Measures of Location and Spread

• Measure of central tendency


  Describes the centre of the data
• Measures of central tendency


  • Mode or modal class
  • Median
  • Mean
• Mode or modal class


  The value or class which occurs most often
• Median


  The middle value when the data values are put in order
• Mean


  Calculated using Σx/n
• Calculating mean from cumulative frequency table


  Σxf/Σf
• Mean uses all values in the data therefore it gives a true measure of data, but it is affected by extreme values
• Calculating mean, median and modal class for continuous data in grouped frequency table


  Find the midpoint of each class interval
• Median (Q2)


  Splits the data into two equal halves (50%)
• Lower quartile (Q1)


  One quarter of the way through the dataset
• Upper quartile (Q3)


  Three quarters of the way through the dataset
• Percentiles


  Split the data set into 100 parts
• Finding lower and upper quartiles for discrete data


  Divide n by 4 (lower quartile) or find 3/4 of n (upper quartile)

  2. If this is a whole number, the lower or upper quartile is the midpoint between this data point and the number above. If it is not, round up and pick this number.
• Calculating standard deviation from grouped frequency table

  Find Σfx^2, Σfx and Σf

  2. Use formula σ^2 = Σfx^2/Σf - (Σfx/Σf)^2

  3. Square root variance to find standard deviation
• Coding


  Each value in the data can be coded to give a new set of values, which is easier to work with
• Coding using formula y = (x-a)/b


  Mean of coded data: y_bar = (x_bar-a)/b

  Standard deviation of coded data: σ_y = σ_x/b
• Estimating medians, quartiles and percentiles from grouped frequency table


  Assume data values are evenly distributed within each class

  Q1 = (n/4)th data value

  Q2 = (n/2)th data value

  Q3 = (3n/4)th data value
• Measures of spread


  • Range
  • Interquartile range (IQR)
  • Interpercentile range
• Variance (σ^2)


  Shows how spread out the data is
• Calculating variance


  σ^2 = Σ(x-x_bar)^2/n

  2. σ^2 = Σx^2/n - (Σx/n)^2

  3. σ^2 = Σf(x-x_bar)^2/Σf = Σfx^2/Σf - (Σfx/Σf)^2
• Standard deviation (σ)


  Square root of variance
• Calculating standard deviation from grouped frequency table


  σ = √(Σf(x-x_bar)^2/Σf) = √(Σfx^2/Σf - (Σfx/Σf)^2)