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Questions and Answers
What are the pros of using the mean as a measure of central tendency?
Easily put in equations & manipulated algebraically
What are the cons of using the mean as a measure of central tendency?
Influenced by extreme scores, called Outliers (especially when n is small)
What is resistance in statistics?
Resistance: a measure’s sensitivity to outlying values
How does the median handle outlying values?
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How is the mode affected by outlying values?
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Study Notes
Measures of Central Tendency
- Central tendency refers to the "middle" or "center" of a variable's distribution, described by a single score that best represents the entire distribution.
- There are three main measures of central tendency: Mean, Median, and Mode.
The Mean
- The mean is the mathematical center or average value, also known as the "balancing point" of the distribution.
- The mean is calculated by summing all the values and dividing by the total number of values (n).
- The formula for the mean is: ̄𝑿 = (∑ 𝑿𝒊) / 𝒏 for samples, and 𝝁= 𝒏 (∑ 𝑿 𝒊 ) / 𝑵 for populations.
- The mean is easily manipulated algebraically and plays a critical role in inferential statistics, but it is influenced by extreme scores (outliers) and cannot be used for nominal or ordinal data.
The Median
- The median is the score that divides the distribution in half, corresponding to the 50th percentile, and is often referred to as the "middle location" in a distribution.
- The median is calculated by finding the middle value when the data is arranged in order, and is more resistant to outlying values than the mean.
- The median is not easily used in equations or algebraic manipulations.
The Mode
- The mode is the most frequently occurring value in a distribution.
- The mode is not affected by outlying values, but it is not easily used in algebraic manipulations or equations.
Comparing Measures of Central Tendency
- Each measure of central tendency has its advantages and disadvantages, and the choice of which to use depends on the context and type of data.
The Harmonic Mean
- The harmonic mean is another measure of central tendency, and it is used in specific contexts, such as in the relationship between the arithmetic, geometric, and harmonic means.
- The relationship between the arithmetic, geometric, and harmonic means is: 𝐻 ≤ 𝐺 ≤ 𝑥̅.
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Test your understanding of calculating harmonic mean using a frequency table in statistics and probability theory. Practice finding the harmonic mean for different class intervals and frequencies.
