Statistical Reasoning for everyday life

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Statistical Reasoningfor everyday life

• Intro to Probability and Statistics
• Mr. Spering Room 113

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4.3 Measures of Variation
Top of the Muffin to you! ?????

• Variation
• Describes how widely data are spread out about
  the center of a distribution.
• ????How would you expect the variation to differ
  between the running times of theatre movies
  compared to running times for television
  sitcoms????
• Theatre movie times more variation
• Television sitcoms less variation usually 30 or
  60 minutes

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4.3 Measures of Variation

• How do we investigate variation?
• Study all of the raw data
• Range
• Quartiles
• Five-number summary (BOXPLOT or BOX-and-WHISKER)
• Interquartile range
• Semi-quartile range
• Percentiles
• MAD
• Variance Standard Deviation

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4.3 Measures of Variation
65th Percentile!

• Today
• Semi-quartile range
• Percentiles
• MAD
• Variance Standard Deviation

MAD???
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4.3 Measures of Variation

• Semi-quartile range
• The semi-quartile range is another measure of
  spread. It is calculated as one half the
  difference between the Upper Quartile (often
  called Q3) and the Lower Quartile (Q1). The
  formula for semi-quartile range is
• (Q3Q1) 2.
• Since half the values in a distribution lie
  between Q3 and Q1, the semi-quartile range is
  one-half the distance needed to cover half the
  values. In a symmetric distribution, an interval
  stretching from one semi-quartile range below the
  median to one semi-quartile above the median will
  contain one-half of the values. However, this
  will not be true for a skewed distribution.
• The semi-quartile range is not affected by higher
  values, so it is a good measure of spread to use
  for skewed distributions, but it is rarely used
  for data sets that have normal distributions. In
  the case of a data set with a normal
  distribution, the standard deviation is used
  instead. We will discuss standard deviation
  later.

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4.3 Measures of Variation

• EXAMPLE Find the Semi-quartile range of the
  data.
• Semi-quartile
• 4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3, 11.0
• Q1 5.6
• Q3 8.5
• Semi-quartile (8.5 5.6) 2
• 1.45

(Q3Q1) 2
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4.3 Measures of Variation

• Percentiles
• The nth percentile of a data set is (an estimate)
  of a value separating the bottom values from the
  top (100 n). A data value that lies between
  two percentiles is often said to lie in the lower
  percentile. You can approximate the percentile
  of any data value with the following formulas

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4.3 Measures of Variation

• EXAMPLE Percentiles.
• What percentile is the lowest score, Q1, Q2, Q3,
  and highest score?
• 4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3,
  11.0
• Lowest number 0 percentile
• Q1 5.6 25th percentile
• Q2 (7.7 - 7.2)/2 7.45 50th percentile
• Q3 8.5 75th percentile
• Highest number 100th percentile

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4.3 Measures of Variation

• EXAMPLE Percentiles.
• What percentile is the 9.3?
• 4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3,
  11.0
• 9.3 is the 9th number out of ten, after the
  numbers are set in ascending order. Therefore,
  it is larger than 9 out of ten numbers, or the
  90th percentile.

Note One quartile is equivalent to 25 percentile
while 1 decile is equal to 10 percentile and 1
quintile is equal to 20 percentile Think about
it P25 Q1, P50 D5 Q2 median value, P75
Q3, P100 D10 Q4, P10 D1, P20 D2, P30
D3, P40 D4, P60 D6, P70 D7, P80 D8, P90
D9
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4.3 Measures of Variation

• MAD Mean Absolute Deviation
• MAD is the mean of the absolute differences
  between the sample mean and the data values.

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4.3 Measures of Variation

• Example Find the MAD (Mean Absolute Deviation).
• DATA 10, 1, 3, 3, 3, 4, 5, 6, 5, 10
• Mean 5
• ? 5, 4, 2, 2, 2, 1, 0, 1, 0, 5 22
• MAD 22/10 2.2

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4.3 Measures of Variation

• Variance
• The variance of a random variable is a measure of
  statistical dispersion/distribution found by
  averaging the squared distance of its possible
  values from the mean. Whereas the mean is a way
  to describe the location of a distribution, the
  variance is a way to capture its scale or degree
  of being spread out. The unit of variance is the
  square of the unit of the original variable.

Variance
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4.3 Measures of Variation

• Example Find the Variance.
• DATA 10, 1, 3, 3, 3, 4, 5, 6, 5, 10
• Mean 5
• ? 25, 16, 4, 4, 4, 1, 0, 1, 0, 25
• Variance (s2) 80/9 8 8/9 8.89

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4.3 Measures of Variation

• Standard Deviation
• Universally accepted as the best measure of
  statistical dispersion/distribution.
• Standard deviation is developed because there is
  a problem with variances. Recall that the
  deviations were squared. That means that the
  units were also squared. To get the units back
  the same as the original data values, the square
  root must be taken.

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4.3 Measures of Variation

• Example Find the Standard Deviation
• DATA 10, 1, 3, 3, 3, 4, 5, 6, 5, 10
• Mean 5
• ? 25, 16, 4, 4, 4, 1, 0, 1, 0, 25
• Variance (s2) 80/9 8 8/9 8.89
• Standard Deviation
• 2.981

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4.3 Measures of Variation

• The Range Rule of Thumb
• The is approximately related to the range
  of distribution by the following
• We can use this rule of thumb to estimate the low
  and high values
• low value mean 2 standard deviation
• high value mean 2 standard deviation
• The range rule of thumb does not work well when
  low and high values are extreme outliers.
  Therefore, use careful judgment in deciding
  whether the range rule of thumb is applicable.

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4.3 Measures of Variation

• The Range Rule of Thumb
• EXAMPLE
• The mean score on the mathematics SAT for women
  is 496, and the standard of deviation is 108.
  Use the range rule of thumb to estimate the
  minimum and maximum scores for women on the
  mathematics SAT.
• low value mean 2 standard deviation
• 496 (2108) 280 minimum
• high value mean 2 standard deviation
• 496 (2108) 712 maximum
• Is this reasonable?
• Of course, scores below 280 and above 712 are
  unusual on SATs.

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4.3 Measures of Variation

• HOMEWORK
• Pg 174 3
• Pg 175 9, 10, and 14
• Pg 176 24, pg 176 25-27 all (Letters c, d
  only)