GTECH 201 Lecture 12

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GTECH 201Lecture 12
Intro to Descriptive Statistics
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Topics for Today

• Measures of Central Tendency
• Mean, Median, Mode
• Sample and Population Mean
• Weighted Means
• Selecting Appropriate Measures of Central
  Tendency
• Measures of Dispersion
• Variance
• Standard Deviation

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Descriptive vs. Inferential

• Descriptive Statistics
• Methods for organizing and summarizing
  information
• Inferential Statistics
• Methods for drawing and measuring the reliability
  of conclusions about a population based on
  information obtained from a sample of the
  population

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Looking at This Data Set
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Overview

• Mean
• Median
• Mode
• Sample and Population Mean
• Weighted Means
• Selecting Appropriate Measures of Central
  Tendency
• Applying these measures

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Mean

• The mean of a set of n observations is the
  arithmetic average
• Mean of n observations x1, x2,x3,.xn is

In Excel, AVERAGE(insert range)
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Median

• The data value that is exactly in the middle of
  an ordered list if the number of pieces of data
  is odd
• The mean of the two middle pieces of data in an
  ordered list if the number of pieces of data is
  even
• The median is a typical value it is the midpoint
  of observations when they are arranged in an
  ascending or descending order

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Mode

• The most frequent data value i.e., any value
  having the highest frequency among the
  observations
• In Excel,you use the functions
• MEDIAN (insert range)
• MODE (insert range)
• Unimodal, Bimodal, Multimodal data sets
• Outliers

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Sample and Population Means

• Mean of a data set
• Population mean if data set includes entire
  population
• Sample mean if data set is only a sample of the
  population

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Weighted Means

• To calculate the mean when your information is
  available only in the form of summary data
• C Interval Freq
• 25 29.9 4
• 30 34.9 5
• 35 39.9 12

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Skewed Distributions
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Skewed Distributions

• When there is one mode and the distribution is
  symmetric
• mean, median, mode are the same
• Positive skew
• mean moves towards the positive tail
• median also pulls towards the positive tail
• Negative skew
• mean moves towards the negative tail
• median also moves towards the negative tail

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Selecting Appropriate Measures

• Mean
• affected by extreme values
• includes all observations, therefore
  comprehensive (useful for interval/ratio data)
• Median
• not affected by the number of observations
• reveals typical situations (used for ordinal
  data)
• Mode
• useful for nominal variables

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Other Useful Calculations

• In addition to the sum of data, Sxwe need to be
  able to calculate

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Variability or Spread

• Mean and the median - limits
• Range coarse measure of variability
• Percentiles
• kth percentile is the point at which k percent of
  the numbers fall below it and the rest are fall
  above it
• 25th percentile (lower quartile)
• 50th percentile (median)
• 75th percentile (upper quartile)
• Interquartile range (difference between the 25th
  percentile value and the 75th percentile value)

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Describing the Spread

• A five number summary
• Median
• Quartiles
• Extremes
• Variance and Standard Deviation
• Measures spread about the mean
• Standard deviation cannot be discussed without
  the mean

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Calculating Percentiles

• In the list of twelve observations
• 4 7 11 11 11 11 14 16 16 24 29
• Compute median, 25th and 75th percentiles

The lower quartile is the median of the 6
observations that fall below the median
The upper quartile is the median of the 6
observations that fall above the median
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Five Number Summary

• Median 11
• Lower Quartile 9
• Upper Quartile 16
• Extremes are 2 and 29
• Can compute the range 27
• In a symmetric distribution, the lower and upper
  quartiles are equally distant from the median

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Variance

• Is the mean of the squares of the deviations of
  the observations from their mean
• Population variance
• Sample variance

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Example
The heights, in inches for five starting players
in a mens college basket ball team
are 67 72 76 76 84 Compute the mean and
standard deviation.
75
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Standard Deviation

• Standard deviation is positive square root of the
  variance
• Variance in our basketball example

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Formulas Standard Deviation
Standard deviation of a sample
Standard deviation of a population
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Example (Continued)
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Short Cut Simpler Formula
Standard Deviation of a sample
Sum of the squares of data values, i.e., you
square each data value and then sum those squared
values
Square of the sum of data values, i.e., you sum
all the data values and then square that sum
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Example (using the short cut)
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Interpreting Std. Deviation

• s and s 2 will be small when all the data are
  close together
• The deviations from the mean
• Will be both positive and negative
• Sum will always be 0
• s is always 0 or a positive number
• s 0 means no spread as s value increases, the
  spread of the data increases
• The units of s are the same as the original
  observations
• s is heavily influenced by outliers

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Coefficient of Variation
CV is the standard deviation described as a
percent of the mean
CV
CV is useful when comparing different sets of
data where sample size and standard deviation are
different