What are the main measures of central tendency in statistics?

Measures of Central Tendency

Central tendency refers to statistical measures that identify a single value as representative of an entire dataset. The main measures are:

• Mean: The arithmetic average.
• Median: The middle value when data are ordered.
• Mode: The most frequently occurring value.

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1. Mean

The mean of $n$ numbers $x_1, x_2, \ldots, x_n$ is:

$$ \text{Mean} = \bar{x} = \frac{1}{n} sum_{i=1}^{n} x_i $$

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2. Median

The median is the middle value when data are arranged in order. If $n$ is odd, it is the middle value; if $n$ is even, it is the average of the two middle values.

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3. Mode

The mode is the value that appears most often in the dataset.

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Worked Example

Suppose we have the dataset: $4, 7, 7, 10, 12$ **Step 1: Mean** %%MATH_DISPLAY_1%% **Step 2: Median** - Ordered data: $4, 7, 7, 10, 12$ - $n = 5$ (odd), so median is the third value: $7$ **Step 3: Mode** - $7$ appears twice, more than any other value. - Mode = $7$

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Takeaways

• The mean is sensitive to extreme values (outliers).
• The median is robust to outliers and skewed data.
• The mode is useful for categorical or discrete data.