Standard Deviation

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?

A Quick Example Long JumpShot Put Mean6.16m13.29m SD.23m1.24m n2628 Kluft6.78m14.77m Skujyte6.30m16.4m  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 ( )/0.23 = 2.70 standard deviations better than the mean

A Quick Example (Cont.) Long JumpShot Put Mean6.16m13.29m SD.23m1.24m n2628 Kluft6.78m14.77m Skujyte6.30m16.4m  Skujyte’s winning shot put was = 3.11 meters longer than the mean shot put distance.  That is 3.11/1.24=2.51 standard deviations better than the mean

A Quick Example (Cont.) Long JumpShot Put Mean6.16m13.29m SD.23m1.24m n2628 Kluft6.78m14.77m Skujyte6.30m16.4m  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?

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

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

Just Checking (pg 107)

The lower of your two tests will be dropped. You got a 90 on test 1, an 80 on test 2 You 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 a)Which one will be dropped? b)Does this seem “fair” ?

On first test, mean = 88, sd = 4 z= (90-88)/4 =.5 On second test, mean=75, sd=5 z=(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.

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?

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

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

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!

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

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

- Mean would increase to SD is still 100 pts

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

 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 Z-Scores

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!

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