What are the main measures of central tendency and dispersion in statistics?

Measures of Central Tendency and Dispersion

Definition

Central tendency refers to statistical measures that identify the center or typical value in a dataset. The main measures are:

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

Dispersion (or variability) describes how spread out the data are. The main measures are:

• Range: The difference between the largest and smallest values.
• Variance: The average squared deviation from the mean.
• Standard deviation: The square root of the variance.

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Worked Example

Suppose we have the dataset: $4, 7, 7, 10, 12$ **Step 1: Mean** %%MATH_DISPLAY_0%% **Step 2: Median** Order the data: $4, 7, 7, 10, 12$ There are $5$ values (odd), so the median is the third value: $7$. **Step 3: Mode** $7$ appears twice, more than any other value. So, mode $= 7$. **Step 4: Range** %%MATH_DISPLAY_1%% **Step 5: Variance and Standard Deviation** First, find the deviations from the mean ($8$): - $(4-8)^2 = 16$ - $(7-8)^2 = 1$ - $(7-8)^2 = 1$ - $(10-8)^2 = 4$ - $(12-8)^2 = 16$ \sum: $16 + 1 + 1 + 4 + 16 = 38$

Sample variance:

$$ s^2 = \frac{38}{5-1} = \frac{38}{4} = 9.5 $$

Sample standard deviation:

$$ s = \sqrt{9.5} \approx 3.08 $$

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Takeaways

• Central tendency (mean, median, mode) summarizes the "center" of data.
• Dispersion (range, variance, standard deviation) describes data spread.
• Both types of measures are essential for understanding and comparing datasets.