Q: Mathematical Properties of Arithmetic Mean and Median
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Mathematical Properties of Arithmetic Mean and Median
Arithmetic Mean:
- Definition and Calculation: The arithmetic mean, commonly known as the average, is calculated by summing all the values in a dataset and then dividing by the number of values. Mathematically, if ( x_1, x_2, \ldots, x_n ) are the values in the dataset, the mean ( \bar{x} ) is given by:
[
\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i
]
Where ( n ) is the number of values. - Sensitivity to Outliers: The arithmetic mean is highly sensitive to extreme values or outliers. A single extremely high or low value can disproportionately affect the mean, making it unrepresentative of the majority of the data.
- Additivity Property: The mean of the sum of two or more datasets is the sum of their means. If you combine datasets ( A ) and ( B ), then the mean of the combined dataset is: [
\bar{x}_{AB} = \frac{n_A \bar{x}_A + n_B \bar{x}_B}{n_A + n_B}
]
Where ( n_A ) and ( n_B ) are the sizes of datasets ( A ) and ( B ), respectively. - Mathematical Expectation: In probability theory, the arithmetic mean is used to represent the expected value of a random variable. It is a measure of the central tendency around which the values of the random variable are distributed.
Median:
- Definition and Calculation: The median is the middle value in a sorted dataset. For an odd number of observations, it is the central value. For an even number of observations, it is the average of the two central values. For a dataset ( x_1, x_2, \ldots, x_n ) arranged in ascending order:
- If ( n ) is odd: ( \text{Median} = x_{\frac{n+1}{2}} )
- If ( n ) is even: ( \text{Median} = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2} )
- Robustness to Outliers: Unlike the mean, the median is not affected by extreme values or outliers. It provides a better measure of central tendency when the dataset contains outliers or is skewed.
- Positional Property: The median divides the dataset into two equal halves. Fifty percent of the values are below the median and fifty percent are above it.
- Comparison with Mean: In a symmetric distribution, the mean and median are equal. However, in skewed distributions, the median provides a better measure of central tendency because it is less affected by skewness and outliers.
In summary, the arithmetic mean and median are fundamental statistical measures with distinct mathematical properties. The mean is sensitive to extreme values and provides a measure of the average, while the median offers a robust measure of central tendency that is resistant to outliers.