Explanation of Variables, Distribution, and Symmetry as They Pertain to Statistics

  1. Explain and give an example for each of the following types of variables:
    1. Equal-Interval: Variables in which there are equal distances between each value. An example of equal-interval variables would be S.A.T. scores.    
    1. Rank-Order: Variables in which the values represent relative standing to other values. An example of rank-order variables would be the order in which a set of runners finished a race.
    1. Nominal: Variables in which scores represent categories or names rather than numerical values. An example of nominal variables would be what each student at a high school chose for lunch one day.
    1. Ratio Scale: A ratio scale constitutes a set of equal-interval variables with the addition of an absolute zero point. An example would be the number of times that a set of student used the rest room during a class.  
    1. Continuous: A continuous variable, in contrast to a discrete variable, is a variable in which the value can constitute an infinite number of values. An example would be the time that it would take to run one mile (i.e. 1.24 minutes).
  2. Following are the speeds of 40 cars clocked by radar on a particular road in a 35-mph zone on a particular afternoon:
    1. Describe the general shape of the distribution: The distribution is unimodal because there is only one major peak. The distribution is more symmetrical than skewed, but does favor a center-right peak. Also, as to kurtosis the distribution it is heavy-tailed.
  3. Give an example of something having these distribution shapes:
    1. Bimodal: The age of all individuals at a university. The instructors would probably on average be much older than the students, with fewer in the middle.
    1. Approximately Rectangular: The age at which most people receive their driver’s license in the state of Texas. Since the legal age is 16 most people will get their driver’s license around that age. 
    1. Positively Skewed: Race times for the 40-yard dash in high school were positively skewed. Most people finished at about the same time, but there were those that were quite a bit slower. On a graph this would cause a tail on the right of the graph.
  4. Find an example in a newspaper or magazine of a graph that misleads by failing to use equal interval sizes or by exaggerating proportions.
Graph1
Graph1

The second graph is misleading when it comes to the data, because it does not accurately represent “Thing 3”. The last variable looks like it might be only twice the amount of variable 2 when in fact the score is quite a bit more.

Graph2
Graph2
  • Nownes (2000) surveyed representatives of interest groups who were registered as lobbyists of three U.S. state legislatures. One of the issues he studied was whether interest groups are in competition with each other. Table 1–10 shows the results for one such question.
    • Using this table as an example, explain the idea of a frequency table to a person who has never had a course in statistics. This frequency table shows the results of a survey of lobbyists in a logical, straightforward format. The different categories of variables are: no competition, some competition, and a lot of competition. In a single column the results of the actual number of lobbyist for each category is listed, 118, 342, and 131, respectively. In another column the percentage of each category for the total surveyed is listed, 20%, 58%, and 22%, respectively. Finally at the bottom the total surveyed, 591, and the total percentage, which is of course 100%, is listed. 
    • Explain the general meaning of the pattern of results. The pattern of the results basically explains how the scores are distributed over the different possible variables. For instance, in the abovementioned example the pattern goes from low to high to low, 20% to 58% to 22%, respectively. In this example the distribution is unimodal, symmetrical, and comes very close to a normal curve.
  • Mouradian (2001) surveyed college students selected from a screening session to include two groups (Perpetrators and Comparisons):
    • Using this table as an example, explain the idea of a frequency table to a person who has never had a course in statistics. The table splits the distribution of scores between male and female, in order to isolate gender perception and involvement. The table then splits the gender groups into perpetrators, students who reported at least one violent act in their most recent relationship, and comparisons, who reported no violent acts in their last three relationships. Once these distinctions are made in the table, the participants are then asked to write as many examples of circumstances in which a person might engage in acts of this sort with or towards their significant other. The responses are then categorized and the frequency and percentage of each response for each group and subgroup is arranged in columns.   
    • Explain the general meaning of the pattern of results. The distribution for this table would be bimodal in nature, with pronounced spikes at “Control motives/Expression aggression” and “Rejection of perpetrator or act”. The pattern would be slightly more positively skewed than negatively skewed because of the deficit of frequency in the bottom right-hand part of the table.
Frequency Table (15)
SpeedFrequencyPercent
140 
1512.5
2012.5
2312.5
2412.5
2712.5
2812.5
3037.5
3137.5
3237.5
3325
34410
3525
36615
3725
3812.5
4037.5
4212.5
4512.5
4612.5
5225
530 
66840100
Histogram
Histogram
Frequency Polygon
Frequency Polygon

References

Aron, A., Aron, E., & Coups, E. (2006). Statistics for psychology (4th ed.). Upper Saddle River, NJ: Pearson/Allyn Bacon.

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