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Standard Deviation

In this exploration you will investigate standard deviation and deveop a better undertstanding of how it can help you measure the spread of a data set. This exploration will reinforce the concepts in Lesson 2.2  of Discovering Advanced Algebra: An Investigative Approach.

Sketch

This sketch shows 12 data points, with values that you can adjust from 0 to 30 by dragging. The numerical values of the data points and their sum are shown. You can see the mean by pressing Show Mean. You can also press buttons to see the deviations from the means and their sum, the squares of the deviations, and the standard deviation. When the standard deviation is displayed, vertical lines will enclose all points that lie within one standard deviation of the mean.

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Investigate

  1. For the data points shown, make a guess about where the mean is. Click to show the mean. Were you close? Now hide the mean, change the data points, and try again.
  2. For the data points you have, make a guess about which points lie from one standard deviation below the mean to one standard deviation above the mean. Click to show the standard deviation. Were you close? Now hide the standard deviation, change the data points, and try again.
  3. Click to show the deviations. What is their sum? Will this always be the case? Drag some points to test your conjecture. Can you explain why your conjecture is true?
  4. Click to show the mean and the standard deviation. How can you make the standard deviation large? Small?
  5. Drag some points so that you have outliers. What effect do outliers have on the mean? On the standard deviation? For which is the effect greater? Explain.
  6. If all of the data points were shifted 10 units to the right, what would happen to the mean? To the standard deviation? Explain.
  7. Compare and contrast the information you get from the mean with that from the standard deviation.
  8. If you have already studied box plots, compare and contrast the information gained from a box-and-whiskers plot with this graph of standard deviation.