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Standard deviation

Standard deviation. Information. Consistency. The table shows the number of cars sold by the sales teams at two car dealerships in April 2012. How can we compare the consistency of the two sales teams?.

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Standard deviation

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  1. Standard deviation

  2. Information

  3. Consistency The table shows the number of cars sold by the sales teams at two car dealerships in April 2012. How can we compare the consistency of the two sales teams? Find the mean sales at dealerships A and B and discuss how each salesperson compares to this mean.

  4. What is standard deviation? The standard deviation of a set of data is a measure of the distribution of the values. It indicates how far away the data values are from the mean. A small standard deviation compared to the data values indicates that the data is packed closely around the mean. A larger standard deviation results from a more widely distributed set of data. Here are some symbols you will encounter when calculating standard deviation: x is a data value Σ means “the sum of” x is used for the mean n is the number of values in the data set σ is used for the standard deviation

  5. The mean The table shows the number of home runs scored by ten baseball players over the 2011 season. The mean number of home runs is: 39 + 30 + 29 +...+ 2 10 230 10 23 x= = =

  6. Calculating standard deviation 1 The standard deviation (σ) is a measure of how far each data value is from the mean. x –x What do you think the first step is when finding the standard deviation? 16 7 The first step is to subtract xfrom each data value. 6 5 2 Here, the mean is 23, so we subtract 23 from every piece of data. 1 –1 –5 What do you notice about this column? –10 –21 All of these values add up to 0.

  7. Calculating standard deviation 2 What can we do now to ensure that all these values are positive? x –x (x–x)2 256 49 We can square all of the values. 36 The second step is to find the valuesof . 25 (x–x)2 4 1 1 25 100 441

  8. Calculating standard deviation 3 (x– x)2 The third step is to find the mean value of . x –x (x–x)2 The mean value of the last column is: 938 93.8 = 10 This value is known as the variance: the mean of the squared deviations from the mean.

  9. √93.8 Calculating standard deviation 4 In the process of finding the variance, we squared the values to make them all positive. x –x (x–x)2 We therefore need to take the square root of the variance to find the standard deviation. variance = 93.8 standard deviation = = 9.69 (to nearest hundredth)

  10. What does it mean? Here is the original table of data: The mean number of home runs is 23. The standard deviation is 9.69. Discuss what the standard deviation shows about the data. Another group of players had a standard deviation of 5.81. Which group had the better results? Justify your answer.

  11. Dot plots

  12. General formulas 1 The varianceof a set of data is given by the formula: å(x –x)2 variance = n The standarddeviationis the square root of the variance: å(x –x)2 standard deviation = √ n

  13. General formulas 2 In practice, the following equivalent formulas are usuallyused for finding the mean and the variance: å x2 – variance = x 2 n å x2 standard deviation = – x 2 n The key quantity in these two formulas is: . å x2 This is the sum of the squares of each piece of data.

  14. √93.8 Using the general formula å To use the formula, we need to find . x2 x2 Substituting and into the formula for the variance gives: 6228 å x2 x 23 = = 1521 900 841 å x2 variance = x 2 – 784 n 625 6228 232 = – 576 10 484 = 93.8 324 169 standard deviation = 4 =9.69 (to nearest hundredth) 6228

  15. Practice

  16. Using a calculator

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