Section 14.2: Descriptive Statistics

14.2 Outline

  1. Measures of Central Tendency
    1. mean
    2. median
    3. mode
    4. weighted mean
  2. Measures of Position
    1. quartiles
    2. deciles
    3. percentiles
  3. Measures of Dispersion
    1. range
    2. boxplot
    3. variance of the population
    4. variance of a random sample
    5. standard deviation


14.2 Essential Ideas

Averages or measures of central tendency are:

  1. Mean The number found by adding the data and then dividing by the number of data values.
  2. Median The middle number when the numbers in the data values are arranged in order of size.  If there are two middle numbers (in the case of an even number of data), the median is the mean of the two middle numbers.
  3. Mode The value that occurs most frequently.  If no number occurs more than once,  there is no mode.  It is possible to have more than one mode.

The mean is the most sensitive average.  It reflects the entire distribution and is the most common average.  The median gives the middle value.  It is useful when there are a few extraordinary values to distort the mean.  The mode is the average that measures “popularity.”  It is possible to have no mode or more than one mode.

Measures of position include quartiles, deciles, and percentiles.

Measures of dispersion include the range, variance, and standard deviation.

The standard deviation of a sample, denoted by s, is the square root of the variance.  To find it, carry out these steps:

  1. Determine the mean of the set of numbers.
  2. Subtract the mean from each number in the set.
  3. Square each of these differences.
  4. Find the sum of the squares of the differences.
  5. Divide this sum by one less than the number of pieces of data. This is the variance of the sample.
  6. Take the square root of the variance. This is the standard deviation of the sample.