Measures of Central Tendency and Variability

Central Tendency

Many Variables tend to be normally distributed

• number of siblings
• salary of workers in one job category
• number of sunny days per month
• satisfaction with internet browser

By this we mean:

• the values tend to cluster around a ‘middle’ or ‘central’ value
• the further you get from this central value, the fewer observations there are

A Normally Distributed Variable:

• can be quite well-described with two values
• a measure of central tendency
• a measure of variability

So...

You need to know:

• where the ‘middle’ is
• how variable the data are

Measuring the middle:

• arithmetic average (mean)
• middle score (median)
• most frequent score (mode)

Mean

• Can be thought of as the ‘balance point’
• To calculate the mean:
  • Add up the individual values
  • Divide by the number of values

Median

• This is the ‘middle’ score
• To find the median:
  • line up all the scores from lowest to highest
  • count up the number of scores
  • identify the middle score
  • if the # of scores is uneven, the middle score is N/2+.5
  • if the # of scores is even, the middle score is exactly between N/2 and N/2+1
  • Count through your lined-up scores until you find the middle one
  • This is your median
  • If the middle falls between 2 scores, add them together and take the average

Mode

• Put the scores in ‘piles’ with identical scores piled together
• Look for the tallest pile, or the most frequently-occurring score
• This is the mode

Note: If the Mean, Median, and Mode are equal, then the data are normally distributed.

Non-Normal Distributions

• The three values are NOT the same
  • Mean: 13.3
  • Median: 14.75
  • Mode: 15
• This tells us that the data are NOT symmetrical, or normally distributed

• In fact, by the ordering of these values, we know HOW the data are not symmetrical
  • when mean<median<mode
    • there is extra ‘weight’ on the low side
  • when mean>median>mode
    • there is extra ‘weight’ on the high side
  • this is called ‘skew’

Variability

• in addition to identifying the ‘middle’, we need to talk about how variable the data are
• to do this:
  • we use measures of variability
  • these measures tell us how much the individual scores differ from each other

Range

• the simplest measure of variability
• it just tells us the extent of the distribution - how different the largest value is from the smallest
• to calculate the range, just subtract the smallest value from the largest
• when the range of one set of data is greater than another, it may suggest that the data with the greater range may be more variable

The Inter-Quartile Range

• is a more complex measure of variability
• to calculate this value:
  • line up all the scores from lowest to highest
• divide the total number of scores into 4 equal-sized groups:
  • 1 2 3 4
• the median is the dividing line between groups 2 and 3
  • you can also identify the dividing line between groups 1 and 2, and groups 3 and 4
    • the first gives you a score above which 3/4 of your scores fall
    • the second gives you a score above which 1/4 of your scores fall
    • the difference between these is the inter-quartile range

Standard Deviation

• Is the ‘arithmetic’ measure of variability
• Takes into account the difference between each individual score and the group mean
• Calculates a sort of ‘average’ deviation

You can ‘pair’ these measures

• mean-standard deviation
• median-inter-quartile range
• mode-range
  • The first pair is the most ‘powerful’, and most frequently used
  • The second pair is good for skewed data
  • The third pair is rarely used, only for simple descriptive purposes

Questions? email ajohnson@uwo.ca
This page was last updated on Sunday, February 21, 1999 at 08:28:40 PM