Measures of Central
Tendency
Measures of Central Tendency

Central Tendency = values that summarize/
represent the majority of scores in a
distribution
Three main measures of central tendency:
Mean ( X = Sample Mean; μ = Population Mean)
Median
Mode
Measures of Central Tendency
Frequency
Mode = most frequently occurring data point
40
35
30
25
20
15
10
5
0
1
2
3
4
5
DV
6
7
8
9
Measures of Central Tendency
Mode = (3+4)/2 = 3.5
Data Point
Frequency
0
2
1
5
2
7
3
14
4
15
5
8
6
5
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 = (7+11)/2 = 9
Measures of Central Tendency
Median


Median Location = (N +1)/2 = (56 + 1)/2 = 28.5
Median = (3+4)/2 = 3.5
Data Point
Frequency
0
2
1
5
2
7
3
14
4
15
5
8
6
5
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
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. 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?
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
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
Measures of Central Tendency
Advantages and Disadvantages of the
Measures:

Mode
1. Typically a number that actually occurs in dataset
2. Has highest probability of occurrence
3. Applicable to Nominal, as well as Ordinal,
Interval and Ratio Scales
4. Unaffected by extreme scores
5. But not representative if multimodal with peaks
far apart (see next slide)
Measures of Central Tendency
Mode
60
Frequency
50
40
30
20
10
0
1
2
3
4
5
DV
6
7
8
9
Measures of Central Tendency
Advantages and Disadvantages of the
Measures:

Median
1. Also unaffected by extreme scores
Data: 5 8 11 Median = 8
Data: 5 8 5 million Median = 8
2. Usually its value actually occurs in the data
3. But cannot be entered into equations, because there is no
equation that defines it
4. And not as stable from sample to sample, because
dependent upon the number of scores in the sample
Measures of Central Tendency
Advantages and Disadvantages of the
Measures:

Mean
1.
2.
3.
4.
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
Measures of Central Tendency
Advantages and Disadvantages of the
Measures:

Mean
Sums/totals vs. average or mean values


i.e. Basketball player has 134 total points this season,
while average of other players is 200 points
What would most people reasonably conclude?
Measures of Central Tendency



What if he played fewer games than other players (due to
injury)?
Looking at averages, the player actually averaged ~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