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Chapter 3

Measures of Central Tendency

Understanding Central Tendency

• Central tendency

– A statistic that identifies the center of a distribution

– Most typical, common, or frequently occurring score

– Measured as

o Mode o Median o Mean

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The Mode

• Most common or frequently occurring score

– Examine the frequency table to find the most frequent response

– If using a figure, the mode is the highest bar or peak

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The Median

• The value of the variable in the center of the distribution

– Sort the scores from highest to lowest

– Calculate the median score

o Odd number of scores:

– Determine the value of the median score

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The Median

– Sort the scores from highest to lowest

– Calculate the median score

o Even number of scores:

– Determine the value of the median score

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The Median

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The Mean

• The average value of the variable

– To calculate:

o Sum the scores o Divide by the total number of scores

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The Mean

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The Mean

• Calculating the mean from a frequency table

– To calculate:

o Multiply each score (X) by its frequency (f) o Sum the products (fX) o Divide by the total number of scores

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The Mean as a Balancing Point

• In the following set of 9 scores, M = 3.00

• If we subtract the mean from each score:

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Score Mean Score – Mean 2 3.00 -1.00 3 3.00 .00 1 3.00 -2.00 4 3.00 1.00 3 3.00 0.00 1 3.00 -2.00 4 3.00 1.00 6 3.00 3.00 3 3.00 .00

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The Mean as a Balancing Point

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The Sample Mean vs. the Population Mean: Statistics vs. Parameters

• A sample estimates the characteristics of the population

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Target Numeric

characteristic Mean

Sample Statistic

Population parameter 

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Comparison of the Mode, Median, and Mean

• Strengths of the mode

– Easy to determine

– Nominal data

– Actual value

– Multiple modes

– Not affected by outliers

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Comparison of the Mode, Median, and Mean

• Weaknesses of the mode

– Oversimplified use of just one score

– Thus, cannot be used in hypothesis testing

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Comparison of the Mode, Median, and Mean

• Strengths of the median

– More accurate measure in skewed distributions

• Weaknesses of the median

– Relies on only the middle one or two scores

– Cannot be used in hypothesis testing

– Inaccurate for bimodal distributions

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Comparison of the Mode, Median, and Mean

• Strengths of the mean

– Calculated from all the scores in the distribution

– Thus, can be used in hypothesis testing

• Weaknesses of the mean

– Affected by outliers

– May not describe bimodal distributions

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Comparison of the Mode, Median, and Mean

• Unimodal and symmetric distribution

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Comparison of the Mode, Median, and Mean

• Unimodal and asymmetric distributions

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Comparison of the Mode, Median, and Mean

• Bimodal distribution

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Comparison of the Mode, Median, and Mean

• Bimodal distribution

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Measures of Central Tendency: Drawing Conclusions

• Researchers are interested in modality, symmetry and variability

– Focused on helping researchers understand modality

– Helps us know something about symmetry

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Looking Ahead

• Measures of central tendency are used to understand what a set of scores have in common.

• How can we quantify the degree to which a set of scores are different?

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Summary

• Measure of central tendency

• Mode

• Median

• Mean

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Summary

• Sample mean

• Population mean

• Statistic

• Parameter

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