Collecting and Organizing Data
This module describes methods of organizing data in an orderly way to facilitate the discovery of trends, and the use of statistical procedures to evaluate results. As you read this module, keep in mind the following questions.
Even a simple experiment can result in the accumulation of large quantities of data
It is quite common to make as many as 10,000 observations in a single experiment. Making that many observations is an impressive task in itself, even more so when we recall that each observation must be recorded, organized, summarized, analyzed, and interpreted 10,000 times each. Fortunately, one can save a great deal of time and reduce the probability of errors simply by deciding in advance how the data will be recorded and organized.
For example, assume we are carrying out a simple animal study which requires that we record the time it takes a rat to run down an alley (the elapsed time is referred to as a latency). Four groups, of 12 rats each, will run the alley six times a day for seven days. For 10 days (acquisition days) each rat is to receive a reinforcer (food or water) for running the alley and for the other seven days (extinction days) the rats receive no reinforcers. Thus we will make 4,896 observations of latency, calculated as follows:
10 acquisition days X 48 rats X 6 trials = 2,880 7 extinction days X 48 rats X 6 trials = 2,016 ________ Total Latencies measurements 4,896 We will also record each rat's daily weight (48 X 17) 816 ________ Total observations 5,712
Even in a relatively uncomplicated study such as the one described here, it is worthwhile to record the observations in a way that will facilitate further analysis. One would probably record each day's data for each group on a different card or sheet, such as the one illustrated in Figure 3.
Group X (Blue) Date: Sept.30 S# Weight N 1 322.7 2.41 2.22 1.93 2.15 2.11 2.18 2 319.5 2.31 2.14 2.83 2.52 198.16 2.91 3 309.6 2.68 2.23 2.37 2.21 6.39 2.81 4 321.1 2.07 1.82 1.80 2.04 4 60 2.17 5 318.3 2.27 2.62 2.80 2.44 2.65 2.41 6 327.2 3.67 3.41 3.84 5.07 6.80 90.88 7 341.1 1.71 1.77 1.82 3.15 2.99 2.94 8 357.2 2.82 2.08 2.13 2.10 1.96 139.05 9 338.8 5.98 2.57 2.32 2.94 3.14 94.50 10 338.1 1.92 2.09 2.44 2.15 6.90 2.73 11 332.8 1.72 1.76 1.63 1.75 2.01 2.15 12 314.6 2.00 1.99 2.17 2.39 2.06 1.76
Figure 3. A sample data card containing raw latency scores for Group X on the second day of extinction training
Proper recording, as observations are made, can greatly facilitate subsequent tasks
Note that the experimenter is finding the average daily latency, or time it takes each animal to run each day's trials. Also note that the card is dated and that the X condition has the word "blue" beside it. This information is derived from the fact that all animals in this group have blue markings on their cages. Each animal in a group should be given an identifying mark or number. Some experimenters notch their animals' ears according to prescribed numbering schemes; others identify rats by different colored felt- pen rings on their tails. A number scheme is necessary for rats because they are not as different as a typical mother cat's kittens.
The observations made during the study would fill 17 such tables, for a total of 5,712 records. Now we have the problem of processing these records into meaningful evidence.
Group X (Blue) Date: Sept.30 Rat# N N N N N N M Mdn 1 .41 .45 .52 .47 .47 .46 .46 .47 2 .43 .47 .35 .40 .01 .34 .33 .38 3 .37 .45 .42 .45 .16 .36 .37 .40 4 .48 .55 .56 .49 .22 .46 .46 .49 5 .44 .38 .36 .41 .38 .41 .40 .40 6 .28 .29 .26 .20 .15 .01 .20 .23 7 .58 .56 .55 .32 .33 .34 .45 .45 8 .35 .48 .47 .48 .51 .01 39 .48 9 .17 .37 .43 .34 .32 .01 .28 .33 10 .52 .48 .41 .47 .11 .37 .39 .44 11 .58 .57 .61 .57 .50 .47 .55 .57 12 .48 .50 .46 .42 .49 .57 .49 .49 MM = .40 Mdn Mdn =.45
Figure 4. A data card containing the reciprocals (1/Latency) of the raw data shown in Figure 3
In Figure 4, the latency data from Figure 3 has been transformed to speed data by taking the reciprocal of each latency. One divided by 2.41 equals .41; one divided by 2.22 equals .45; and so forth. Experimenters sometimes carry out transformations on their data to normalize the scale. Summarizing data as it is collected allows us to carry out statistical calculations with less effort at the completion of the research.
A graph can provide a visual image of a pattern or trend
Finally, after a study is completed, we can graph the data to get an overall view of the results. As usual, we follow the convention of graphing the dependent variable (the response being measured) along the ordinate (y- axis). The length of the ordinate should be 60-75% the length of the abscissa. A typical line graph of the acquisition results from the animal study might appear as in Figure 5. Notice that for all groups the response becomes faster over days of acquisition training. These are learning curves.
Figure 5 (not important, and not shown here) Mean running speed across blocks of acquisition trials (shows a number of lines going up over time).
One of the most informative and frequently used ways of describing data is called the frequency distribution. Figure 6A is an example of a frequency distribution. In this graph, the scores on a final exam have been arranged on the x-axis, from low to high. A single tally mark has been used to represent each of the scores obtained. The graph shows us in detail how the students are distributed among the various score values. Note that the pattern is the same as in the frequency polygon (Figure 6B). The base line in Figure 6B is the same as in Figure 6A, but the vertical axis indicates the number of exams receiving each score. The frequency polygon is constructed by simply connecting the tops of the columns of x's.
The shape of the graphs in Figure 6 suggests a very important psychological fact. The majority of the individuals tested fall in the middle range; there are very few at either of the extremes. This picture is typical of human traits. People are not easily categorized as bright or stupid, or as anxious or calm. Most individuals fall somewhere in between.
x xxx xxxxxx xxxxxxxxxx x xxxxxxxxxxxx xxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx vYvYxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxx xxxxxxxxxxxxxxxxxxxxxxxxx 79 80 81 82 83 84 85 86 87 88 89 90 (A)
Figure 6. Two forms of frequency distribution. Figure A uses a tally mark (x) to indicate the performance of each individual. Such figures are seldom used because they are tedious to construct. Figure B is a frequency polygon of the same data.
The "normal curve " is one of the most important concepts in psychological research
All frequency polygons constructed from real measurements will show
some unevenness. In general, the more measurements we take, the
smoother the graph becomes, because many of the irregularities are the
result of chance variations in the population measured. It can be shown both
mathematically and experimentally that most ideal frequency polygons
eventually approach a shape known as the "normal curve" as the number of
measurements increases and as other conditions of measurement are
controlled. The normal curve approximates a bell shape, as can be seen in
Figure 7. In fact, the terms "normal curve" and "bell-shaped curve" are used
Figure 7. Normal, or bell-shaped curves. Distributions of speed measured at two
stages near the end of training, when a reward of 12 pellets was (1) given immediately
if speed was than .20 (5 sec. elapsed time) and was (2) withheld if speed was greater
than .20 (after Logan, 1960).
Figure 7. Normal, or bell-shaped curves. Distributions of speed measured at two stages near the end of training, when a reward of 12 pellets was (1) given immediately if speed was than .20 (5 sec. elapsed time) and was (2) withheld if speed was greater than .20 (after Logan, 1960).
Figure 8 shows the pattern (called the U-shaped curve) typical of immediate recall memory studies. We tend to remember items at the beginning and end of a list better than those in the middle, which produced the U-shaped curve. This phenomenon occurs in our daily lives. Listen to the words sung in the middle of the "Star-Spangled Banner" at football games. Most of us begin this song convincingly and end convincingly, but muddle through the middle. The "forget-words," words that subjects are instructed to not remember, produce a flat-rectangular or uniform distribution.
Figure 8. Graphic representation of a distribution. Immediate recall probabilities for remember-words and forget-words, as a function of serial position (after Bjork and Woodward, 1973).
figure 9. Typical graphic representation of a J-curve. The number of years of schooling believed "necessary " by those in different socio-economic neighborhoods (after Sherif and Sherif, 1964).
Still other types of social phenomena produce a third pattern, which is called a J-curve because of its resemblance to the letter for which it is named (see Figure 9). This pattern illustrates a rapidly increasing frequency in a small portion of a population. For instance, most people never commit homicide, but a few will become mass murderers; or, most drivers collect few if any traffic violations per year, but a few will amass hundreds.
The histogram, illustrated in Figure 10, is another way to show data graphically. A histogram is a kind of bar graph in which the height of a column indicates the frequency with which a particular value was observed. Notice that the histogram of Figure 10 has a normal shape. If we connected the midpoints of each bar we could convert the histogram into a frequency polygon like that illustrated in Figure 7.
Figures 11A and 11B are, respectively, negatively and positively skewed distributions. The word "skew" means slanted or unsymmetrical. Notice that the J-curve is a special instance of a positively skewed distribution. The negatively skewed distribution represents grades on an easy exam. This type of distribution would have more A's than the normal distribution. Grading on "the curve," (that is, the normal curve) is not always the best of all possible means. In Figure 11A, for example, such a grading system would award a C, the "average" grade, to those who scored at the 90-100% level on the too- easy examination. Who, then, could get an A?
Figure 11A. Graphic representation of negatively skewed distribution. The frequency of examination grades on a statistics test that proved to be too easy for the students.
Figure 11B Graphic representation of a positively skewed distribution. A bar graph of ordinally scaled examination scores on a too-difficult test in statistics.
In Figure 12 we have a bimodal distribution. This shape could emerge when, as shown here, two distinct populations are studied rather than one. The total population of abnormal patients can be broken down into two different diagnostic groups. One group, labeled schizophrenic patients, tends toward introversion, while manic-depressive patients tend toward extroversion. These could be plotted on different graphs, but showing them together provides a comparison of the same characteristic for two groups.
Figure 12. Graphic representation of a bimodal distribution. Distribution of scores on a test of introversion-extroversion in an abnormal population (after Neymann and Yacorzynski 1942).
1. Before generating data from a research study, one should have an orderly plan for recording the data. a. True b. False 2. It is useful to summarize data as it is collected. a. True b. False 3. What shape would a distribution of the heights of women probably take? a. U-shaped b. Positively skewed c. J-curve d. Nommal 4. The independent variable is graphed along the: a. ordinate. b. y-axis. c. abscissa. d. both (b) and (c) 5. The ordinate should be approximately ___________________% the length of the abscissa. a. 20-50 b. 60-75 c. 100-110 d. 50 6. Bimodal distributions: a. may represent more than one population of subjects. b. often occur when we randomly sample from a single population. c. (either of the above may be true) d. (neither of the above) 7. Match. 1) Histogram_____ 2) Frequency polygon______ a. Chart on which frequencies are represented by points connected by lines b. Chart on which all frequencies are represented by columns
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Graphs are not really a statistical device, but they aid experimenters in interpreting their results. The ordinate should be 60-75% as long as the abscissa. Independent variables are plotted along the abscissa. Dependent variables, the behavioral measures, are plotted along the ordinate. Graphs should also be labelled.
Trends emerge from data when they are summarized in a graph. Many variables are normally distributed. This indicates that most scores are in the middle ranges and that there are few extreme scores. For instance, most people have IQ's around 100; very few people have IQ's greater than 140 or less than 60. In other words, the intellectually gifted and the mentally disadvantaged are in a minority.
If we plot the grip strength of a group of 6-year-olds and a group of 9-year-olds on one graph, the distribution will be bimodal. These values represent the average scores for age 6 and age 9 children in the population. In this case, bimodal distribution represents two populations which, if graphed separately, would produce two normal curves.
A class of forty students participated in an experiment. They tried to learn a list of 10 words f lashed rapidly on a screen before them. Here are the scores made by the students on a retention test taken after three learning trials. 9 8 8 7 7 7 7 6 6 6 6 6 6 6 5 5 5 5 6 5 5 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 2 2 1
Construct a frequency distribution table for these scores by counting the number of times each score is achieved. SCORE | FREQUENCY | | | | | | |__________________________| Table A
ln a frequency polygon, frequencies are represented by points connected by lines. Using your frequency distribution table (Table A), draw a frequency polygon for the experiment.
ln a histogram, frequencies are represented by the heights of columns. Draw a histogram for the frequencies in the experiment.
"Skew" refers to a lack of bilateral symmetry. If we drew a line down the middle of a skewed distribution, the two halves would not be mirror images of each other as they would be in a symmetrical distribution.
Skewed distributions are negatively or positively skewed. The distribution above is negatively skewed since the tail of the distribution, where there are few scores. is to the left, or negative end, of the score scale. The right side of the distribution has the greatest number of scores or frequencies. If the tail of the distribution was on the right side, then we would have a positively skewed distribution.
Flat-rectangular, U-shaped, and J-curve distributions are self- descriptive. In flat-rectangular distributions, the scores are evenly distributed along the score scale. In the J-curve, the most scores have small values. In the U distribution, we have many scores at the upper and lower values on the score scale with few frequencies in the middle.
For the following examples, determine whether a graph will be bimodal, negatively skewed, positively skewed, normal, flat- rectangular, U-shaped, or a J-curve. Using the space provided below, draw a stylized graph for each of the hypothetical situations presented.
a. annual income of U.S males
b. grip strength of men and women
c. grades on a difficult exam
d. the number of people that come to a complete stop, slow down, or continue at usual speed at a stop sign
e. distribution of the numbers 1, 2, 3, 4, 5 and 6 as they occur on 600 throws of a die.
ANSWERS a. Positively skewed
c. Positively skewed
Some people find it easier to keep the terms "ordinal" and "abscissa" straight by using a memory aid. As we say the word "ordinal" we open our mouths up and high ( ). The ordinal is the vertical or y-axis. As we say the word "abscissa" our mouths are open wide (a). The abscissa is the horizontal or x-axis.
Match the following.
1, Dependent variable_____
2. Independent variable______
e. horizontal axis
f. vertical axis
NOW TAKE PROGRESS CHECK 2
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Unit 13 Table of Contents