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Section 3-3

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Section 3-3

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    1. Section 3-3 Measures of Variation

    2. WAITING TIMES AT DIFFERENT BANKS

    3. RANGE

    4. STANDARD DEVIATION FOR A SAMPLE

    5. STANDARD DEVIATION FORMULAS

    6. EXAMPLE

    7. PROPERTIES OF THE STANDARD DEVIATION The standard deviation is a measure of variation of all values from the mean. The value of the standard deviation s is usually positive. It is zero only when all of the data values are the same number. (It is never negative.) Also, larger values of s indicate greater amounts of variation. The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers. The units of the standard deviation s are the same as the units of the original data values.

    8. STANDARD DEVIATION OF A POPULATION

    9. FINDING THE STANDARD DEVIATION ON THE TI-81/84 Press STAT; select 1:Edit…. Enter your data values in L1. (You may enter the values in any of the lists.) Press 2ND, MODE (for QUIT). Press STAT; arrow over to CALC. Select 1:1-Var Stats. Enter L1 by pressing 2ND, 1. Press ENTER. The sample standard deviation is given by Sx. The population standard deviation is given by sx.

    10. SYMBOLS FOR STANDARD DEVIATION

    11. EXAMPLE

    12. VARIANCE

    13. ROUND-OFF RULE FOR MEASURES OF VARIATON

    14. STANDARD DEVIATION FROM A FREQUENCY DISTRIBUTION

    15. EXAMPLE

    16. RANGE RULE OF THUMB —PART 1

    17. RANGE RULE OF THUMB —PART 2

    18. EXAMPLES Heights of men have a mean of 69.0 in and a standard deviation of 2.8 in. Use the range rule of thumb to estimate the minimum and maximum “usual” heights of men. In this context, is it unusual for a man to be 6 ft, 6 in tall? The shortest home-run hit by Mark McGwire was 340 ft and the longest was 550 ft. Use the range rule of thumb to estimate the standard deviation.

    19. MORE PROPERTIES OF THE STANDARD DEVIATION The standard deviation measures the variation among the data values. Values close together have a small standard deviation, but values with much more variation have a larger standard deviation. The standard deviation has the same units of measurement as the original values. For many data sets, a value is unusual if it differs from the mean by more than two standard deviations. When comparing variation in two different data sets, compare the standard deviations only if the data sets use the same scale and units and they have means that are approximately the same.

    20. THE EMPIRICAL (OR 68-95-99.7) RULE

    22. EXAMPLE

    23. CHEBYSHEV’S THEOREM

    24. EXAMPLE

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