Variance, Standard Deviation, and Empirical Rule

Range

• The range of a data set is the difference between the largest value and the smallest value.
• Example: San Francisco temperatures
  • Largest value: 63
  • Smallest value: 51
  • Range: $63 - 51 = 12$

Variance and Standard Deviation

• Means and medians measure the center of data.
• Variance and standard deviation measure how spread out data are.
• Small spread: values close to the mean.
• Large spread: values far from the mean.

Variance

• Measure of how far the values in a data set are from the mean on average.
• Computed differently for populations and samples.

Population Variance

• Let $x<em>1, x</em>2, x<em>3, …, x</em>n$ denote the values in a population of size $n$.
• Let $\mu$ denote the population mean.
• Population variance denoted by $\sigma^2$ is given by: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$
• Example: Compute the population variance for San Francisco temperatures.
  1. Compute the population mean $\mu$: $\mu = \frac{\sum x_i}{n} = 57.5$
  2. For each population value $x<em>i$, compute $x</em>i - \mu$.
  3. Square the deviations: $(x_i - \mu)^2$.
  4. Sum the squared deviations: 169.
  5. Divide the sum by the population size $n$ to obtain the population variance: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = \frac{169}{12} = 14.083$

Sample Variance

• When data values come from a sample, the variance is called the sample variance.
• The mean $\mu$ is replaced by the sample mean $\bar{x}$, and the denominator is $n-1$ instead of $n$.
• Sample variance is denoted by $s^2$: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$
• Why divide by $n-1$?
  • Deviations using the sample mean tend to be a bit smaller than the deviations using the population mean.
  • Dividing by $n-1$ provides a correction to avoid underestimating the population variance.
• Example: Battery lifetimes (hours) of six batteries: 3, 4, 6, 5, 4, 2.
  1. Sample mean: $\bar{x} = \frac{3 + 4 + 6 + 5 + 4 + 2}{6} = 4$
  2. Sample variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{(3-4)^2 + (4-4)^2 + (6-4)^2 + (5-4)^2 + (4-4)^2 + (2-4)^2}{6-1} = \frac{1 + 0 + 4 + 1 + 0 + 4}{5} = \frac{10}{5} = 2$

Standard Deviation

• The standard deviation is the square root of the variance.
• The units of the standard deviation are the same as the units of the data.
• Sample standard deviation: $s$
• Population standard deviation: $\sigma$
• Example: San Francisco temperatures population variance: $\sigma^2 = 14.083$
  • Population standard deviation: $\sigma = \sqrt{\sigma^2} = \sqrt{14.083} = 3.753$
• Example: Battery lifetimes sample variance: $s^2 = 2$
  • Sample standard deviation: $s = \sqrt{s^2} = \sqrt{2} = 1.414$

Resistance

• A statistic is resistant if its value is not affected much by extreme values.
• The standard deviation is not resistant.

Excel

• Population variance: =VAR.P(range)
• Population standard deviation: =STDEV.P(range) or SQRT(variance)
• Sample variance: =VAR.S(range)
• Sample standard deviation: =STDEV.S(range) or SQRT(variance)

Empirical Rule

• Applies to data sets with approximately bell-shaped histograms.
• Approximately 68% of the data will be within one standard deviation of the mean: $[\mu - \sigma, \mu + \sigma]$.
• Approximately 95% of the data will be within two standard deviations of the mean: $[\mu - 2\sigma, \mu + 2\sigma]$.
• Almost all of the data will be within three standard deviations of the mean: $[\mu - 3\sigma, \mu + 3\sigma]$.
• Example: U.S. Census Bureau projections for the percentage of the population aged 65 and over.
  • $\mu = 13.25$
  • $\sigma = 1.683$
  • $\mu - \sigma = 11.57$
  • $\mu + \sigma = 14.93$
  • $\mu - 2\sigma = 9.88$
  • $\mu + 2\sigma = 16.61$
  • $\mu - 3\sigma = 8.2$
  • $\mu + 3\sigma = 18.3$
  • Approximately 68% of the data values are between 11.57 and 14.93.
  • Approximately 95% of the data values are between 9.88 and 16.61.
  • Almost all of the data values are between 8.20 and 18.30.
• Example: Average number of days between when a bill was sent out and when the payment was made is 32 with a standard deviation of 7 days.
  • Approximately 68% of the number of days will be between 25 and 39. ($32 \pm 7$).
  • Approximately 95% of the number of days will be between 18 and 46. ($32 \pm 2*7$).
  • Almost all of the data lie between 11 and 53. ($32 \pm 3*7$).