Part Two Descriptive Statistics

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Part Two Descriptive Statistics
2
Chapter Two

• Organizing and Graphing Data

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Two Types of Variables

• Qualitative (categorical)
• Quantitative
• a) Discrete
• b) Continuous

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Organizing data

• Class
• Frequency
• Relative frequency (probability, proportion,
  percent,area)
• Cumulative frequency
• Cumulative relative frequency

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Graphs for Qualitative Data

• Pie Chart (Circle graph)
• Bar Chart
• Demo - Bar/Pie Graphs
• Demo - Breakfast Drinks

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Circle Graph (Pie Chart)

• 1) Establish your class
• 2) Count the frequency of each class
• 3) Compute the relative frequency of each class
• 4) Multiply the relative frequency by 360 degrees
• 5) Mark off the resultant number of degrees for
  each class on the circle

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Graphs for Quantitative Data- 2.2

• Pie charts
• Bar Charts
• Time Series
• Line Charts
• Dot Plots
• Stem and Leaf Plots
• Relative Frequency Histograms

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Numerical Measures of Central Tendency - mean

• Arithmetic Mean the sum of the measurements
  divided by the total number of measurements
• Population symbol for mean is µ
• ( µ is a parameter)
• The symbol for the mean of a sample is x bar
  ( x bar is a statistic)

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Other Measures of Central Tendency

• Mode the most popular ( the data item that
  occurs the most often)
• Median when data is ranked, the median is the
  middle piece of data. When there is an even
  amount of data, the two middle pieces of data are
  added together and divided by two.
• Midrange the highest plus lowest data entries
  divided by two.
• Demo-Mean vs. Median Applet

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Measures of Variability

• Range the highest data item minus the lowest
  data item
• Variance
• Population variance - s² (parameter)
• Sample variance - s² (statistic)
• Standard Deviation
• Population Std Dev. - s (parameter)
• Sample Std. Dev. s (statistic)
• Demo-Standard Deviation

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Histograms With Different Characteristics

• Normal
• Approximately normal
• Skewed right
• Skewed left
• Uni-modal
• Bi-modal
• Gap
• Data Far from others
• How Class Width Affects a Histogram

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Chebyshevs Rule

• Applies to ANY data set
• No useful information is provided for the
  fraction of measurements that fall within one
  standard deviation of the mean.
• At least ¾ of all data will fall within 2
  standard deviations of the mean.
• At least 8/9 of all of the measurements will fall
  within 3 standard deviations of the mean.
• Generally, of the measurements
• will lie within k standard deviations of the
  mean

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Empirical Rule

• Applies to a normal or approximately normal data
  set
• 68.2 of data will fall within 1 standard
  deviation of the mean
• 95.4 of the data will fall within 2 standard
  deviations of the mean
• 99.7 of the data will fall within 3 standard
  deviations of the mean
• Empirical Rule

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Approximating the Standard Deviation

• The standard deviation should be somewhere
  between ¼ and 1/6 of the range

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Numerical Measures of Relative Standing

• Percentiles for any set of n measurements (
  arranged in ascending or descending order), the
  pth percentile is a number such that p of all
  measurements fall below the pth percentile and
  (100-p) fall above the number.
• Z-scores for a normal or approximately normal
  distribution, the z-score of a measurement is the
  distance between the score and the mean of the
  distribution expressed in standard deviations
• Finding area using Z-scores

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Example of Finding a Z-Score

• The length of time that a pregnancy
• lasts has a mean of 266 days and a standard
  deviation of 16 days. Find the z-score for a
  pregnancy that lasts 294 days.

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Outliers Section 2.8

• Outlier an observation that is unusually large
  or small relative to the other values in the data
  set.
• Causes of Outliers
• a) the measurement is incorrectly observed,
  recorded, or entered into a computer
• b) the measurement comes from a different
  population
• c) the measurement is correct but represents a
  rare event

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Definitions Section 2.8

• Lower Quartile ( also called first quartile)
  the First Quartile is the same as the 25th
  percentile.
• Median, Second Quartile, or Middle Quartile the
  same as the 50th percentile
• Upper Quartile, Third Quartile the same as the
  75th percentile
• Interquartile range Third Quartile minus the
  First Quartile

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Box Plot Method to Determine Outliers Section
2.8

• Rectangle with ends (hinges) drawn at the first
  and Third Quartiles
• 1.5IQR from each hinge are the inner fences
• 3IQR from each hinge are the outer fences
• Mark the data that is between the hinges and the
  inner fences and is closest to the inner fence as
  the adjacent values ( draw a line or whisker from
  the hinge to the adjacent value)
• Mark data that is between inner and outer fences
  as suspect outliers
• Mark data that is beyond the outer fences as
  extreme (highly suspect) outliers
• Mark the median

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Graphing Bivariate relationships

• A bivariate relationship is an association
  between two variables that implies that one
  variable has an effect on the other.
• Scatter plot (scatter gram) is a way to show
  visually the relationship between two variables
• Positive relationship
• Negative relationship
• No relationship

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Homework Chapter 2

• Page 26, 2.5
• Pages 37-40, 2.16, 2.28, 2.29
• Page 43, 2.33, 2.34
• Pages 50-53, 2.37, 2.41, 2.42, 2.48
• Page 60, 2.58, 2.59, 2.61, 2.64
• Page 66, 2.66, 2.67, 2.68, 2.69
• Page 73, 2.82
• Page 83, 2.97
• Page 89, 2.108, 2.110

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