Lesson 2
Describing Central Tendency, Variability and Skew

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Describing the Central Tendency of a Set of Scores

• Set A has greater spread (over 80 points from 12 to 93)
• Set B has a higher center: the center of A is around 50, whereas the center of B is around 155

There are three common measures of the center of a distribution:
• The mode

  The mode is the most frequent score in a set of measures. Some distributions have no mode, e.g., 123, 154, 167, 132. What is the mode of Set B, above?
  The mode is very unstable--minor fluctuations in the data can change it substantially; for this reason it is seldom calculated. Its principal use is colloquial:"The distribution of X is 'bimodal.'"

• The median

  The median is the middle score in the distribution. In the distribution 32, 35, 36, 43, 74, the median is 36. By convention, the median of the distribution 123, 154, 160, 187 is taken to be half way between the two center values: Median = (154 + 160)/2 = 157.
  

  Watch how a median is determined in a series of steps that make clear its definition and calculation (due to John Behrens).

• The mean

  The mean (average, or arithmetic mean) is the sum of the scores in the distribution divided by the number of such scores. The mean of 12, 13, 23, 43, 32 is
  
  Mean = (12 + 13 + 23 + 43 + 32)/5 = 24.6

  In this statistical calculation studio created by John Behrens, you can enter numbers and see how the mean is computed.

  The mean (or aritmetic mean as it is called) has the property that it is the point on the number line which minimizes the sum of squared distances to all the points in the sample. That is, if the numbers 2, 4, 5, 6 and 8 have a mean of 5, then if we substitute the mean, 5, into the following formula, the sum will be as small as it can possibly be for any value of M:

  Sum = (2 - M)² + (4 - M)² + (5 - M)² + (6 - M)² + (8 - M)²

  When M is the mean, 5, the Sum equals 9 + 1 + 0 + 1 + 9 = 20. Try putting any number but 5 in place of M and see if you can get a smaller Sum than 20; and remember, a "minus times a minus is a plus."

  View a demonstration of the "minimum sum of squares" property of the mean.

        The Mean of Combined Groups

• The Range

  The range is the distance from the smallest to the largest score: in Set A at the top of this page, Range = 93 - 12 = 81.

• The Hinge Spread

  The Hinge Spread, H, is the distance from the 75th Percentile to the 25th Percentile.

• The Semi-Interquartile Range

  The Semi-Interquartile Range is half the Hinge Spread
  Q = H/2 In distributions that are not too terribly strange in their configuration, the Median plus and minus Q creates an interval that contains approximately the middle 50% of the distribution. If you learned that for heights of American males Median = 67" and Q = 1.5", then you could calculate that about half of all American males are between the heights of 65.5" and 68.5"

• The Variance

  The two most common measures of variability, namely the Variance and its close descendant the Standard Deviation, owe their popularity to the importance of the Normal Distribution, which we shall study later. Normal distributions, which play an important role in both descriptive and inferential statistics, are completely determined by two "parameters": their mean and their variance.
  The variance describes the heterogeneity of a distribution and is calculated from a formula that involves every score in the distribution. It is typically symbolized by the letter s with a superscript "2". The formula is

  Variance = s² = Sum (Scores - Mean)²/(n - 1)

  The square root (the positive one) of the variance is known as the "standard deviation." It is symbolized by s with no superscript.
  

  Sq. Root of Variance = Standard Deviation, denoted by s

  Use the formula for the variance and standard deviation to calculate both for these scores: 8, 10, 12, 14, 16. Note that n = 5. Make the calculations on a piece of paper. You should get a variance of 10 and a standard deviation equal to the square root of 10 which is 3.16.

  Here's an online demonstration of how the standard deviaiton is calculated, from John Behrens's stats course.
              Properties of the Standard Deviation As the scores in a distribution become more heterogeneous, more "spread out" and different, the value of the standard deviation grows larger.

  If I told you that the standard deviation of 6th grade students Reading grade equaivalent scores was 1.68 yrs (in Grade Equivalent units) and the standard deviation of their Math scores was 0.94 yrs, you would know that the students are more varied in Reading performance than in Math performance. Can you come up with an educationally sound explanation of why this might be so?

  In normal distributions, roughly two-thirds of the scores lie within a distance of one standard deviation of the mean; 95% lie within two standard deviations of the mean; and 99.7% lie within 3 standard deviations.

Exercises
1. Enter the following numbers into the online calculator and observe the mean, median, variance and standard deviation:

   12.3 21.4 34.5 32.8 42.3 18.6 25.2 28.3 27.1 24.3 31.7

2. Here's the same set of scores as in #1 above except, the largest score, 42.3, has been increased to 68.2. What will happen to the mean and median of thisgroup of scores compared to the group of scores in #1?

   12.3 21.4 34.5 32.8 68.2 18.6 25.2 28.3 27.1 24.3 31.7