Summarizing Data

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Summarizing Data
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Overview

• To frame our discussion, consider

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Outline

• Distribution
• Measures of Central Tendency
• Measures of Variability

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Distribution

• The distribution of a variable tells us the
  values it takes, and how often it takes the
  values. The distribution is a summary of the
  frequency of individual values or ranges of
  values for a variable.

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Description

• A distribution is described by its shape, its
  center, and its spread.

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Histogram Shapes
symmetric unimodal
bimodal
positively skewed
negatively skewed
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Measures of Central Tendency

• Mode
• Median
• Mean

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Mode

• The mode of a set of measurements is the
  measurement that occurs most often. The
  measurement with the highest frequency.

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Median
The sample median, is the middle value in a
set of data that is arranged in ascending order.
For an even number of data points the median is
the average of the middle two.
Population median
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Mean

• The mean of a set of measurement is the sum of
  the measurements divided by the total number of
  measurements.

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The Mean
The average (mean) of the n numbers
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Three Different Shapes for a Population
Distribution
symmetric
positive skew
negative skew
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Summary

• Mode and median are not influenced by outliers.
• Outliers influence the mean.
• Mode is useful for qualitative and quantitative
  data.
• Median and mean are useful for quantitative data
  only.

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Measures of Variability

• Range
• Percentile
• Interquartile Range
• Variance
• Standard Deviation

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Range

• The range of a set of measurements is the
  difference between the largest and the smallest
  measurement.

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Percentile

• The pth percentile of a set of ordered
  measurements is the value that has p below it
  and (100-p) above it.

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Interquartile Range

• The interquartile range of a set of measurements
  is the difference between the upper and lower
  quartiles.
• IQR 75 percentile - 25 percentile

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Upper and Lower Fourths
After the n observations in a data set are
ordered from smallest to largest, the lower
(upper) fourth is the median of the smallest
(largest) half of the data, where the median
is included in both halves if n is odd. A
measure of the spread that is resistant to
outliers is the fourth spread fs upper
fourth lower fourth (IQR).
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Outliers
Any observation farther than 1.5fs from the
closest fourth is an outlier. An outlier is
extreme if it is more than 3fs from the nearest
fourth, and it is mild otherwise.
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Boxplots
upper fourth
lower fourth
Scale
median
extreme outliers
mild outliers
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Variance

• The variance of a set of n measurements is the
  sum of the squared deviations divided by either n
  or n-1.
• The choice of n or n-1 depends on whether we are
  dealing with a population or a sample from that
  population.

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Sample Variance
Variance is a measure of the spread of the data.
The sample variance of the sample x1, x2, xn of
n values of X is given by
We refer to s2 as being based on n 1 degrees of
freedom.
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Standard Deviation
Standard deviation is a measure of the spread of
the data using the same units as the data.
The sample standard deviation is the square root
of the sample variance
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Formula for s2
An alternative expression for the numerator of s2
is
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Properties of s2
Let x1, x2,,xn be any sample and c be any
nonzero constant.
where is the sample variance of the xs and
is the sample variance of the ys.
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Standard Deviation

• The standard deviation of a set of measurements
  is the positive square root of the variance.

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Example
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Distribution
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Distribution (3)
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Distribution (4)