance for each value, call it w n </li></ul><ul><li>Our formula </li></ul>
  6. 6. Weight Mean: For example Weight 1300 990 990 SAT 1300 1290 1190 1100 1050 1500 1200
  7. 7. Mean from a Frequency Table <ul><li>To calculate the mean from a frequency table </li></ul><ul><li>Find the class’s midpoints </li></ul><ul><li>Multiply by the frequency </li></ul><ul><li>Add the products </li></ul><ul><li>Divide by the total frequencies </li></ul>Total Midpoint Freq Classes 1 4 63 121 11 75-79 70-74 65-69 60-64 55-59
  8. 8. Median <ul><li>Then preferred measure of center when there are outliers </li></ul><ul><li>The middle value of an ordered list of values </li></ul><ul><li>Represented with the notation “x-tilde” </li></ul><ul><li>If the number of items in the list is odd, the median is the middle number in the list. </li></ul><ul><li>If the number of items is even, there will be two values in the middle. the median is the average of the two. </li></ul>
  9. 9. Median Example 1440 1390 1310 1300 1260 1210 1210 1170 1100 1050 1440 1390 1310 990 1300 1260 1210 1210 1170 1100 1050
  10. 10. Mean vs. Median <ul><li>Consider the following test scores: </li></ul><ul><li>92 81 75 84 77 81 79 70 </li></ul><ul><li>Mean: </li></ul><ul><li>Median: </li></ul>
  11. 11. Mode <ul><li>The most common value </li></ul><ul><li>Maybe none (if every value in the list is unique), one, or several </li></ul><ul><li>Only measure of center to use for nominal data </li></ul><ul><ul><li>E.g., M&M colors </li></ul></ul>
  12. 12. Midrange <ul><li>The value midway between the minimum and maximum values in the list </li></ul>1440 1390 1310 990 1300 1260 1210 1210 1170 1100 1050
  13. 13. Live example <ul><li>Calculate the four measures of center </li></ul>72 82 86 86 80 90 74 58 28 58 88 98
  14. 14. Your Turn <ul><li>Calculate the four measures of the following data (Daily low temperature for Burlington, Vt for January 2010 </li></ul>-1 -4 -1 12 28 32 37 14 2 9 14 27 31 31 32 27 34 18 -3 1 17 -6 -3 16 27 21 10 15 17 17 9

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