Measures of Spread

1
Measures of Spread

• Chapter 3.3 Tools for Analyzing Data
• Mathematics of Data Management (Nelson)
• MDM 4U
• Author Gary Greer (with K. Myers)

2
What is spread?

• spread tells you how widely the data are
  dispersed
• for example, the two histograms have identical
  mean and median, but the spread is significantly
  different

3
Why worry about spread?

• spread indicates how close the values cluster
  around the middle value
• less spread means you have greater confidence
  that values will fall within a particular range.
• for example
• if you were making car parts, you might be
  concerned about certain measurements falling into
  a certain range
• while 2 shipments of parts might have the same
  mean dimensions, you would be much happier to
  have the shipment with the smaller spread

4
Vocabulary

• spread and dispersion seem to refer to much the
  same thing
• range is the difference between the largest and
  smallest value
• a quartile is one of three numerical values that
  divide a group of numbers into 4 equal parts
• the interquartile range is the range between the
  first and third quartiles

5
Quartiles Example

• 26 28 34 36 38 38 40 41 41 44 45 46 51 54 55
• range 55 26 29
• Q2 41 Median
• Q1 36 Median of lower half of data
• Q3 46 Median of upper half of data
• IQR 46 36 10 (contains 50 of data)
• if a quartile occurs between 2 values, it is
  calculated as the average of the two values

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A More Useful Measure of Spread

• the interquartile range is a somewhat useful
  measure of spread
• standard deviation is more useful
• To calculate it we need to find the mean and the
  deviation
• Mean is easy, as we have done that before
• Deviation is the difference between a particular
  point and the mean

7
Deviation

• The mean of these numbers is 48
• The deviation for 24 is 48-24 24
• 24
• 12 24 36 48 60 72 84
• -36
• The deviation for 84 is 48 84 -36

8
Standard Deviation

• deviation is the distance from the piece of data
  you are examining to the mean
• variance is a measure of spread found by
  averaging the squares of the deviation calculated
  for each piece of data
• Taking the square root of variance, you get
  standard deviation
• Standard deviation is a very important and useful
  measure of spread

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Standard Deviation

• s² is used to represent variance
• s is used to represent standard deviation
• s is commonly used to measure the spread of data,
  with larger values of s indicating greater spread
• we are using a population standard deviation

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Example of Standard Deviation

• 26 28 34 36
• mean (26 28 34 36) / 4 31
• s² (2631)² (28-31)² (34-31)² (36-31)²
• 4
• s² 17
• s v17 4.12

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Standard Deviation with Grouped Data

• grouped mean (22 36 46 52) / 16 3.5
• deviations
• 2 2 3.5 -1.5
• 3 3 3.5 -0.5
• 4 4 3.5 0.5
• 5 5 3.5 1.5
• s² 2(-1.5)² 6(-0.5)² 6(0.5)² 2(1.5)²
• 16
• s² 0.7499
• s v0.7499 0.866

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Exercises

• read through the examples on pages 164-167
• try page 168 2b, 3b, 4, 6, 7, 10
• you are responsible for knowing how to do simple
  examples by hand
• however, we will use technology to calculate any
  serious examples (Fathom/TI83)
• have a look at your calculator and see if you
  have this feature

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Normal Distribution

• Chapter 3.4 Tools for Analyzing Data
• Mathematics of Data Management (Nelson)
• MDM 4U
• Author Gary Greer (ideas from K. Myers)

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Histograms

• histograms may be skewed...

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Histograms

• ... or symmetrical

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Normal?

• a normal distribution creates a histogram that is
  symmetrical and has a bell shape, and is used
  quite a bit in statistical analyses
• also called a Guassian Distribution
• it is symmetrical with equal mean, median and
  mode that fall on the line of symmetry of the
  curve

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A Real Example

• the heights of 600 randomly chosen Canadian
  students from the Census at School data set
• the data approximates a normal distribution

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The 68-95-99.7 Rule

• area under curve is 1 (ie it represents 100 of
  the population surveyed)
• approx 68 of the data falls within 1 standard
  deviation
• approx 95 of the data falls within 2 standard
  deviations
• approx 99.7 of the data falls within 3 standard
  deviations
• http//www.ruf.rice.edu/lane/hyperstat/A25329.htm
  l

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Distribution of Data
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Why is the normal distribution so important?

• many psychological and educational variables are
  distributed approximately normally
• reading ability, memory, etc.
• normal distributions are statistically easy to
  work with
• all kinds of statistical tests are based on it
• Lane (2003)

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Normal Distribution

• the notation above is used to describe the normal
  distribution where x is the mean and s² is the
  variance (square of the deviation)
• the area under any normal curve is 1
• the percent of data that lies between two values
  in a normal distribution is equivalent to the
  area under the normal curve between these values
• see examples 2 and 3 on page 175

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An example

• suppose the time before burnout for an LED
  averages 120 months with a standard deviation of
  10 months (and is approximately normally
  distributed)
• what is the length of time a user might expect an
  LED to last?
• 95 of the data will be within 2 standard
  deviations of the mean
• this will mean that 95 of the bulbs will be
  between 120 210 months and 120 210
• so 95 of the bulbs will last 100-140 months

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Example continued

• suppose you wanted to know how long 99.7 of the
  bulbs will last?
• this is the area covering 3 standard deviations
  on either side of the mean
• this will mean that 99.7 of the bulbs will be
  between 120 310 months and 120 310
• so 99.7 of the bulbs will last 90-150 months
• this assumes that all the bulbs are produced to
  the same standard

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Example continued
99.7
95
34
34
13.5
13.5
2.35
2.35
120
140
150
100
90
months
months
months
months
months
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Exercises

• try page 176 1, 3b, 6, 8, 9, 10

26
References

• Lane, D. (2003). What's so important about the
  normal distribution? Retrieved October 5, 2004
  from http//davidmlane.com/hyperstat/normal_distri
  bution.html
• Wikipedia (2004). Online Encyclopedia. Retrieved
  September 1, 2004 from http//en.wikipedia.org/wik
  i/Main_Page