Week 6: Descriptive Data Analysis Flashcards
What is descriptive statistics?
Descriptive statistics = set of techniques for summarizing and displaying data.
LETS ASSUME…
Quantitative data
With scores on one or more variables - for several study participants.
It is important to DESCRIBE each variable individually
DESCRIBE the distribution of a Variable
The Distribution of a Variable
EVERY VARIABLE - has a distribution
= scores are distributed ACROSS the levels of that variable
EXAMPLE
Sample of 100 university students :)
the distribution of the variable “number of siblings” might be such that…
- 10 of them have no siblings
- 30 have one sibling
- 40 have two siblings, and so on.
In the same sample, the distribution of the variable “sex” might be such that…
44 have a score of “male”
56 have a score of “female.”
Describe a frequency table and why it is important?
Frequency Tables
One way to DISPLAY the distribution of a variable is in a frequency table.
EXAMPLE
Frequency table showing a hypothetical distribution of scores on the Rosenberg Self-Esteem Scale for a sample of 40 college students.
The first column lists the values of the variable—the possible scores on the Rosenberg scale
The second column lists the frequency of each score.
This table shows that there were three students who had self-esteem scores of 24, five who had self-esteem scores of 23, and so on.
From a frequency table like this, one can quickly see several important aspects of a distribution
- including the RANGE of scores (from 15 to 24)
- the most and least common scores (22 and 17, respectively)
- EXTREME scores that stand out from the rest.
Explain 2 important points about frquency tables…
IMPORTANT - FREQUENCY TABLES
- the levels listed in the first column usually go from the highest at the top to the lowest at the bottom
- and they usually do not extend beyond the highest and lowest scores in the data.
EXAMPLE
ALTHOUGH scores on the Rosenberg scale can VARY from a high of 30 to a low of 0
THE TABLE ONLY includes levels from 24 to 15 because that range includes all the scores in this particular data set.
With MANY different scores across a WIDE range of values, it is often better to create a grouped frequency table,
—– in which the first column lists ranges of values
—– the second column lists the frequency of scores in each range.
Attached table - grouped frequency table showing a hypothetical distribution of simple reaction times for a sample of 20 participants.
GROUPED FREQUENCY TABLE - The ranges must all be of equal width, and there are usually between…
5 - 15
FREQUENCY TABLES Can also be used for categorical variables
Levels = category labels.
The order = somewhat arbitrary
Listed from the most frequent at the top to the least frequent at the bottom.
Describe and explain - HISTOGRAM
HISTOGRAM
= graphical display of a distribution.
It presents the SAME information as — a frequency table
BUT
is quicker and easier to grasp.
When the variable is quantitative = no gap between the bars.
When the variable is categorical, however, there is usually a SMALL gap between them.
SHAPE = Unimodal
Describe and explain - Distribution shapes
Distribution Shapes
When the distribution of a quantitative variable is displayed in a histogram — it has a shape.
The shape of the distribution of self-esteem scores in is typical.
There is a peak somewhere near the middle of the distribution and “tails” that taper in either direction from the peak.
IMAGE BELOW…
SHAPE = bimodal
meaning they have two distinct peaks
Distributions can also have more than two distinct peaks, = relatively rare in psychological research.
Symmetrical or skewed?
Describe the shape of positive, negative skew and symmetrical…
Symmetrical or skewed?
Describe outliers
An OUTLIER is an extreme score that is much higher or lower than the rest of the scores in the distribution.
Sometimes outliers represent truly extreme scores on the variable of interest.
EXAMPLE
On the Beck Depression Inventory
= a single clinically depressed person might be an outlier in a sample of otherwise happy and high-functioning peers.
However, outliers can also represent errors or misunderstandings on the part of the researcher or participant, equipment malfunctions, or similar problems.
Describe and explain - Central Tendency
Central Tendency
The central tendency of a distribution is its middle
the point around which the scores in the distribution tend to cluster.
(Another term for central tendency is average.)
Three most common measures of central tendency:
the mean, the median, and the mode.
The mean of a distribution (symbolized M)
= is the sum of the scores divided /
by the number of scores
M=ΣX/N
Σ (the Greek letter sigma) = is the summation sign
X = means to sum across the values of the variable
N = represents the number of scores.
The mean…
— provides a good indication of the central tendency of a distribution
— is easily understood by most people.
— has statistical properties FOR inferential statistics.
Describe median
MEDIAN = is the middle score
Find the median…
8 4 12 14 3 2 3
Rearrange low to high
2 3 3 4 8 12 14
Median = 4
EXAMPLE
If the distribution were…
2 3 3 4 8 12 14 15
Median = 6
Describe mode and what we can learn from different sidtribution shapes
MODE = is the most frequent score in a distribution.
The mode is the ONLY measure of central tendency that can also be used for categorical variables
EXAMPLE -
Data set - 2 3 3 4 8 12 14
Mode = 3
When distribution = unimodal and symmetrical
the mean, median, and mode will be very close to each other at the peak of the distribution.
BIMODAL AND ASSYMETRICAL distribution
the mean, median, and mode can be quite different.
In a bimodal distribution
the mean and median will tend to be between the peaks,
WHILE the mode will be at the tallest peak.
In a skewed distribution
the mean will DIFFER from the median in the direction of the skew (i.e., the direction of the longer tail).
For highly skewed distributions, the mean can be pulled so far in the direction of the skew that it is no longer a good measure of the central tendency of that distribution.
HIGHLY SKEWED = Researchers prefer MEDIAN
EXAMPLE (for understanding…)
Imagine, for example, a set of four simple reaction times of 200, 250, 280, and 250 milliseconds (ms). The mean is 245 ms. But the addition of one more score of 5,000 ms—perhaps because the participant was not paying attention—would raise the mean to 1,445 ms. Not only is this measure of central tendency greater than 80% of the scores in the distribution, but it also does not seem to represent the behavior of anyone in the distribution very well. This is why researchers often prefer the median for highly skewed distributions (such as distributions of reaction times).
Desribe and explain - Variability
Measures of Variability
The variability of a distribution is the EXTENT to which the scores vary AROUND their central tendency.
REF IMAGE -
both of which have the SAME central tendency.
The mean, median, and mode of each distribution are 10.
Notice, however, that the two distributions differ in terms of their variability.
The top = LOW variability, with all the scores relatively close to the center.
The bottom one has relatively HIGH variability, with the scores are spread across a much greater range.
Describe range
RANGE = difference between the highest and lowest scores in the distribution.
Although the range is easy to compute and understand, it can be misleading when there are outliers.
Explain the standard deviation
By far the most common measure of variability is the standard deviation.
The standard deviation of a distribution is the average distance between the scores and the mean.
Computing the standard deviation involves a slight complication.
- Specifically, it involves finding the difference between each score and the mean
- Squaring each difference
- finding the mean of these squared differences
- finding the square root of that mean.
Look at table - mentally go through the process of calculating the standard deviation
The computations for the standard deviation are illustrated for a small set of data in Table 12.3.
—- although the differences can be negative, the squared differences are always positive—meaning that the standard deviation is always positive.
—- Although the variance is itself a measure of variability, it generally plays a larger role in inferential statistics than in descriptive statistics.
Why use N − 1 instead of N
N or N − 1
If you have already taken a statistics course, you may have learned to divide the sum of the squared differences by N − 1 rather than by N when you compute the variance and standard deviation. Why is this?
By definition, the standard deviation is the square root of the mean of the squared differences. This implies dividing the sum of squared differences by N, as in the formula just presented. Computing the standard deviation this way is appropriate when your goal is simply to describe the variability in a sample. And learning it this way emphasizes that the variance is in fact the mean of the squared differences—and the standard deviation is the square root of this mean.
However, most calculators and software packages divide the sum of squared differences by N − 1. This is because the standard deviation of a sample tends to be a bit lower than the standard deviation of the population the sample was selected from.
Dividing the sum of squares by N − 1 corrects for this tendency and results in a better estimate of the population standard deviation.
Because researchers generally think of their data as representing a sample selected from a larger population—and because they are generally interested in drawing conclusions about the population—it makes sense to routinely apply this correction.
Describe and explain - percentile rank
In many situations, it is useful to have a way to describe the location of an individual score within its distribution.
Percentile rank of a score = the percentage of scores in the distribution that are LOWER than that score.
EXAMPLE
Any score in the distribution, we can find its percentile rank by counting the number of scores in the distribution that are lower than that score and converting that number to a percentage of the total number of scores.
Notice, for example, that five of the students represented by the data in Table 12.1 had self-esteem scores of 23.
In this distribution, 32 of the 40 scores (80%) are lower than 23.
Thus each of these students has a percentile rank of 80. (It can also be said that they scored “at the 80th percentile.”)
Percentile ranks are often used to report the results of standardized tests of ability or achievement.
Another approach is the z score. The z score for a particular individual is the difference between that individual’s score and the mean of the distribution, divided by the standard deviation of the distribution:
z = (X−M)/SD
A z score indicates how far above or below the mean a raw score is, but it expresses this in terms of the standard deviation. For example, in a distribution of intelligence quotient (IQ) scores with a mean of 100 and a standard deviation of 15, an IQ score of 110 would have a z score of (110 − 100) / 15 = +0.67. In other words, a score of 110 is 0.67 standard deviations (approximately two thirds of a standard deviation) above the mean. Similarly, a raw score of 85 would have a z score of (85 − 100) / 15 = −1.00. In other words, a score of 85 is one standard deviation below the mean.
There are several reasons that z scores are important. Again, they provide a way of describing where an individual’s score is located within a distribution and are sometimes used to report the results of standardized tests. They also provide one way of defining outliers. For example, outliers are sometimes defined as scores that have z scores less than −3.00 or greater than +3.00. In other words, they are defined as scores that are more than three standard deviations from the mean. Finally, z scores play an important role in understanding and computing other statistics, as we will see shortly.
Online Descriptive Statistics
Although many researchers use commercially available software such as SPSS and Excel to analyze their data, there are several free online analysis tools that can also be extremely useful. Many allow you to enter or upload your data and then make one click to conduct several descriptive statistical analyses. Among them are the following.
Rice Virtual Lab in Statistics
http://onlinestatbook.com/stat_analysis/index.html
VassarStats
http://faculty.vassar.edu/lowry/VassarStats.html
Bright Stat
http://www.brightstat.com
For a more complete list, see http://statpages.org/index.html
Differences between groups or conditions are usually described in terms of the mean and standard deviation of each group or condition. For example, Thomas Ollendick and his colleagues conducted a study in which they evaluated two one-session treatments for simple phobias in children (Ollendick et al., 2009)[1]. They randomly assigned children with an intense fear (e.g., to dogs) to one of three conditions. In the exposure condition, the children actually confronted the object of their fear under the guidance of a trained therapist. In the education condition, they learned about phobias and some strategies for coping with them. In the wait-list control condition, they were waiting to receive a treatment after the study was over. The severity of each child’s phobia was then rated on a 1-to-8 scale by a clinician who did not know which treatment the child had received. (This was one of several dependent variables.) The mean fear rating in the education condition was 4.83 with a standard deviation of 1.52, while the mean fear rating in the exposure condition was 3.47 with a standard deviation of 1.77. The mean fear rating in the control condition was 5.56 with a standard deviation of 1.21. In other words, both treatments worked, but the exposure treatment worked better than the education treatment. As we have seen, differences between group or condition means can be presented in a bar graph like that in Figure 12.5, where the heights of the bars represent the group or condition means. We will look more closely at creating American Psychological Association (APA)-style bar graphs shortly.
Conceptually, Cohen’s d is the difference between the two means expressed in standard deviation units. (Notice its similarity to a z score, which expresses the difference between an individual score and a mean in standard deviation units.) A Cohen’s d of 0.50 means that the two group means differ by 0.50 standard deviations (half a standard deviation). A Cohen’s d of 1.20 means that they differ by 1.20 standard deviations. But how should we interpret these values in terms of the strength of the relationship or the size of the difference between the means? Table 12.4 presents some guidelines for interpreting Cohen’s d values in psychological research (Cohen, 1992)[2]. Values near 0.20 are considered small, values near 0.50 are considered medium, and values near 0.80 are considered large. Thus a Cohen’s d value of 0.50 represents a medium-sized difference between two means, and a Cohen’s d value of 1.20 represents a very large difference in the context of psychological research. In the research by Ollendick and his colleagues, there was a large difference (d = 0.82) between the exposure and education conditions.
Sex Differences Expressed as Cohen’s d
Researcher Janet Shibley Hyde has looked at the results of numerous studies on psychological sex differences and expressed the results in terms of Cohen’s d (Hyde, 2007)[3]. Following are a few of the values she has found, averaging across several studies in each case. (Note that because she always treats the mean for men as M1 and the mean for women as M2, positive values indicate that men score higher and negative values indicate that women score higher.)
Hyde points out that although men and women differ by a large amount on some variables (e.g., attitudes toward casual sex), they differ by only a small amount on the vast majority. In many cases, Cohen’s d is less than 0.10, which she terms a “trivial” difference. (The difference in talkativeness discussed in Chapter 1 was also trivial: d = 0.06.) Although researchers and non-researchers alike often emphasize sex differences, Hyde has argued that it makes at least as much sense to think of men and women as fundamentally similar. She refers to this as the “gender similarities hypothesis.”
Correlations Between Quantitative Variables
As we have seen throughout the book, many interesting statistical relationships take the form of correlations between quantitative variables. For example, researchers Kurt Carlson and Jacqueline Conard conducted a study on the relationship between the alphabetical position of the first letter of people’s last names (from A = 1 to Z = 26) and how quickly those people responded to consumer appeals (Carlson & Conard, 2011)[4]. In one study, they sent emails to a large group of MBA students, offering free basketball tickets from a limited supply. The result was that the further toward the end of the alphabet students’ last names were, the faster they tended to respond. These results are summarized in Figure 12.6.
Such relationships are often presented using line graphs or scatterplots, which show how the level of one variable differs across the range of the other. In the line graph in Figure 12.6, for example, each point represents the mean response time for participants with last names in the first, second, third, and fourth quartiles (or quarters) of the name distribution. It clearly shows how response time tends to decline as people’s last names get closer to the end of the alphabet. The scatterplot in Figure 12.7, shows the relationship between 25 research methods students’ scores on the Rosenberg Self-Esteem Scale given on two occasions a week apart. Here the points represent individuals, and we can see that the higher students scored on the first occasion, the higher they tended to score on the second occasion. In general, line graphs are used when the variable on the x-axis has (or is organized into) a small number of distinct values, such as the four quartiles of the name distribution. Scatterplots are used when the variable on the x-axis has a large number of values, such as the different possible self-esteem scores.
The data presented in Figure 12.7 provide a good example of a positive relationship, in which higher scores on one variable tend to be associated with higher scores on the other (so that the points go from the lower left to the upper right of the graph). The data presented in Figure 12.6 provide a good example of a negative relationship, in which higher scores on one variable tend to be associated with lower scores on the other (so that the points go from the upper left to the lower right).
Both of these examples are also linear relationships, in which the points are reasonably well fit by a single straight line. Nonlinear relationships are those in which the points are better fit by a curved line. Figure 12.8, for example, shows a hypothetical relationship between the amount of sleep people get per night and their level of depression. In this example, the line that best fits the points is a curve—a kind of upside down “U”—because people who get about eight hours of sleep tend to be the least depressed, while those who get too little sleep and those who get too much sleep tend to be more depressed. Nonlinear relationships are not uncommon in psychology, but a detailed discussion of them is beyond the scope of this book.
As we saw earlier in the book, the strength of a correlation between quantitative variables is typically measured using a statistic called Pearson’s r. As Figure 12.9 shows, its possible values range from −1.00, through zero, to +1.00. A value of 0 means there is no relationship between the two variables. In addition to his guidelines for interpreting Cohen’s d, Cohen offered guidelines for interpreting Pearson’s r in psychological research (see Table 12.4). Values near ±.10 are considered small, values near ± .30 are considered medium, and values near ±.50 are considered large. Notice that the sign of Pearson’s r is unrelated to its strength. Pearson’s r values of +.30 and −.30, for example, are equally strong; it is just that one represents a moderate positive relationship and the other a moderate negative relationship. Like Cohen’s d, Pearson’s r is also referred to as a measure of “effect size” even though the relationship may not be a causal one.
The computations for Pearson’s r are more complicated than those for Cohen’s d. Although you may never have to do them by hand, it is still instructive to see how. Computationally, Pearson’s r is the “mean cross-product of z scores.” To compute it, one starts by transforming all the scores to z scores. For the X variable, subtract the mean of X from each score and divide each difference by the standard deviation of X. For the Y variable, subtract the mean of Y from each score and divide each difference by the standard deviation of Y. Then, for each individual, multiply the two z scores together to form a cross-product. Finally, take the mean of the cross-products. The formula looks like this:
Table 12.5 illustrates these computations for a small set of data. The first column lists the scores for the X variable, which has a mean of 4.00 and a standard deviation of 1.90. The second column is the z-score for each of these raw scores. The third and fourth columns list the raw scores for the Y variable, which has a mean of 40 and a standard deviation of 11.78, and the corresponding z scores. The fifth column lists the cross-products. For example, the first one is 0.00 multiplied by −0.85, which is equal to 0.00. The second is 1.58 multiplied by 1.19, which is equal to 1.88. The mean of these cross-products, shown at the bottom of that column, is Pearson’s r, which in this case is +.53. There are other formulas for computing Pearson’s r by hand that may be quicker. This approach, however, is much clearer in terms of communicating conceptually what Pearson’s r is.
As we saw earlier, there are two common situations in which the value of Pearson’s r can be misleading. One is when the relationship under study is nonlinear. Even though Figure 12.8 shows a fairly strong relationship between depression and sleep, Pearson’s r would be close to zero because the points in the scatterplot are not well fit by a single straight line. This means that it is important to make a scatterplot and confirm that a relationship is approximately linear before using Pearson’s r. The other is when one or both of the variables have a limited range in the sample relative to the population. This problem is referred to as restriction of range. Assume, for example, that there is a strong negative correlation between people’s age and their enjoyment of hip hop music as shown by the scatterplot in Figure 12.10. Pearson’s r here is −.77. However, if we were to collect data only from 18- to 24-year-olds—represented by the shaded area of Figure 12.11—then the relationship would seem to be quite weak. In fact, Pearson’s r for this restricted range of ages is 0. It is a good idea, therefore, to design studies to avoid restriction of range. For example, if age is one of your primary variables, then you can plan to collect data from people of a wide range of ages. Because restriction of range is not always anticipated or easily avoidable, however, it is good practice to examine your data for possible restriction of range and to interpret Pearson’s r in light of it. (There are also statistical methods to correct Pearson’s r for restriction of range, but they are beyond the scope of this book).
As we have seen throughout this book, bar graphs are generally used to present and compare the mean scores for two or more groups or conditions. The bar graph in Figure 12.11 is an APA-style version of Figure 12.4. Notice that it conforms to all the guidelines listed. A new element in Figure 12.11 is the smaller vertical bars that extend both upward and downward from the top of each main bar. These are error bars, and they represent the variability in each group or condition. Although they sometimes extend one standard deviation in each direction, they are more likely to extend one standard error in each direction (as in Figure 12.11). The standard error is the standard deviation of the group divided by the square root of the sample size of the group. The standard error is used because, in general, a difference between group means that is greater than two standard errors is statistically significant. Thus one can “see” whether a difference is statistically significant based on a bar graph with error bars.
Line graphs are used when the independent variable is measured in a more continuous manner (e.g., time) or to present correlations between quantitative variables when the independent variable has, or is organized into, a relatively small number of distinct levels. Each point in a line graph represents the mean score on the dependent variable for participants at one level of the independent variable. Figure 12.12 is an APA-style version of the results of Carlson and Conard. Notice that it includes error bars representing the standard error and conforms to all the stated guidelines.
In most cases, the information in a line graph could just as easily be presented in a bar graph. In Figure 12.12, for example, one could replace each point with a bar that reaches up to the same level and leave the error bars right where they are. This emphasizes the fundamental similarity of the two types of statistical relationship. Both are differences in the average score on one variable across levels of another. The convention followed by most researchers, however, is to use a bar graph when the variable plotted on the x-axis is categorical and a line graph when it is quantitative.
Scatterplots are used to present correlations and relationships between quantitative variables when the variable on the x-axis (typically the independent variable) has a large number of levels. Each point in a scatterplot represents an individual rather than the mean for a group of individuals, and there are no lines connecting the points. The graph in Figure 12.13 is an APA-style version of Figure 12.7, which illustrates a few additional points. First, when the variables on the x-axis and y-axis are conceptually similar and measured on the same scale—as here, where they are measures of the same variable on two different occasions—this can be emphasized by making the axes the same length. Second, when two or more individuals fall at exactly the same point on the graph, one way this can be indicated is by offsetting the points slightly along the x-axis. Other ways are by displaying the number of individuals in parentheses next to the point or by making the point larger or darker in proportion to the number of individuals. Finally, the straight line that best fits the points in the scatterplot, which is called the regression line, can also be included.
Expressing Descriptive Statistics in Tables
Like graphs, tables can be used to present large amounts of information clearly and efficiently. The same general principles apply to tables as apply to graphs. They should add important information to the presentation of your results, be as simple as possible, and be interpretable on their own. Again, we focus here on tables for an APA-style manuscript.
The most common use of tables is to present several means and standard deviations—usually for complex research designs with multiple independent and dependent variables. Figure 12.14, for example, shows the results of a hypothetical study similar to the one by MacDonald and Martineau (2002)[1] (The means in Figure 12.14 are the means reported by MacDonald and Martineau, but the standard errors are not). Recall that these researchers categorized participants as having low or high self-esteem, put them into a negative or positive mood, and measured their intentions to have unprotected sex. They also measured participants’ attitudes toward unprotected sex. Notice that the table includes horizontal lines spanning the entire table at the top and bottom, and just beneath the column headings. Furthermore, every column has a heading—including the leftmost column—and there are additional headings that span two or more columns that help to organize the information and present it more efficiently. Finally, notice that APA-style tables are numbered consecutively starting at 1 (Table 1, Table 2, and so on) and given a brief but clear and descriptive title.
Another common use of tables is to present correlations—usually measured by Pearson’s r—among several variables. This kind of table is called a correlation matrix. Figure 12.15 is a correlation matrix based on a study by David McCabe and colleagues (McCabe, Roediger, McDaniel, Balota, & Hambrick, 2010)[2]. They were interested in the relationships between working memory and several other variables. We can see from the table that the correlation between working memory and executive function, for example, was an extremely strong .96, that the correlation between working memory and vocabulary was a medium .27, and that all the measures except vocabulary tend to decline with age. Notice here that only half the table is filled in because the other half would have identical values. For example, the Pearson’s r value in the upper right corner (working memory and age) would be the same as the one in the lower left corner (age and working memory). The correlation of a variable with itself is always 1.00, so these values are replaced by dashes to make the table easier to read.
As with graphs, precise statistical results that appear in a table do not need to be repeated in the text. Instead, the writer can note major trends and alert the reader to details (e.g., specific correlations) that are of particular interest.
Now you are ready to enter your data in a spreadsheet program or, if it is already in a computer file, to format it for analysis. You can use a general spreadsheet program like Microsoft Excel or a statistical analysis program like SPSS to create your data file. (Data files created in one program can usually be converted to work with other programs.) The most common format is for each row to represent a participant and for each column to represent a variable (with the variable name at the top of each column). A sample data file is shown in Table 12.6. The first column contains participant identification numbers. This is followed by columns containing demographic information (sex and age), independent variables (mood, four self-esteem items, and the total of the four self-esteem items), and finally dependent variables (intentions and attitudes). Categorical variables can usually be entered as category labels (e.g., “M” and “F” for male and female) or as numbers (e.g., “0” for negative mood and “1” for positive mood). Although category labels are often clearer, some analyses might require numbers. SPSS allows you to enter numbers but also attach a category label to each number.
If you have multiple-response measures—such as the self-esteem measure in Table 12.6—you could combine the items by hand and then enter the total score in your spreadsheet. However, it is much better to enter each response as a separate variable in the spreadsheet—as with the self-esteem measure in Table 12.6—and use the software to combine them (e.g., using the “AVERAGE” function in Excel or the “Compute” function in SPSS). Not only is this approach more accurate, but it allows you to detect and correct errors, to assess internal consistency, and to analyze individual responses if you decide to do so later.
