Standard deviation Knowledge-Based Study

Study notes

1. Measuring Spread Around the Center

   When analyzing a dataset, measures of central tendency such as the mean tell us where the center of the data lies. However, two datasets can have identical means while having completely different distributions.

   Consider two groups of test scores:

   • Group A: 70, 70, 70, 70, 70 (Mean = 70)
   • Group B: 50, 60, 70, 80, 90 (Mean = 70)

   Although both sets share a mean of 70, the scores in Group B are much more spread out. Standard deviation is a numerical measure that quantifies the typical distance between data points and the arithmetic mean. A small standard deviation indicates that data values cluster closely around the mean, whereas a large standard deviation indicates that the values are widely dispersed.

   Population vs. Sample Standard Deviation

   To calculate standard deviation, we first compute deviations from the mean: $x_i - \mu$ or $x_i - \bar{x}$. Because the sum of deviations around the mean is always zero, squaring each deviation eliminates negative values and emphasizes larger distances.

   The formula depends on whether the data represents an entire population or a sample:

   Population Standard Deviation ($\sigma$)

   When data from every individual in the population is known:

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

   where $N$ is the population size and $\mu$ is the population mean.

   Sample Standard Deviation ($s$)

   When estimating the variability of a population from a representative sample, we divide by $n - 1$ (Bessel's correction) to obtain an unbiased estimate of the population variance:

   $$s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}$$

   where $n$ is the sample size and $\bar{x}$ is the sample mean.

   Calculating Sample Standard Deviation

   Let us calculate the sample standard deviation for the dataset: 4, 8, 6, 5, 7.

   1. Calculate the sample mean ($\bar{x}$):
      $\bar{x} = \frac{4 + 8 + 6 + 5 + 7}{5} = \frac{30}{5} = 6$
   2. Determine the deviations from the mean ($x_i - \bar{x}$):
      • $4 - 6 = -2$
      • $8 - 6 = 2$
      • $6 - 6 = 0$
      • $5 - 6 = -1$
      • $7 - 6 = 1$
   3. Square each deviation:
      • $(-2)^2 = 4$
      • $2^2 = 4$
      • $0^2 = 0$
      • $(-1)^2 = 1$
      • $1^2 = 1$
   4. Sum the squared deviations:
      $\sum (x_i - \bar{x})^2 = 4 + 4 + 0 + 1 + 1 = 10$
   5. Divide by $n - 1$ to find the sample variance ($s^2$):
      $s^2 = \frac{10}{5 - 1} = \frac{10}{4} = 2.5$
   6. Take the non-negative square root:
      $s = \sqrt{2.5} \approx 1.58$

   Important Properties of Standard Deviation

   • Non-negativity: Standard deviation is always greater than or equal to zero ($s \ge 0$). It equals zero if and only if all values in the dataset are identical.
   • Sensitivity to Outliers: Because deviations are squared, extreme values exert a strong influence on standard deviation, making it non-resistant to outliers.
   • Same Units as Data: Unlike variance, which is expressed in squared units, standard deviation shares the exact unit of measurement as the original data (e.g., kilograms, seconds).
   • Effects of Linear Transformations:
     • Adding a constant: Adding or subtracting a constant $c$ to every data value shifts the center but does not change the spread. Thus, the standard deviation remains unchanged.
     • Multiplying by a constant: Multiplying every value by a constant $k$ scales the spread; the new standard deviation becomes $|k| \cdot s$.