Elementary Statistics for AP Psychology (page 2)
Practice questions for this study guide can be found at:
A large amount of data can be collected in research studies. Psychologists need to make sense of the data. Qualitative data are frequently changed to numerical data for ease of handling. Quantitative data already is numerical. Numbers that are used simply to name something are said to be on a nominal scale and can be used to count the number of cases. For example, for a survey, girls can be designated as "1," whereas boys can be designated as "2." These numbers have no intrinsic meaning. Numbers that can be ranked are said to be on an ordinal scale, and can be put in order. For example, the highest scorer can be designated as "1," the second highest as "2," the third highest as "3," etc. These numbers cannot be averaged. Number 1 could have scored 50 points higher than 2. Number 2 may have scored 4 points higher than 3. If there is a meaningful difference between each of the numbers, the numbers are said to be on an interval scale. For example, the difference between 32° Fahrenheit (F) and 42°F is 10°F. The difference between 64°F and 74°F is also 10°F. However, 64°F is not twice as hot as 32°F. When a meaningful ratio can be made with two numbers, the numbers are said to be on a ratio scale. The key difference between an interval scale and a ratio scale is that the ratio scale has a real or absolute zero point. For quantities of weight, volume, and distance, zero is a meaningful concept, whereas the meaning of 0°F is arbitrary.
Statistics is a field that involves the analysis of numerical data about representative samples of populations.
Numbers that summarize a set of research data obtained from a sample are called descriptive statistics. In general, descriptive statistics describe sets of interval or ratio data. After collecting data, psychologists organize the data to create a frequency distribution, an orderly arrangement of scores indicating the frequency of each score or group of scores. The data can be pictured as a histogram—a bar graph from the frequency distribution—or as a frequency polygon—a line graph that replaces the bars with single points and connects the points with a line. With a very large number of data points, the frequency polygon approaches a smooth curve. Frequency polygraphs are shown in Figure 6.1.
Measures of Central Tendency
Measures of central tendency describe the average or most typical scores for a set of research data or distribution. Measures of central tendency include the mode, median, and mean. The mode is the most frequently occurring score in a set of research data. If two scores appear most frequently, the distribution is bimodal; if three or more scores appear most frequently, the distribution is multimodal. The median is the middle score when the set of data is ordered by size. For an odd number of scores, the median is the middle one. For an even number of scores, the median lies halfway between the two middle scores. The mean is the arithmetic average of the set of scores. The mean is determined by adding up all of the scores, then dividing by the number of scores. For the set of quiz scores 5, 6, 7, 7, 7, 8, 8, 9, 9, 10; the mode is 7; the median is 7.5; the mean is 7.6. The mode is the least used measure of central tendency, but can be useful to provide a "quick and dirty" measure of central tendency especially when the set of data has not been ordered. The mean is generally the preferred measure of central tendency because it takes into account the information in all of the data points; however, it is very sensitive to extremes. The mean is pulled in the direction of extreme data points. The advantage of the median is that it is less sensitive to extremes, but it doesn't take into account all of the information in the data points. The mean, mode, and median turn out to be the same score in symmetrical distributions. The two sides of the frequency polygon are mirror images as shown in Figure 6.1a. The normal distribution or normal curve is a symmetric, bell-shaped curve that represents data about how many human characteristics are dispersed in the population. Distributions where most of the scores are squeezed into one end are skewed. A few of the scores stretch out away from the group like a tail. The skew is named for the direction of the tail. Figure 6.1b pictures a negatively skewed distribution, and Figure 6.1c shows a positively skewed distribution. The mean is pulled in the direction of the tails, so the mean is lower than the median in a negatively skewed distribution, and higher than the median in a positively skewed distribution. In very skewed distributions, the median is a better measure of central tendency than the mean.
Measures of Variability
Variability describes the spread or dispersion of scores for a set of research data or distribution. Measures of variability include the range, variance, and standard deviation. The range is the largest score minus the smallest score. It is a rough measure of dispersion. For the same set of quiz scores (5, 6, 7, 7, 7, 8, 8, 9, 9, 10), the range is 5. Variance and standard deviation (SD) indicate the degree to which scores differ from each other and vary around the mean value for the set. Variance and standard deviation indicate both how much scores group together and how dispersed they are. Variance is determined by computing the difference between each value and the mean, squaring the difference between each value and the mean (to eliminate negative signs), summing the squared differences, then taking the average of the sum of squared differences. The standard deviation of the distribution is the square root of the variance. For a different set of quiz scores (6, 7, 8, 8, 8, 8, 8, 8, 9, 10), the variance is 1 and the SD is 1. Standard deviation must fall between 0 and half the value of the range. If the standard deviation approaches 0, scores are very similar to each other and very close to the mean. If the standard deviation approaches half the value of the range, scores vary greatly from the mean. Frequency polygons with the same mean and the same range, but a different standard deviation, that are plotted on the same axes show a difference in variability by their shapes. The taller and narrower frequency polygon shows less variability and has a lower standard deviation than the short and wider one.
Since you don't bring a calculator to the exam, you won't be required to figure out variance or standard deviation.
Scores can be reported in different ways. One example is the standard score or z score. Standard scores enable psychologists to compare scores that are initially on different scales. For example, a z score of 1 for an IQ test might equal 115, while a z score of 1 for the SAT I might equal 600.The mean score of a distribution has a standard score of zero. A score that is one standard deviation above the mean has a z score of 1. A standard score is computed by subtracting the mean raw score of the distribution from the raw score of interest, then dividing the difference by the standard deviation of the distribution of raw scores. Another type of score, the percentile score, indicates the percentage of scores at or below a particular score. Thus, if you score at the 90th percentile, 90% of the scores are the same or below yours. Percentile scores vary from 1 to 99.
A statistical measure of the degree of relatedness or association between two sets of data, X and Y, is called the correlation coefficient. The correlation coefficient (r) varies from –1 to +1. One indicates a perfect relationship between the two sets of data. If the correlation coefficient is –1, that perfect relationship is inverse; as one variable increases, the other variable decreases. If the correlation coefficient (r) is +1, that perfect relationship is direct; as one variable increases the other variable increases, and as one variable decreases, the other variable decreases. A correlation coefficient (r) of 0 indicates no relationship at all between the two variables. As the correlation coefficient approaches –1 or +1, the relationship between variables gets stronger. Correlation coefficients are useful because they enable psychologists to make predictions about Y when they know the value of X and the correlation coefficient. For example, if r = .9 for scores of students in an AP Biology class and for the same students in AP Psychology class, a student who earns an A in biology probably earns an A in psychology, whereas a student who earns a D in biology probably earns a D in psychology. If r = .1 for scores of students in an English class and scores of the same students in AP Calculus class, knowing the English grade doesn't help predict the AP Calculus grade.
Correlation does not imply causation. Correlation indicates only that there is a relationship between variables, not how the relationship came about.
The strength and direction of correlations can be illustrated graphically in scattergrams or scatterplots in which paired X and Y scores for each subject are plotted as single points on a graph. The slope of a line that best fits the pattern of points suggests the degree and direction of the relationship between the two variables. The slope of the line for a perfect positive correlation is r = +1, as in Figure 6.2a. The slope of the line for a perfect negative correlation is r = –1, as in Figure 6.2b. Where dots are scattered all over the plot and no appropriate line can be drawn, r = 0 as in Figure 6.2c, which indicates no relationship between the two sets of data.
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