11.7 Summary (EMBKJ)

• Histograms visualise how many times different events occurred. Each rectangle in a histogram represents one event and the height of the rectangle is relative to the number of times that the event occurred.

• Frequency polygons represent the same information as histograms, but using lines and points rather than rectangles. A frequency polygon connects the middle of the top edge of each rectangle in a histogram.

• Ogives (also known as cumulative histograms) show the total number of times that a value or anything less than that value appears in the data set. To draw an ogive you need to add up all the counts in a histogram from left to right.

  • The first count in an ogive is always zero.
  • The last count in an ogive is always the sum of all the counts in the data set.

• The variance and standard deviation are measures of dispersion.

  • The standard deviation is the square root of the variance.
  • Variance: \(\sigma^2 = \frac{1}{n}\sum_{i=1}^n (x_i - \overline{x})^2\)
  • Standard deviation: \(\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^n (x_i - \overline{x})^2}\)
  • The standard deviation is measured in the same units as the mean and the data, but the variance is not. The variance is measured in the square of the data units.

• In a symmetric distribution

  • the mean is approximately equal to the median; and
  • the tails of the distribution are balanced.

• In a right (positively) skewed distribution

  • the mean is greater than the median;
  • the tail on the right hand side is longer than the tail on the left hand side; and
  • the median is closer to the first quartile than the third quartile.

• In a left (negatively) skewed distribution

  • the mean is less than the median;
  • the tail on the left hand side is longer than the tail on the right hand side; and
  • the median is closer to the third quartile than the first quartile.
• An outlier is a value that is far away from the rest of the data.