Lesson 4: Measuring Center and Spread

Lesson 4 consists of four topics.  To see their titles, click on the Contents tab in ActivStats.

Section notes:

4-1 The Center of a Distribution:  This interactive tool activity describes a partially complete method for finding the median.  See the book, p. 28, for the complete method.  Another way:  if there are an odd number of values, the median is the middle value.  If there are an even number of values, the median is the average of the two middle values.

4-1 First Data Desk activity:  This activity shows two ways to calculate the mean using Data Desk:

  1. By entering the formula for mean into Data Desk.  This is not how one would do this ordinarily, but is shown here to illustrate how formulas can be used to create derived data in Data Desk, analogous to the use of formulas in Excel.  Write down in your notebook how this is done, so you can do it later with a different formula.  Remember that the hyperview menu of a window is pulled down by clicking on the arrow in the upper left-hand corner of the window.


    2. Note: formulas are case-insensitive, and if the name of a dataset has no embedded blanks, you don't need to enclose it in quotes.  The formula shown, Sum('Interval')/NumNum('Interval') is easier to write as sum(interval)/numnum(interval) .
     

  2. By letting the "built-in" mean function calculate it, as shown.  This is quicker, but still rather awkward!  The Data Desk activity at the end of 4-1 shows a better way.
4-2 The Spread of a Distribution: For the definitions of lower quartile (Q1) and upper quartile (Q3), see the book, p. 32, or use the ActivStats glossary.  Unfortunately, for most sets of data, the defined values exist only approximately.  There are different methods for approximating quartiles; they all give about the same answers.  Here's one for you to use when you compute them by hand:
  1. Sort the data from smallest to largest.
  2. Find the middle item.  If there are an even number of items, the middle item will be an imaginary item "between" the two middle items.
  3. The middle item will define two groups of items: those to its left, and those to its right.
  4. the median of the left-hand group = Q1
  5. the median of the right-hand group = Q3
4-2 Second Data Desk activity:  Again, this gives you practice using formulas with Data Desk.  The last activity will show you how to use the built-in standard deviation function calculate it.

4-4 Standardizing:  Pay particular attention to z-scores (standardized scores); we will be using them extensively.   Raw data are expressed in their natural units: # of bushels of corn, grades on tests, the length of a leaf in inches.  Z-scores re-express these "raw scores" in terms of their distance from the mean (either positive or negative) in units of standard deviations.  For example, 200 bushels of corn might have a z-score of 2.2, meaning that 200 is 2.3 standard deviations above the mean for that particular set of data.  A z-score of -1.5 would correspond to a raw score that is 1.5 standard deviations below the mean.

Homework:
Practice (keep) Credit (hand in)
4.1 4.2
4.3 (1)-(5) 4.4
4.5 4.6 By hand. In part (a) omit software verification.
4.7 By hand.  Use the above method to find the quartiles.
4.8 By hand
4.11 ActivStats H/W 9.  Note: the upper quartile as computed by the above = 57.6 4.12 ActivStats H/W 10
4.19 4.18 Skip (a) and (b). 

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