Chapter 5: Describing Distributions Numerically

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Chapter 5Describing Distributions Numerically
2
Center Finding the Median

• Look for the center of the distribution
• Middle value that divides the histogram into two
  equal areas

3
Spread

• Range the difference between the maximum and
  minimum values.
• Quartiles the median of the upper half of data
  (Q3) and the median of the lower half (Q1)
• Q3 the 75th percentile
• Q1 the 25th percentile

• Interquartile Range (IQR)
• IQR upper quartile lower quartile
• Upper Fence
• Q31.5IQR
• Lower Fence
• Q1 1.5IQR

4
Five-Number Summary

• The median, quartiles, minimum, and maximum of
  the distribution

5
Comparing Groups TI Tips

• Performance of 4th grade boys and girls on an
  agility test.
• How do the 4th graders compare on the agility
  test?
• Use a comparison of the 5-number summaries of
  both distributions.

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Summarizing Symmetric Distributions

• Averaging
• Mean or Median?
• Mean is the point at which the histogram would
  balance.
• Median splits the histogram into two equal areas.
• For a skewed distribution, report the median (as
  the measure of center) rather than the mean.

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Standard Deviation (Spread)

• Standard Deviation
• How far each data value is from the mean.
• Variance is the average of the sum of the squared
  deviations.
• Standard deviation is the square root of the
  variance.

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Variation

• Variation is a fundamental concept in Statistics
• Measures of Spread
• Help us to be precise about what we dont know.
• Tell how well other summaries describe the data.

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Shape, Center, Spread

• Report the following about a distribution
• Shape of the distribution
• Its center
• Its spread

• If the distribution is skewed, report
• Median
• IQR
• If the distribution is symmetric, report
• Mean
• Standard Deviation
• If there are outliers, report mean and standard
  deviation with and without the outliers.

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Summarizing a DistributionTI Tips

• Agility for 4th grade boys
• Box Plot

• Histogram
• Slightly skewed right

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Summarizing a DistributionTI Tips

• 1-Var Stats
• STAT
• CALC 1-var Stats

• Scroll down

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Summarizing a DistributionTI Tips

• Mean equals 21.64
• Median equals 22
• Standard Deviation equals 3.31
• The count is 11
• Minimum is 17
• Maximum is 29
• Q1 equals 18
• Q3 equals 23

• What would be your interpretation?
• Mean or Median?
• Does skew affect your decision regarding measures
  of center and spread?
• What measures would you report as part of your
  interpretation?

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What Can Go Wrong??

• Do a Reality Check.
• Make sure the calculated summaries make sense.
• Sort the values before finding median or
  percentiles.
• Numerical summaries are used with a quantitative
  variable.
• Watch out for multiple modes.
• Try separating the data into different groups.
• If unable to separate, dont summarize the center
  and spread.

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What Can Go Wrong??

• Beware of outliers.
• Consider removing the outlier and recalculating
  and reporting the outliers individually.
• If you can discover the reason behind the
  outlier, remove or correct it.
• If neither option is available, use the median
  and IQR to summarize center and spread.

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What Can Go Wrong??

• Make a Picture!!!!!
• Always draw a histogram or boxplot to check for
  skew and outliers.
• Dont accept a mean or standard deviation without
  some discussion of the distribution (including a
  picture).
• Be careful when comparing groups that have very
  different spreads.
• Try re-expressing to equalize the spread of
  groups.