Variation

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Variation

• Measures of variation quantify how spread out the
  data is.
• Variation is one of the core ideas in Statistics

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Super-simple measure of variation

• Range highest value lowest value
• Not good for much, but gives us some idea how
  spread out the data is.

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Standard Deviation

• Standard Deviation is a measure of variation
  based on the mean
• Because of this, it can be strongly influenced by
  outliers, just like the mean.
• Standard Deviation is always positive or 0 (zero
  only if all the data are the same)
• The standard deviation has the same units as the
  data

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Calculating Standard Deviation

• Definitional formula

Notice we are measuring variation of the data
from the mean. This formula is for the sample
standard deviation, and is based on the sample
mean and sample size
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Calculating Standard Deviation

• Shortcut Formula

The advantage No need to calculate the mean
first The disadvantage Doesnt make as much
sense
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Example Definitional Form
Data x
7 7-12.3 -5.3 28.09
8 8-12.3 -4.3 18.49
10 10-12.3 -2.3 5.29
11 11-12.3 -1.3 1.69
13 13-12.3 0.7 .49
25 25-12.3 12.7 161.29
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Example Shortcut Form
Data x
7 49
8 64
10 100
11 121
13 169
25 625
74 1128
Sums
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Population Standard Deviation

• If we have the population data, we can calculate
  the population standard deviation. To
  distinguish it, we use a different symbol.

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Variance

• Sample Variance
• Population Variance

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Understanding Standard Deviation

• Main idea
• Bigger value, data is more spread out.
• Smaller value, data is closer together.

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Rule of Thumb

• To very roughly approximate s,
• Rough interpretation
• Most data will be within two standard
  deviations of the mean. In other words,
• Approximate highest value
• Approximate lowest value

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Empirical Rule

• For data sets with a bell-shaped distribution,

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Example

• For a particular fast-food store, the time people
  have to wait at the drive-through has a
  bell-shaped distribution with

Then about 68 of people wait between
About 95 of people wait between
Almost everyone (99.7) of people wait between
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Homework

• 2.5 3, 9, 21, 23, 25, 33