Lecture Notes

Chapter 2 Organization of Data, Graphs, and Central Tendency
(Modified: 2003-07-09)

• Questionnaire
  1. I can usually achieve what I want when I work hard for it.
  2. Once I make plans I am almost certain to make them work.
  3. I prefer games involving some luck over games of pure skill.
  4. I can learn almost anything if I set my mind to it.
  5. My major accomplishments are entirely due to my hard work and ability.
  6. I usually do not set goals because I have a hard time following through on them.
  7. Bad luck has sometimes prevented me from achieving things.
  8. Almost anything is possible for me if I really want it.
  9. Most of what will happen in my career is beyond my control.
  10. I find it pointless to keep working on something that is too difficult for me.

Answer using the following scale:

1. =Disagree strongly
2. =Disagree
3. =Disagree slightly
4. =Neither agree or disagree
5. =Agree slightly
6. =Agree
7. =Agree strongly

Notice that these answers are in a 7 point Likert Scale format.

Scoring:

On items 1, 2, 4, 5, and 8 add the numbers.

On items 3, 6, 7, 9, 10 reverse the point value. (1=7, 2=6, etc.), add the scores.

Finally, add both sets to get your final score._______________

• Simple Frequency Distributions
  • An ordered arrangement that shows the frequency of each score
• Grouped Frequency Distributions See Appendix B (we will not cover)
  • Uses class intervals to condense simple frequency distribution
    • Recommended Conventions for Class Intervals
      • use 10-20 classes
      • choose a convenient size for interval (i)
      • begin each class interval with a multiple of i
        • if 31 is first score and i = 3, then first interval should start at 30
        • when i =5, make the multiples fall at midpoint
      • the largest scores go at the top
  • Converting unorganized scores into a grouped frequency distribution
    • Find the high and low score
    • Find the range (use upper and lower limits!)
    • Determine i by trial and error (smaller i is better)
    • Begin bottom interval with a multiple of i
    • work your way up the distribution as with the simple frequency distribution
• Graphing Rules
  • Summary of Graphic Rules
    • Line graphs are favored over other types (but not with discontinuous data).
    • Graphs with common scales are favored over graphs with multiple scales.
    • Two-dimensional are favored over three-dimensional.
    • Bar graphs can becoming confusing when multiple independent variables are used.
    • Avoid pie charts and area graphs.
    • Avoid the temptation of adding computer-graphic elements such as shading and color.
• Types of Graphs
  • Frequency Distributions
    • use frequency polygon or histogram for numerical data
    • use bar graph for qualitative data
  • Frequency Polygons
    • use line to connect midpoints of intervals
    • Example
  • Histogram
    • use raised bars to display midpoints of intervals
    • Example
  • Bar Graph
    • use raised bar (but put spaces in between!) to display frequencies of categories
  • Line Graph
    • most common (and clearest, see above)
    • use with numerical data only
  • Box Plots
    • a newer and informative type of graph
    • Example
  • Summary Page
• Describing Distributions
  • Bell Shaped Distributions--largest frequencies in the middle, normal curve
    • In the normal curve, the mean, median, and mode all have the same value!
  • Skewed Distributions
    • positive--largest frequencies in low scores
    • In positively skewed distributions, the mean is greater than the median.
    • negative--largest frequencies in high scores
    • In negatively skewed distributions, the median is greater than the mean.
  • Bimodal Distributions--have two humps
  • Rectangular Distributions--all have similar frequencies
  • J-curves--extremely skewed distributions
• Measures of Central Tendency
  • Overview of Central Tendency

  Knowing the point in a distribution that best describes its center is accomplished by three measures of central tendency. The mean or arithmetic average is the best known of these measures. The mean is computed by dividing the total of all of the data points in a distribution by the number of cases in that distribution. The median is another common measure of central tendency. The median describes the point in the distribution at which half of the cases fall above and below. The median is preferred to the mean in distributions that contain extreme values or outliers because the median is less sensitive to such data points. Related to the median are such measures as quartiles, noniles, and deciles. These divide the distribution into fourths, ninths, and tenths, respectively. The mode is simply the most common value in a distribution. The mode is the least precise of all the measures of central tendency. Also, distributions may have two or more modes. A distribution with two modes is called bimodal. Finally, in the theoretical normal curve, all three measures of central tendency described above fall at the same point.

  • Statistics and Parameters Again
    • Sample measures of central tendency are only estimates of population parameters
• Calculation of Mean and Median
  • Mean

    For ungrouped data

    m or X=SX/N

    For grouped data

    m or X=Sf X/N

  • Median
    • use interpolation

    For ungrouped data

    • count up the scores until you come to the 50% point, that is the median

    For grouped data See Appendix B (we will not cover)

    • divide N/2
    • from the bottom of distribution, go up until you find interval with median in it
    • subtract from N/2 the total frequencies of all intervals below the interval that contains the median
    • take the difference above and divide by the number of frequencies in the interval that contains the median
    • multiply the proportion above by i
    • add the product above to the lower limit of the interval that contains the median
    • you did it!
    • Remember:
      • how big is the interval which contains the median?
      • how many scores are in that interval?
      • assume that the scores in that interval are evenly spaced
      • then interpolate using i / N of scores
  • Examples
    • Mean for ungrouped data
      • find the mean for these scores:

      10, 15, 8, 4, 7, 11, 12, 19, 8, 7, 6, 10, 5

      (9.385)

    • Mean for grouped data See Appendix B (we will not cover)

    Score (X)	Frequency (f)	Midpoint	fX
    4-5	2	4.5	9
    6-7	3	6.5	19.5
    8-9	2	8.5	17
    10-11	3	10.5	31.5
    12-13	1	12.5	12.5
    14-15	1	14.5	14.5
    16-17	0	16.5	0
    18-19	1	18.5	18.5
    S	13		122.5

    Then, using:

    • m or X=Sf X/N

      or 122.5/13 = 9.42

  • Calculate the median of the grouped scores above See Appendix B (we will not cover)
    • divide N/2 (13/2 = 6.5)
    • from the bottom of distribution, go up until you find interval with median in it (8-9)
    • subtract from N/2 the total frequencies of all intervals below the interval that contains the median (6.5-5 = 1.5)
    • take the difference above and divide by the number of frequencies in the interval that contains the median (1.5/2 = 0.75)
    • multiply the proportion above by i (0.75 * 2 = 1.50)
    • add the product above to the lower limit of the interval that contains the median
    • (7.5 + 1.50 = 9)
    • See another example?
  • Weighted means
    • To compute means of means
    • GPA example from text (p. 52)
    • See example below:

• Weighted Average or Mean of Means is:
  • 4800/100 = 48

• URLs
  • HyperStat Online (Rice)--very nice page works, 2003-07-07
  • Minard's Map--Tufte's example of a early classic of visualization works, 2003-07-07
• Quiz