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  # Measures of Variation - PowerPoint PPT Presentation
  
  The standard deviation is just the square root of the variance. Measures of Variation. As well as the Central Tendency of the data in a population or sample a second important characteristic of the data is it variability about some center. . Measures of Variation include: The range
  
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  Presentation Transcript
  
  Measures of Variation
  
  As well as the Central Tendency of the data in a population or sample a second important characteristic of the data is it variability about some center.
  
  • Measures of Variation include:
  • The range
  • The Variance
  • The Standard Deviation
  • The Mean Absolute Deviation
  
  Measures of Variation
  
  Standard Deviation of a Population
  
  We will label the population variance to be σ2
  
  And define σ2 = Σi(xi – μ)2/N
  
  Where
  
  μ is the population mean
  
  N is the size of the population
  
  Σi(xi – μ)2 is the sum of the squares of the difference between each item in the population and the mean.
  
  Measures of Variation
  
  {82, 68, 74, 86, 90, 88, 62, 75, 80, 55}
  
  For this student, these grades are the total population of her scores that are used to calculate her mean or average grade. We obtain:
  
  μ = (82 + 68 + 74 + 86 + 90 + 88 + 62 + 75 + 80 + 55)/10
  
  = 760/10 = 76
  
  The mean of this population is 76
  
  Measures of Variation
  
  Having obtained the mean, we can now calculate the variance
  
  {82, 68, 74, 86, 90, 88, 62, 75, 80, 55} and μ =76
  
  σ2 = Σi(xi – μ)2/N
  
  = {(82-76)2 + (68-76)2 + (74-76)2 + (86-76)2 + (90-76)2 +
  
  (88-76)2 + (62-76)2 + (75-76)2 + (80-76)2 + (55-76)2 }/10
  
  = (36 + 64 + 4 +100 + 196 + 144 + 196 + 1 + 16 + 441)/10
  
  = 119.8
  
  μ = 76
  
  σ
  
  σ
  
  Measures of Variation
  
  We find the standard deviation in this population data by taking the square root of the variance.
  
  σ2 = Σi(xi – μ)2/N = 119.8
  
  σ= (119.8)½ = 10.94
  
  If we display the data on a dot plot, we can visualize the use of the standard deviation as a measure of variation in the data
  
  {82, 68, 74, 86, 90, 88, 62, 75, 80, 55}
  
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  55 60 65 70 75 80 85 90 95 100
  
  Mean = 76
  
  Measures of Variation
  
  Chebyshev’s Theorem
  
  The proportion of any set of data lying within K standard deviations of the mean is always at least 1 – 1/K2, for all K greater than or equal to 2.
  
  Chebyshev’s Inequality tells us that in any statistical distribution at least ¾ of the values will lie within 2 standard deviations of the mean, and at least 8/9 of all values will lie within 3 standard deviations of the mean.
  
  In the previous example we found μ = 76 and σ= 10.94
  
  μ - 2σ= 76 – 2(10.94) = 54.12
  
  μ + 2σ= 76 + 2(10.94) =97.88
  
  We find that 100% of the values lie within 2σ of the mean
  
  Measures of Variation
  
  The Sample Standard Deviation
  
  The standard deviation of a sample is denoted by the letter s. The sample standard deviation is an estimate of the population standard deviation σ
  
  _
  
  s2 = Σi(xi – x)2/(n – 1)
  
  Where x bar in the previous formula denotes the sample mean. The sample standard deviation is obtained by taking the square root of the variance.
  
  Note! To calculate the sample variance we divide by the number of degrees of freedom (n – 1) instead of the sample size n. We have already calculated the sample mean when we use the same sample data to obtain a second statistic. Only n-1 of those values are considered free – the nth value is fixed since the sum must equal n times the mean.
  
  Measures of Variation
  
  The formula for the standard deviation can be transformed into a form that slightly simplifies the computation.
  
  s = (nΣi(xi)2 – (Σixi)2)/n(n – 1))½
  
  On first sight it is not clear that we have simplified the calculation, but if we assume that the previous 10 grades were a sample taken from a larger number of students enrolled in a course, then we will illustrate how the two formula are used to calculate the standard deviation.
  
  Measures of Variation
  
  Using the original formula and treating the previous data a sample data with a mean of 76 we get:
  
  _
  
  s = (Σi(xi – x)2/(n – 1))½
  
  {82, 68, 74, 86, 90, 88, 62, 75, 80, 55}
  
  s = (((82-76)2 + (68-76)2 + (74-76)2 + (86-76)2 + (90-76)2 +
  
  (88-76)2 + (62-76)2 + (75-76)2 + (80-76)2 + (55-76)2)/(n-1))½
  
  = (1198/9)½ = 133.11½ = 11.54
  
  Measures of Variation
  
  To use the modified formula, we first construct the following table
  
  {82, 68, 74, 86, 90, 88, 62, 75, 80, 55} n = 10
  
  x x2
  
  82 6724
  
  68 4724
  
  74 5476
  
  86 7396
  
  90 8100
  
  88 7744
  
  62 3844
  
  75 5625
  
  80 6400
  
  55 3025
  
  760 58958
  
  s2 = ((10)(58958)-7602)/(10)(9)
  
  = (589580-577600)/(10)(9)
  
  = 133.11
  
  s = 133.11½ = 11.54
  
  In this second method we find the total of the sample items and the total of the square of each of these items.
  
  Measures of Variation
  
  Finding the standard deviation for tabulated or weighted data
  
  Recall the table we constructed for finding the mean of a sample of September temperature readings in the Central Tendency lecture notes.
  
  Class Midpoint (x)Total (f)f*xx2f*x2
  
  64.5 - 69 .5 67 6 402 4489 26934
  
  69.5 – 74.5 72 11 792 5184 57024
  
  74.5 – 79.5 77 20 1540 5929 118580
  
  79.5 – 84.5 82 13 1066 6724 87412
  
  84.5 – 89.5 87 9 783 7569 68121
  
  89.5 – 94.5 92 1928464 846460 4675 366535
  
  We have augmented the previous table by adding two additional columns that will be used for calculating the sample standard deviation of these grouped data.
  
  Measures of Variation
  
  The formula for obtaining the standard deviation of weighted or tabulated data is:
  
  s = (nΣi(fi * xi2) – (Σi fi * xi)2)/n(n – 1))½
  
  From the previous table we have
  
  nΣi(fi * xi2) = (60)(366535) = 21992100
  
  (Σi fi * xi)2 =(4675)2 = 21855625
  
  s = ((21992100 – 21855625)/(60)(59))½ = 38.55½ = 6.21
  
  6.21
  
  6.21
  
  2s
  
  2s
  
  Measures of Variation
  
  We construct an ogive from the previous table
  
  frequency
  
  60
  
  55
  
  50
  
  45
  
  40
  
  35
  
  30
  
  25
  
  20
  
  15
  
  10
  
  5
  
  0
  
  Mean = 79.183
  
  s = 6.21
  
  x
  
  x
  
  2s = 12.42
  
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  x
  
  64.5 69.5 74.5 79.5 84.5 89.5 94.5
  
  Temperature
  
  2σ  95% of values
  
  3 σ  99.8 % of values
  
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  Measures of Variation
  
  The Normal Distribution
  
  • Continuous
  • Symmetric
  • Mean = Median = Mode (all the same value)
  
  mean
  
  σ  68% of values
  
  Measures of Variation
  
  Other measures of variation
  
  Using the range to estimate the standard deviation
  
  s ~ range/4
  
  On an earlier slide we found for a population of student grades:
  
  {82, 68, 74, 86, 90, 88, 62, 75, 80, 55}
  
  μ = 76 and σ= 10.94
  
  The range of this population = 90 – 55 = 35
  
  This gives us an estimate of σ= 35/4 = 8.75
  
  In the tabulated data for the temp readings we have range = 92 – 65 = 27  s = 27/4 = 6.15 which agrees fairly well with the calculated value of s = 6.21
  
  Measures of Variation
  
  The Coefficient of Variation (CV)
  
  Define: For either a population or a sample the Coefficient of Variation is defined to be the ratio of the standard deviation over the mean
  
  CV = s/ x’ for a sample
  
  Where x’ denotes x bar the sample mean
  
  CV = σ/ μ for a population
  
  The CV for the population of grades from the previous page:
  
  CV = 10.94/76 = 0.144
  
  Part 2
  
  Measures of Relative Standing
  
  Relative Standing
  
  A z score is the number of standard deviations that a raw score, x, is above or below the mean.
  
  A raw score x taken from a population is converted to a standardized z score by the formula
  
  z = (x – μ)/σ
  
  In a sample the z score of a value x is given by
  
  z = (x – x’)/s where x’ denotes the sample mean
  
  Relative Standing
  
  Percentiles
  
  percentile of value x = ((number of values < x)/ total number of values)*100
  
  (round the result to the nearest whole number
  
  Suppose that in a class of 25 people we have the following averages (ordered in ascending order)
  
  42, 59, 63, 67, 69, 69, 70, 73, 73, 74, 74, 74, 77, 78, 78, 79, 80, 81, 84, 85, 87, 89, 91, 94, 98
  
  If you received a 77, what percentile are you?
  
  percentile of 77 = (12/25)*100 = 48
  
  Relative Standing
  
  Quartiles
  
  Instead offinding the percentile of a single data value as we did on the previous page, it is often useful to group the data into 4, or more, (nearly) equal groups. When grouping the data into four equal groupings, we call these groupings quartiles.
  
  Let n = number of items in the data set
  
  k = percent desired (ex. k= 25)
  
  L = locator  the value separating the first k percent of the data from the rest
  
  L = (k/100) * n
  
  L25
  
  Q2
  
  Q1
  
  Q3
  
  7
  
  13
  
  19
  
  Relative Standing
  
  Let’s separate the 25 class grades into four quartiles.
  
  • Step 1 – order the data in ascending order
  
  42, 59, 63, 67, 69, 69, 70, 73, 73, 74, 74, 74, 77, 78, 78, 79, 80, 81, 84, 85, 87, 89, 91, 94, 98
  
  Now find the 3 locators L25, L50, L75,
  
  Round fraction part up to the next integer
  
  L25 = (25/100) * 25 = 6.25
  
  L50 = (50/100) * 25 = 12.5
  
  L75 = (75/100) * 25 = 18.75
  
  Measures of variation
  
  Measure of central tendency
  
  Relative Standing
  
  • Other measures of relative standing include
  • Interquartile range (IQR) = Q3 - Q1
  • Semi-interquartile range = (Q3 - Q1)/ 2
  • Midquartile = (Q3 +Q1)/2
  • 10 – 90 percentile range = P90 - P10
  
  For the data on the previous page we have:
  
  IQR = 84 – 70 = 16
  
  Semi IQR = (84 – 70)/2 = 8
  
  Midquartile = (84 + 70)/2 = 77
  
  L25
  
  median
  
  L75
  
  69 73 77 81 85 89
  
  92
  
  Box Diagram
  
  Recall the ordered high temperature readings from an previous lecture
  
  65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73,
  
  74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78,
  
  78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81,
  
  81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92
  
  To construct a box diagram to illustrate the extent to which the extreme data values lie beyond the interquartile range, draw a line with the low and high value highlighted at the two ends. Mark the gradations between these two extremes, then locate the quartile boundaries Q1, Med., and Q3 on this line. Construct a box about these values.
  
  Q1 = (73 + 74)/2 = 73.5
  
  Q1 M Q3
  
  65