U46 Measures of Dispersion: Essential Formulas

This section covers key formulas and concepts for measuring the spread or variability of a data set, including range, inter-quartile range, variance, and standard deviation.

1 Range and Inter-quartile Range (IQR)

Range

The simplest measure of dispersion. For a set of data, the range is the difference between the largest value $x_{\max}$ and the smallest value $x_{\min}$.

$$ \text{Range} = x_{\max} - x_{\min} $$
$x_{\min}$$x_{\max}$Range

Inter-quartile Range (IQR)

A measure of spread that is less affected by extreme values. It is the range of the middle 50% of the data, calculated as the difference between the upper quartile $Q_3$ and the lower quartile $Q_1$.

$$ \text{IQR} = Q_3 - Q_1 $$

2 Variance and Standard Deviation

Variance for Ungrouped Data

The average of the squared differences from the mean $\mu$. For a data set $x_1, x_2, \dots, x_n$ with mean $\mu$, the variance $\sigma^2$ is given by:

$$ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 $$

Alternatively, a more convenient computational formula is:

$$ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - \mu^2 $$

Standard Deviation

The most common measure of dispersion. It is the positive square root of the variance and has the same units as the original data.

$$ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2} $$

3 Effects of Data Transformations

Adding/Subtracting a Constant

If a constant $k$ is added to (or subtracted from) each data value, the measures of spread (range, IQR, standard deviation, variance) remain unchanged.

Let the original data set have standard deviation $\sigma_x$ and variance $\sigma_x^2$. For the new set $y_i = x_i + k$, we have:

$$ \sigma_y = \sigma_x, \quad \sigma_y^2 = \sigma_x^2 $$

Multiplying/Dividing by a Constant

If each data value is multiplied by a constant $k$, the range, IQR, and standard deviation are multiplied by $|k|$, and the variance is multiplied by $k^2$.

For the new set $y_i = kx_i$, we have:

$$ \sigma_y = |k| \sigma_x, \quad \sigma_y^2 = k^2 \sigma_x^2 $$

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