Page 129 - Data Science class 10
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Step 3  Calculate the square of each of the differences.

                    We get:
                    –0.5 × –0.5 = 0.25

                    1.5 × 1.5 = 2.25
                     –2.5 × -2.5 = 6.25

                    1.5 × 1.5 = 2.25

            Step 4  Find the average of squared numbers.
                                              11
                    0.25 + 2.25 + 6.25 + 2.25 =
                                               4
                    = 2.75

            Step 5   Finally, find the square root of the variance. Which results in a standard deviation measure of approximately
                    1.65. The whole calculation is shown in the table given below:


                                Data Point                      Mean      Distance from mean   Square of distances

                                    5                            5.5              -0.5                0.25
                                    7                            5.5              1.5                 2.25
                                    3                            5.5              -2.5                6.25

                                    7                            5.5              1.5                 2.25
                               Mean = sum/n                      5.5
                              Sum of squares                                                          11.00

                         Variance = sum of squares/N                                                  2.75
                Standard Deviation = Square root of the Variance                                      1.65

            The standard deviation 1.65 can be represented graphically as given below:


                                                      Standard deviation curve
                                  0.25

                                   0.2

                                  0.15

                                   0.1


                                  0.05

                                    0
                                      0    0.5    1    1.5    2    2.5    3    3.5    4   4.5


            A few real-life applications of standard deviation include:
               • Grading Tests: If teacher wishes to know whether students are performing at the same level or whether there
              is a higher standard deviation.
               • Calculate the Results of Any Survey: If someone wants to have some measure of the reliability of the responses
              received in the survey, they can predict how a larger group of people may answer the same questions.


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