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Standard Deviation

Standard Deviation. Standard Deviation. Variance ( ): Sum of squared deviations from the mean. Standard Deviation is the square root of the variance. Standard Deviation as a “Ruler”. How can you compare measures – be it scores, athletic performance, etc., across widely different groups?

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Standard Deviation

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  1. Standard Deviation

  2. Standard Deviation • Variance (): Sum of squared deviations from the mean • Standard Deviation is the square root of the variance

  3. Standard Deviation as a “Ruler” • How can you compare measures – be it scores, athletic performance, etc., across widely different groups? • Book Example: Who wins a heptathlon? • The trick to comparing very different looking values is to use the standard deviation • You are asking, in a sense, how far is a given value from the mean?

  4. A Quick Example • Kluft’s 6.78m long jump is 0.62m longer than the mean jump of 6.16m • The SD for the event was 0.23m, so her jump was (6.78-6.16)/0.23 = 2.70 standard deviations better than the mean

  5. A Quick Example (Cont.) • Skujyte’s winning shot put was 16.4-13.29 = 3.11 meters longer than the mean shot put distance. • That is 3.11/1.24=2.51 standard deviations better than the mean

  6. A Quick Example (Cont.) • Kluft’s long jump was 2.70 standard deviations better than the mean • Skujyte’s shot put was 2.51 standard deviations better than the mean. • Who had the more impressive performance?

  7. Standardizing with z-Scores • Expressing distance in standard deviations standardizes performances. • To standardize a value we subtract the mean performance from the individual performance then divide by the standard deviation • These values are called standardized values and are commonly denoted with the letter z, so are often called z-scores • No units

  8. z-Scores • A z-Score of 2 tells us that a data value is 2 standard deviations above the mean. • A z-Score of -1.6 tells us the data value was 1.6 standard deviations below the mean • The farther a data value from the mean the more impressive it is, regardless of sign. • Kluft’s long jump with z-score 2.7 is more impressive than Skujyte’s shot put with z-score 2.51

  9. Just Checking (pg 107)

  10. The lower of your two tests will be dropped. You got a 90 on test 1, an 80 on test 2You are all set to drop the 80… BUT WAIT! Your teacher announces she grades “on a curve.” She standardizes the scores in order to decide the lower one. The mean of the first test was 88 with sd=4, the mean on the second was 75 with sd=5 Which one will be dropped? Does this seem “fair” ?

  11. On first test, mean = 88, sd = 4z= (90-88)/4 = .5On second test, mean=75, sd=5z=(80-75)/5 = 1.0 The first test has a lower z-score so it will be the one that gets dropped No, this doesn’t seem fair. The second test is 1 sd above the mean, farther away than the first, so it’s the better score relative to the class.

  12. Shifting Data • When we standardize data to get z-scores we do two things: • We shift the data by subtracting the mean • We rescale the values by dividing by their standard deviation • What happens to a grade distribution if everyone gets 5 extra points? • If we switch feet to meters, what happens to the distribution of the heights of students?

  13. Shifting • When we shift the data by adding (or subtracting) a constant to each value all measures of position (center, percentiles, min, max) will increase (or decrease) by the same constant • Spread is not affected. • Shape doesn’t change, spread doesn’t change: • Not range, not IQR, not the SD

  14. Rescaling • Converting from something like kilograms to pounds is an act of rescaling the data: • To move from kg to lbs we multiply kg*2.2lbs/kg • This will not change the shape of the distribution • Mean gets multiplied by 2.2 • In fact, all measures of position are multiplied by the same constant

  15. Rescaling • What do you think happens to spread? • The spread of pounds would be larger than the spread of kg after rescaling. • By how much? • 2.2 times larger!

  16. Rescaling • When we multiply or divide all the data values by a constant all measures of position (mean, median, percentiles) are multiplied or divided by that same constant. • The same is true for measures of spread: all measures of spread are multiplied or divided by that same constant

  17. Just Checking (pg 110)Before re-centering some SAT scores, the mean of all test scores was 450- How would adding 50pts to each score affect the mean?- The SD = 100pts, what would it be after adding 50 pts?

  18. - Mean would increase to 500- SD is still 100 pts

  19. Back to z-Scores • Standardizing into z-Scores: • Shift them by the mean • Rescale by the Standard Deviation • When we divide by s, the standard deviation gets divided by s as well • The new SD becomes 1

  20. Z-Scores Z-Scores have a mean of 0 and a standard deviation of 1 Standardizing into z-Scores does not change the shape of the distribution of a variable Standardizing into z-Scores changes the center by making it 0 Standardizing into z-Scores changes the spread by making the SD = 1

  21. When is a z-score BIG? • As a rule, z-scores are big at around 3, definitely big around 6 or 7… • But that isn’t nearly enough!

  22. Homework 129, # 1, 2, 3, 5, 7, 9, 24 Page 130, # 26, 29, 30, 34, 43 (Previously Assigned)

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