WHYshould we take these measurements?

• Answers the question, "What is a typical student in this class doing?"
  
  
• allows you to compare performance of any single student against group
  
  
• parents want to know whether their child better or worse than typical child at this grade level
  
  

Mean

To Calculate:

1. add all the scores
   
2. divide by number of scores
   = arithmatic average



• single most commonly used statistic
• what most of us mean when we say "average" if I wanted to intimidate you, I'd write it as:
  PLACE SCANNED IMAGE HERE OF MATH EQUATION

CHOOSE ANY 5 NUMBERS AT RANDOM FROM FIRST GROUP, AVERAGE THOSE

• very stable measure --> won't vary much taken from any random sample from same population
  
  
• so used a lot in sampling
  
  
• if you have a class of 40 students, can take scores of every second student and do mean on it --> would come out roughly the same
  
  personally, I always add all forty in case it's not random, all smart student's happen to be first half of alphabet, happen to miss the two very highest scores, or lowest, will set you off
  
  
• however, useful trick if you're rushed, have hundreds of averages to calculate on large populations
  
  
• also means that if you change one mark (student appeals) generally won't have to bother recalculating the average because in normal 35 person class changing one person's mark one or two points won't make any significant difference
  
  
• because it's stable across samples usual starting point in calculating other statistics like standard deviation
  
  

However mean is sensitive to extreme scores 100, 5, 4, 3, 2, --> will screw you up

• "The average number of pupils per class" in school district my favorite example of this
  
  
  • total pupils divided by total staff right?
    
  • except principal, clowns in central office have extreme score of 0 students
    
  • so you get figure of "22 kids per class", though there seem to be 36 in your room and every teacher you know....
    
    
• "Average" salary is another one like that
  
  

  Howard Hughes has a 10 billion dollars billion other people have one dollar, mean = 10 dollars which is ten times what a billion people actually have

  So, sometimes 'mean' does not indicate what typical kid is doing in your class because distorted by extreme scores
  
  

  • adult student, or student who is absent 90% of the time
    
    
  • what happens to our average if I add three zeros to our list of grades?

Median

Another way of figuring out what's typical:

• point at which half the scores are above and half are below
  
  "Is my son Johnny in the top half or the bottom half of the class?"
  
  
• not affected by extreme scores
  
• use when you want to minimize impact of extreme score

Disadvantage:

• less stable (varies more between samples selected from same group)
  
• so you have to calculate based on whole class
  
• not used to calculate other stats
  

Mode

Third and final alternative of what's most typical

• most frequent score
  
  
• if there is a tie, then is bimodal, trimodal, multimodal, whatever
  
• if there is one it is unimodal
  
• Easiest measure of central tendency (average) to compute
  
  
  • you just count up
    
  • real fast

• Useful on nominal (naming) scales [or ordinal scales (ranked but not equal distance between points)]
  
  
  • e.g., elections
    
  • male/female
    
    
  • if doing checklists -- can most of the class do it, yes or no?
    
  • can most of the class do it ok, really well, or not at all?
    
  • course evaluation forms -- 3 = undecided
    

• Mean has no meaning on six point scale
  
  
  • useful in class size example --> what does typical teacher actually have in her class? Count the most frequent probably about 30

• Highly unstable -- one or two additional scores can change it

  The Mode of teacher/pupil example = 0
  
  

  • usually group scores using frequency distribution to reduce number of ties