Business Statistics
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Business Statistics

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Shorab< K.M.

Shorab< K.M.

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Business Statistics Business Statistics Presentation Transcript

  • Slides Prepared by JOHN S. LOUCKS St. Edward’s University
  • Chapter 3 Descriptive Statistics: Numerical Methods Part A
    • Measures of Location
    • Measures of Variability
    x   %
  • Measures of Location
    • Mean
    • Median
    • Mode
    • Percentiles
    • Quartiles
    View slide
  • Example: Apartment Rents
    • Given below is a sample of monthly rent values ($)
    • for one-bedroom apartments. The data is a sample of 70
    • apartments in a particular city. The data are presented
    • in ascending order.
    View slide
  • Mean
    • The mean of a data set is the average of all the data values.
    • If the data are from a sample, the mean is denoted by
    • .
    • If the data are from a population, the mean is denoted by  (mu).
  • Example: Apartment Rents
    • Mean
  • Median
    • The median is the measure of location most often reported for annual income and property value data.
    • A few extremely large incomes or property values can inflate the mean.
  • Median
    • The median of a data set is the value in the middle when the data items are arranged in ascending order.
    • For an odd number of observations, the median is the middle value.
    • For an even number of observations, the median is the average of the two middle values.
  • Example: Apartment Rents
    • Median
    • Median = 50th percentile
    • i = ( p /100) n = (50/100)70 = 35.5 Averaging the 35th and 36th data values:
    • Median = (475 + 475)/2 = 475
  • Mode
    • The mode of a data set is the value that occurs with greatest frequency.
    • The greatest frequency can occur at two or more different values.
    • If the data have exactly two modes, the data are bimodal .
    • If the data have more than two modes, the data are multimodal .
  • Example: Apartment Rents
    • Mode
    • 450 occurred most frequently (7 times)
    • Mode = 450
  • Percentiles
    • A percentile provides information about how the data are spread over the interval from the smallest value to the largest value.
    • Admission test scores for colleges and universities are frequently reported in terms of percentiles.
    • The p th percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p ) percent of the items take on this value or more.
      • Arrange the data in ascending order.
      • Compute index i , the position of the p th percentile.
      • i = ( p /100) n
      • If i is not an integer, round up. The p th percentile is the value in the i th position.
      • If i is an integer, the p th percentile is the average of the values in positions i and i +1.
    Percentiles
  • Example: Apartment Rents
    • 90th Percentile
    • i = ( p /100) n = (90/100)70 = 63
    • Averaging the 63rd and 64th data values:
      • 90th Percentile = (580 + 590)/2 = 585
  • Quartiles
    • Quartiles are specific percentiles
    • First Quartile = 25th Percentile
    • Second Quartile = 50th Percentile = Median
    • Third Quartile = 75th Percentile
  • Example: Apartment Rents
    • Third Quartile
    • Third quartile = 75th percentile
    • i = ( p /100) n = (75/100)70 = 52.5 = 53
      • Third quartile = 525
  • Measures of Variability
    • It is often desirable to consider measures of variability (dispersion), as well as measures of location.
    • For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each.
  • Measures of Variability
    • Range
    • Interquartile Range
    • Variance
    • Standard Deviation
    • Coefficient of Variation
  • Range
    • The range of a data set is the difference between the largest and smallest data values.
    • It is the simplest measure of variability.
    • It is very sensitive to the smallest and largest data values.
  • Example: Apartment Rents
    • Range
    • Range = largest value - smallest value
    • Range = 615 - 425 = 190
  • Interquartile Range
    • The interquartile range of a data set is the difference between the third quartile and the first quartile.
    • It is the range for the middle 50% of the data.
    • It overcomes the sensitivity to extreme data values.
  • Example: Apartment Rents
    • Interquartile Range
    • 3rd Quartile ( Q 3) = 525
    • 1st Quartile ( Q 1) = 445
    • Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80
  • Variance
    • The variance is a measure of variability that utilizes all the data.
    • It is based on the difference between the value of each observation ( x i ) and the mean ( x for a sample,  for a population).
  • Variance
    • The variance is the average of the squared differences between each data value and the mean.
    • If the data set is a sample, the variance is denoted by s 2 .
    • If the data set is a population, the variance is denoted by  2 .
  • Standard Deviation
    • The standard deviation of a data set is the positive square root of the variance.
    • It is measured in the same units as the data , making it more easily comparable, than the variance, to the mean.
    • If the data set is a sample, the standard deviation is denoted s .
    • If the data set is a population, the standard deviation is denoted  (sigma).
  • Coefficient of Variation
    • The coefficient of variation indicates how large the standard deviation is in relation to the mean.
    • If the data set is a sample, the coefficient of variation is computed as follows:
    • If the data set is a population, the coefficient of variation is computed as follows:
  • Example: Apartment Rents
    • Variance
    • Standard Deviation
    • Coefficient of Variation
  • End of Chapter 3, Part A