6B

1
6B Measures of Variation

• We have seen that the average value tells about
  the center of the data, but that doesnt show us
  the whole picture.
• The three sets below all have the same mean and
  median, but are clearly very different
• 4, 4, 4, 5, 5, 5, 6, 6 ,6
• 1, 2, 3, 4, 5, 6, 7, 8, 9
• 0, 0, 0, 0, 5, 10, 10, 10, 10
• If these were examples of waiting times on hold,
  which would you rather call?

2
Variation measures Range

• One simple, basic measure of variation is the
  range.
• The range is simply the difference between the
  largest and smallest data values
• 4, 4, 4, 5, 5, 5, 6, 6 ,6 range 6 4
  2
• 1, 2, 3, 4, 5, 6, 7, 8, 9 range 9 1
  8
• 0, 0, 0, 0, 5, 10, 10, 10, 10 range 10 0
  10
• The range only includes the high and low values
  and can often be misleading.
• 1, 10, 10, 10, 10, 10 range 9 more varied??
• 2, 4, 6, 8, 10 range 8

3
Quartiles

• Since the range can be misleading, we can use
  quartiles to show us a more complete picture of
  the spread of the data
• Median 50 above and 50 below
• Using quartiles, we will have 4 values that
  separate the data into fourths instead of halves
  like the median does
• Lower Quartile median of lower half of data
• Upper Quartile median of upper half of data
• LQ MED
  UQ
• Data ...25...25...25......25...

4
Quartiles examples

• Data Set A 1, 2, 2, 3, 4, 6, 8, 9, 10, 11, 12,
  12, 15, 16, 18 median lower quartile upper
  quartile
• Data Set B 24, 32, 41, 44, 53, 59, 64, 72, 76,
  79, 83, 88 median lower quartile upper
  quartile

5
Five-Number Summary

• The five-number summary for a data set is
• Minimum
• Lower quartile (Q1)
• Median (Q2)
• Upper quartile (Q3)
• Maximum

6
Boxplot

• A Boxplot uses the five-number summary to show
  the middle 50 of the data as a box, with lines
  extending out to the extreme values. (sometimes
  called a box and whisker plot)
• Data Set A minimum 1 maximum 18
  Q1 3 Q2 9 Q3
  12
• 1 3 9
  12 18

7
Standard Deviation

• The standard deviation is a single measure of the
  variation that uses every data value.
• It is an averaged measure of how far each data
  value is from the mean.
• The standard deviation is used extensively in
  statistical analyses.
• It is fairly simple to calculate by hand for
  small data sets, and most calculators can
  calculate standard deviations, as well.

8
Standard Deviation Formula
9
Standard Deviation Formula
Translation x mean S sum 1. find mean
( x ) 2. for each x, subtract mean 3. then
square each of those values 4. then add up all
of those values 5. then divide by n 1 6.
then take the square root
10
Standard Deviation Example

• Data Set 2, 4, 6, 8, 10 n 5, x 6
• x x x (x x )2 s 2
  -4 16
• 4 -2 4
• 6 0 0
• 8 2 4
• 10 4 16
• 40 sum

11
Interpreting Standard Deviations

• The larger the standard deviation, the more
  variation there is in the data set.
• The smaller the standard deviation, the less
  variation there is in the data set (more
  consistent)
• EXAM 1 mean 83.8, standard deviation 19.6
• EXAM 2 mean 87.3, standard deviation 13.7
• What does this tell us?

12
Is variation good?

• Manufacturing do we want a lot of variation in
  the size of a product being manufactured?
• Sports do you want a team that will score
  consistently?
• Investing would you rather invest in a fund
  that has big gains and losses or is more
  consistent?

13
Exceptional values

• The general rule of thumb is that any data value
  more than 2 standard deviations above or below
  the mean is exceptional or unusual.
• This is just a general rule that may not be as
  accurate when extreme outliers are present or
  when the data set is skewed.
• If your drive time to school is studied over time
  and has mean 15 minutes, with sd 4 min, would
  a drive time of 20 minutes be exceptionally long?
  How about 30 minutes?
• What would be the typical range of driving times
  you should expect with the measurements above?