Lecture Notes

Chapter 3 Variability
(Modified: 2003-07-16)

• Variability

  The central tendency of a distribution tells much about the distribution. Alone, however, measures of central tendency provide an incomplete picture. Helping to complete that picture are measures of deviance or variability.

  The range of a distribution is a fairly primitive method for assessing variability. The range is computed by subtracting the lowest value in a distribution from the highest and then adding one. The standard deviation is the most common measure of deviation or variability. When used in conjunction with the normal curve, the standard deviation provides a much clearer picture of variability.

   

• Computing the Range

  To compute the range, use the formula:

  • range = X_H - X_L

    X_H = the upper limit of highest score in the distribution

    X_L = the lower limit of the lowest score in the distribution

  • Another formula:

    range = (high score - low score) (new, see p. 51)

    (Can you see that the two formulas are mathematically equivalent?)

• Interquartile Range (IQR) (new)
  • The middle 50% of scores in a distribution
  • The scores from the 25%ile to the 75%ile
  • Calculate by:
    • finding the score representing the 25th%ile (0.25 x N will tell you which score), then find the 25th%ile by counting UP to that score from the BOTTOM of the distribution (interpolate if necessary).
    • finding the score representing the 75th%ile (0.25 x N will tell you which score), then find the 25th%ile by counting DOWN to that score from the TOP of the distribution (interpolate if necessary).
    • Finally, calculate using:
      • IQR = 75th%ile - 25th%ile

     

• The Standard Deviation
  • There are three standard deviations, one for populations s (sigma, a parameter) , one for estimating populations s (s hat), and one for samples S (a statistic). It will take some time before you are comfortable with each type, be patient. We will use two different formula approaches, deviations and raw scores, at first, then we will switch to computer-based calculations.
  • Deviation Approach--think of a deviation as a difference from a standard. In this case, we are looking for differences from the standard of the mean. (This problem appears on p. 57.)
  • Examples: shooting (SDs different, means different), shooting (SDs different, means same), golf, distributions

  Score	Mean	X - m (or X bar)	Deviation (x)	x2
  28	10	28 - 10	18	324
  11	10	11 - 10	1	1
  10	10	10 - 10	0	0
  5	10	5 - 10	-5	25
  4	10	4 - 10	-6	36
  2	10	2 - 10	-8	64
  SX = 60			Sx = 0	SX2 = 450

  • 
  • s or S = ÷S(X-m)/N = ÷Sx²/N = ÷450/6 = 8.66
  • Raw Scores Approach (This problem appears on p. 59.)
    • Formula

X	X2
28	784
11	121
10	100
5	25
4	16
2	4
SX = 60	SX2 = 1050

• Where:
  • SX² = sum of the squared scores

    (SX)² = square of the sum of raw scores

    N = the number of scores (Here N = 6)

• Now, using the data above in the raw scores formula:

  s or S = ÷1050 - (60)²/6/6

  s or S = ÷(1050 - (600)/6)/6

  s or S = ÷(1050 - (600))/6

  s or S = ÷450/6

  s or S = ÷75

  s or S = 8.66

• Estimation the standard deviation of population? Then use: s hat
  • Formula (raw score version recommended)

• (can you see the difference?)
• s hat is being used to estimate s (sigma), however s hat is not a truly unbiased estimator.
• You may also calculate s hat for frequency distributions using the following formula (see p. 62 for an example using both formulae:

• The Variance
  • Mathematically, the variance = the square of the standard deviation. There are two variances, one for populations, s²; and one for estimating populations, s². The formulas are identical to the formulas for the corresponding standard deviations, except that you do not take the square root.

• The variance will become extremely useful to us near the end of the course when we cover the analysis of variance or ANOVA.

• URLs
  • Descriptive Statistics Calculator--does on-line calculations (very slow link)
• Quiz