Central Tendency

  • measures of central tendency summarize the middle or most typical values for a variable

    • mode

    • median

    • mean

  • two factors guide choice of measure for summarizing data

    • level of measurement

    • shape of distribution

mode

  • mode: the value of the most common score of the variable

    • the only and best measure for nominal variables

      • complement with frequencies of each category

      • ex: self-reported employment in n=500 UNLV students

    • could use for ordinal, interval, and ratio if the values are discrete

      • discrete variables can take on a finite number of values

      • ex: survey response to “I am relaxed most of the time”

  • technically mode is the absolute peak

    • sometimes there is another peak slightly lower but is visible in frequency distributions

  • multiple peaks in a distribution often means there are sub-groupings in the sample

    • you could consider separating them for further analyses

what might the distribution shape look like, where would two peaks be?

multimodal

3 and 1/2

  • if quantitative and multimodal consider why

    • are there naturally occurring groups in the data that should be examined separately?

    • were different versions of a measure used?

    • was there some procedural difference that could have created unintended grouping in the data?

median

  • median: the middle value in a frequency distribution

    • the point that falls in the middle of all points when putting the data in chronological order

    • in a cumulative frequency table, the median is the point at the 50th percentile

      • half of the measurements are below it and half are above

    • the ordinal middle of the distribution

  • if there are an odd number of data points:

    • put the points in order

    • count the points (n)

    • median is the value of item at (n+1)/2

  • if there are an even number of data points:

    • put the points in order

    • count the points (n)

    • take points surrounding (n+1)/2

    • median is the midpoint of these two points

  • does not use all of the values in the data

  • could use for ordinal, interval, and ration scales

  • be cautious with averaging of middle two ordinal values

    • should not do arithmetic with values on an ordinal scale because they’re not incremental in value

    • complement with frequencies if small number of levels

  • best when:

    • shape of the distribution for interval/ratio variable is skewed

mean

  • mean: typically what people refer to when they say average

  • the arithmetic center of a distribution

  • a balance point of all scores

  • use the median when distribution is heavily skewed

  • uses all of the values in the data

    • implications

      • may be misleading for multi-modal data

      • misleading for heavily skewed distributions

      • influenced by extreme values

  • best for interval and ratio scales

  • best when

    • shape of the distribution for interval/ratio variable is symmetrical (not heavily skewed) and unimodal (not bi-modal identified sub-groups)