Measures of Central Tendency

• In the first section we talked about collecting data.
• In the second we discussed organizing data.
• We will begin to look at ways to "measure" data
  • Central tendency or average, middle, ....
  • Dispersion or variance (next section)
• Average is a term we usually associate with the mean, but ...
• The arithmetic mean or just mean is x̄

  •     Σ xi
    x̄ = ----
         n
    

  • The mean considers all data.
  • Extreme data points can throw off the mean.
  • Consider salaries.
  • But this is probably the most common measure of central tendency.
• The median is the value in the middle of a set of ranked data.
  • To find the median:
    • Sort the data
    • If n is odd the median is the value x_n/2+1
    • If n is even, the median is (x_n/2+x_n/2+1)/2
  • Sort of takes all data into account.
  • But clustered data will have a bad impact on the median.
  • Consider a group of parents and children, where one parent brings two or more children to an event.
  • The median is good for data like salaries.
• The mode is the value of the data that occurs most frequently.
  • To find the mode
    • Count the number of occurrences of each data value
    • The one occurring most frequently is the mode.
    • in bi modal data, there are two modes.
    • Some books do not allow bi modal data.
    • I talked to a stat person and they said if one data value is not in the set of modes, the the data has a mode.
  • The mode is used very infrequently
  • Except for retail.
• The mid range
  • mid range = (low + high)/2
  • Only uses two pieces of data.
  • Extremes will really throw it off.
• Measures of position
  • The median is really a measurement of what is in the middle of the data.
  • If we split the data into two sets, all data below the median and all data above the median we could find the median of those two sets.
    • These would be the first quartile and the third quartile.
    • 1/4 of the data is below Q₁
    • 1/4 of the data is between Q₁ and Q₂
    • 1/4 of the data is between Q₂ and Q₃
    • 1/4 of the data is above Q₃
  • If we do this with 99 positions instead of 4, we have percentiles.
  • Percentile is used frequently when comparing standardized test results.
  • It does not tell us the score, but where our score lies.
• Do some problems.