Measures of Central Tendency

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Measures of Central Tendency
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Measures of Central Tendency

• Central Tendency values that summarize/
  represent the majority of scores in a
  distribution
• Three main measures of central tendency
• Mean ( Sample Mean µ Population Mean)
• Median
• Mode

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Measures of Central Tendency

• Mode most frequently occurring data point

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Measures of Central Tendency

• Mode (34)/2 3.5

Data Point Frequency
0 2
1 5
2 7
3 14
4 15
5 8
6 5
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Measures of Central Tendency

• Median the middle number when data are arranged
  in numerical order
• Data 3 5 1
• Step 1 Arrange in numerical order
• 1 3 5
• Step 2 Pick the middle number (3)
• Data 3 5 7 11 14 15
• Median (711)/2 9

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Measures of Central Tendency

• Median
• Median Location (N 1)/2 (56 1)/2 28.5
• Median (34)/2 3.5

Data Point Frequency
0 2
1 5
2 7
3 14
4 15
5 8
6 5
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Measures of Central Tendency

• Mean Average ?X/N
• ?X 191 Mean 191/56 3.41

Data Point Frequency X
0 2 0
1 5 5
2 7 14
3 14 42
4 15 60
5 8 40
6 5 30
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Measures of Central Tendency

• Occasionally we may need to add or subtract,
  multiply or divide, a certain fixed number
  (constant) to all values in our dataset
• i.e. this is essentially what is done when
  curving a test
• What do you think would happen to the average
  score if 4 points were added to each score?
• What would happen if each score was doubled?

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Measures of Central Tendency

• Characteristics of the Mean
• Adding or subtracting a constant from each score
  also adds or subtracts the same number from the
  mean
• i.e. adding 10 to all scores in a sample will
  increase the mean of these scores by 10
• ?X 751 Mean 751/56 13.41

Data Point 10 Frequency X
0 10 2 20
1 11 5 55
2 12 7 84
3 13 14 182
4 14 15 210
5 15 8 120
6 16 5 80
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Measures of Central Tendency

• Characteristics of the Mean
• Multiplying or dividing a constant from each
  score has similar effects upon the mean
• i.e. multiplying each score in a sample by 10
  will increase the mean by 10x
• ?X 1910 Mean 1910/56 34.1

Data Point x10 Frequency X
0 0 2 0
1 10 5 50
2 20 7 140
3 30 14 420
4 40 15 600
5 50 8 400
6 60 5 300
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Measures of Central Tendency

• Advantages and Disadvantages of the Measures
• Mode
• Typically a number that actually occurs in
  dataset
• Has highest probability of occurrence
• Applicable to Nominal, as well as Ordinal,
  Interval and Ratio Scales
• Unaffected by extreme scores
• But not representative if multimodal with peaks
  far apart (see next slide)

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Measures of Central Tendency

• Mode

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Measures of Central Tendency

• Advantages and Disadvantages of the Measures
• Median
• Also unaffected by extreme scores
• Data 5 8 11 Median 8
• Data 5 8 5 million Median 8
• Usually its value actually occurs in the data
• But cannot be entered into equations, because
  there is no equation that defines it
• And not as stable from sample to sample, because
  dependent upon the number of scores in the sample

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Measures of Central Tendency

• Advantages and Disadvantages of the Measures
• Mean
• Defined algebraically
• Stable from sample to sample
• But usually does not actually occur in the data
• And heavily influenced by outliers
• Data 5 8 11 Mean 8
• Data 5 8 5 million Mean 1,666,671

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Measures of Central Tendency

• Advantages and Disadvantages of the Measures
• Mean
• Often you will see sums quoted instead of average
  or mean values, you should be wary of these
  statistics because they are easily skewed
• i.e. Statistics for the performance of a
  basketball player are quoted in the newspaper, it
  says that he has 134 points over the course of
  the season, whereas other players average well
  over 200.
• From this you would conclude that he is a
  mediocre player at best, however, it is possible
  that he has played fewer games than other players
  (due to injury)
• Looking at averages, the player actually averages
  50 pts. per game, but has only played three
  games, whereas other players average 20 or less
  pts. over more games
• Using this much richer information, our
  conclusions would be completely different
  AVERAGES ARE ALWAYS MORE INFORMATIVE THAN SIMPLE
  SUMS