Measures of Variability

• Location measures do not provide a proper summary of the nature of a data set.
• We can not make meaningful conclusion without considering sample variability. Example:
  • Each data set contains two samples and the difference in the means is roughly the same.
  • Data set B provides sharper distinction between two populations.

Figure 1.7: Different data sets. Difference in the means is roughly the same.

• The simplest measures of sample variability is the sample range $X_{max}-X_{min}$.
• The range can be very useful in statistical quality control.
• The question ``How far is each x from the mean?''
• The one that is used most often is the sample standard deviation.
• The sample variance, denoted by $\sigma^2$
  $$\sigma^2=\sum_{i=1}^n\frac{(x_i-\bar{x})^2}{n-1}$$

• The sample standard deviation, denoted by $\sigma$, is the positive square root of $\sigma^2$, that is
  $$\sigma=\sqrt{\sigma^2}$$

• The sample variance is measured in squared units. The sample standard deviation is in linear units.

• For a bell-shaped distribution,
  • within one standard deviation of the mean there will be approximately (empirically) 68% of the data;
  • within two standard deviations of the mean there will be approximately 95% of the data;
  • within three standard deviations of the mean there will be approximately 99.7% of the data.
• That is,
  $$x \pm \sigma \approx 68\%,   x \pm 2\sigma \approx 95\%,   x \pm 3\sigma \approx 99.7\%.$$

• This is a rule of thumb. Since the range $R \approx 6\sigma$, the rule is also called the $6\sigma$-rule.
• An observation beyond ( $x - 2\sigma, x + 2\sigma$) can be declared as an outlier.

• The quantity $n-1$ is called the degrees of freedom associated with the variance estimate.
• It depicts the number of independent pieces of information available for computing variability. Only $n-1$ terms can vary freely.
• In general,
  $$\sum_{i=1}^n(x_i-\bar{x})=0$$

• The computation of a sample variance does not involve $n$ independent squared deviations from the mean. For example,
  • for the data set (5, 17, 6, 4), the sample mean is 8.
  • The variance is
    $$(5-8)^2+(17-8)^2+(6-8)^2+(4-8)^2$$
    $$=(-3)^2+(9)^2+(-2)^2+(-4)^2$$

  • The quantities inside parentheses sum to zero.

• Example 1.4. An engineer is interested in testing the ``bias'' in a pH meter. Data are collected on the meter by measuring the pH of a neutral substance (pH = 7.0). A sample of size 10 is taken with results given by
  $$7.07~7.00~7.10~6.97~7.00~7.03~7.01~7.01~6.98~7.08$$
  $$\bar{x}=7.0205$$
  $$\sigma^2=0.001939$$
  $$\sigma=\sqrt{0.001939}=0.0440$$
  with 9 degrees of freedom.

• In statistical inference, we like to draw conclusions about characteristics of populations, called population parameters.
• Population mean and population variance are two important parameters.
• The sample variance is used to draw inferences about the population variance.
• The sample standard deviation and the sample mean are used to draw inferences about the population mean.
• In general, the variance is considered more in inferential theory, while the standard deviation is used more in applications.