This document discusses measures of central tendency including the mean, median, and mode. It provides examples and definitions for each measure. The mean is the average and is calculated by summing all values and dividing by the total number. The median is the middle value when values are ranked in order. The mode is the most frequent value. The best measure depends on the scale of measurement and shape of the distribution, such as whether it is symmetrical or skewed.

1. Measures of Central
Tendency
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2. Statistics
Measures of Central
Tendency
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3. • Measures of central tendency are scores
that represent the center of the
distribution.
• Three of the most common measures of
central tendency are:
– Mean
– Median
– Mode
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4. Mean
• The most commonly used measure of central
tendency
• When people ask about the “average” of a group
of scores, they usually are referring to the mean.
• The mean is the sum of all the scores in the
distribution divided by the total scores (the
mathematical average).
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5. Mean (con’t)
• Mean of a sample
• Mean of a population
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6. Mean (con’t)
Exam Scores
75 82 72 68 89
X sum all scores
91 78 94 88 75 n = total number of
scores for the sample
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7. Mean (con’t)
Performance and Memory S tudy
2 2 3 3 4 6
5
Frequency
4 4 4 4 1 4
3
0 2
1
0
1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5
Number of Words Recalled
The mean includes the weight of every score.
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8. Pros and cons of using mean
• Pros
– Summarizes data in a way that is easy to understand.
– Uses all the data
– Used in many statistical applications
• Cons
– Affected by extreme values
• E.g., average salary at a company
– 12,000; 12,000; 12,000; 12,000; 12,000; 12,000;
12,000; 12,000; 12,000; 12,000; 20,000; 390,000
– Mean = $44,167
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9. Median
• The middle score of the distribution when all the
scores have been ranked.
• If there are an even number of scores, the median
is the average of the two middle scores.
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10. Median (con’t)
2 Number of Words Recalled in Performance Study
2
3
3
4
4
4
4
4
10
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11. Mode
• The most frequent score in the distribution.
• A distribution where a single score is most frequent has one
mode and is called unimodal.
• A distribution that consists of only one of each score has n
modes.
• When there are ties for the most frequent score, the
distribution is bimodal if two scores tie or multimodal if more
than two scores tie.
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13. Mode (con’t)
2
2 Number of Words Recalled in Performance Study
3
3
The mode is 4.
4
4
4
4
4
10
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14. Mode (con’t)
72 72 73 76 78
81 83 85 85 86
87 88 90 91 92
This distribution is bimodal.
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15. Red Blue Green Yellow
Yellow Green Blue Red
Green Red Yellow Blue
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16. Red Blue Green Yellow
Yellow Green Blue Red
Green Red Yellow Blue
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17. • Calculations
• Key: dependent measure is reaction time
– Time it takes to say the color
• Determine the mean, median, and mode
of the datasets in the handout.
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18. The best measure of central
tendency depends on….
• The scale of measurement.
• The shape of the
distribution.
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19. Scales of Measurement
• Nominal scale = mode
• Ordinal scale = median
• Interval scale = mean, median,
or mode
• Ratio scale = mean, median, or
mode
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20. Shape of the Distribution
Skew refers to the general shape of a distribution
when it is graphed.
Symmetrical = zero skew
Scores clustered on the high or low end
of a distribution = skewed distribution
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21. Symmetrical Distribution
16
14
Frequency
12
10
8
6
4
2
0
24.5 29.5 34.5 39.5 44.5 49.5 54.5 59.5 64.5 69.5
Scores
The mean, median, and mode are the same.
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22. • The normal distribution is the “ideal”
symmetrical distribution
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23. Distributions that are skewed have one
side of the distribution where the data
frequency tapers off
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24. Skewed Distribution
Positive Skew
12
10
Frequency
8
6
4
2
0
27 32 37 42 47 52 57 62 67 72 77
Scores
Tail points in the positive direction.
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25. Skewed Distribution
Negative Skew
12
10
Frequency
8
6
4
2
0
27 32 37 42 47 52 57 62 67 72 77
Scores
Tail points in negative direction.
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26. The mean will either underestimate or
overestimate the center of skewed
distributions.
Positive Skew Negative Skew
12
12
10 10
Frequency
Frequency
8 8
6 6
4 4
2 2
0
27 32 37 42 47 52 57 62 67 72 77 0
27 32 37 42 47 52 57 62 67 72 77
Scores Scores
Mode Mode
Median Median
Mean Mean
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27. Kurtosis
• Measure of the flatness or peakedness of
the distribution
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