Statistics Help Guide
Statistics Help Guide
Becki Paynich
I. A Few Important Things:
A. Math is the least important factor in statistics. Knowing which statistics
test to employ based on sample, level of measurement, and what you are trying to understand is the most important part in statistics. However, you need to understand how the math works to better understand why you pick any given test.
B. Statistics is not brain surgery. Lighten up. You’ll do fine.
C. This is a help guide, not a comprehensive treatment of statistics. Please
refer to a good Statistics textbook for a deeper understanding of statistics.
II. Levels of Measurement: Knowing the level of measurement is extremely
important in statistics. Most tests of significance are explicit about the level of measurement they require. If the requirements for the level of measurement are not adhered to, results lose validity. For example, Pearson’s r assumes that variables are at least interval level. If a Pearson’s r is employed on nominal or ordinal level variables, results cannot be directly interpreted (pretty much your results are meaningless).
A. Nominal—a variable that is measured in categories that cannot be ranked.
Usually nominal variables are collapsed into the smallest number of categories possible (0,1) so that they may be used in analysis. A variable in this form (dichotomous) can be used in virtually any type of analysis. For example: the variable RACE may be originally separated into 12 different categories. These 12 might be collapsed into 2 (most likely being Caucasian=1 and Non-Caucasian=0). The important thing to remember is that collapsing categories into larger ones can hide important relationships between them. For example, the difference between African-Americans and Native Americans on a given variable may be important but if they are both collapsed into the category of Non-Caucasian, this relationship will be obscured. Another thing to keep in mind is that a dichotomous or dummy variable (one with only two categories as in above example) is the most powerful form, statistics wise, for a nominal variable to be in.
B. Ordinal—a variable that can be ranked but the exact difference between
categories cannot be identified. For example: The categories of Strongly Agree, Agree, No Opinion, Disagree, and Strongly Disagree, can be differentiated by order but one does not know the exact mathematical difference between two respondents who answer Strongly Agree and Agree respectively. Another example would be categorical ranges. Income, for example, can be grouped into the following ranges: $0-$10,000; $10,0001-$20,000; and $20,000+. While certainly, this variable is measured with numbers, one cannot tell the exact difference between a respondent who selected the 0-10,000 range and a respondent that selected the 10,001-20,000 range. In theory, they could by $1.00 apart or closer to $10,000 apart in income. A final form of ordinal variable which is actually treated as interval-level is when one adds together multiple ordinal level responses. For example, I may want to make an index on how well a student likes the course materials by combining his/her responses to questions about the books, the web page, and the class notes. Thus, if a student answered Strongly Agree to questions about these three items, and the score for Strongly Agree is 5, the student’s total score for this new index is 15.
Ordinal variables can also be and often are collapsed into a smaller number of categories. For example: A variable measuring rank may have 8 or more original categories that can be collapsed into the categories of administration=1 and non-administration=0. Or, there could be three categories with a middle category being middle-management. The problem again becomes sacrificing important relationships that may be obscured by the collapse for ease in analysis. The best rule of thumb is to run analyses with all the categories in their smaller form first to identify important relationships between them and then run analyses again after collapsing the variables to identify important relationships across larger ones.
** When you create nominal and ordinal level questions, make sure they are mutually exclusive (no respondent can fit into more than one category) and exhaustive (you have listed all possibilities). Also, when designing numerical ranges, make sure that the intervals have the same width. In the following example, notice that the width for each category is $10,000:
Question: What is your income range?
1 = 0-10,000
2 = Above 10,000 to 20,000
3 = Above 20,000 to 30,000
C. Interval—a variable or scale that uses numbers to rank order. This is the best type of variable to have as you can keep it in interval form or turn it into an ordinal or nominal level variable and do virtually any type of analysis. Furthermore, the exact difference between two respondents can be identified. For example: with the variable age, when respondents report their actual age, one can mathematically determine the differences between the ages of respondents. If however, the variable is grouped into age ranges (0-5, 6-10, 11-15, 16-20…) then the variable is ordinal and not interval scale. Interval level variables can be collapsed into ordinal or even nominal categories but this is usually avoided as most statistical tests designed for interval level variables are more powerful and provide more understanding of the nature of the relationships than those tests designed for ordinal or nominal variables.
D. Ratio—a ratio scale variable is identical to interval in almost every aspect
except for it has an absolute zero. Age and Income have absolute zeros. If a variable can be considered on a ratio scale, then it is also interval scale. **Many researchers in criminal justice ignore the difference between interval and ratio as the tests of significance that we use for ratio and interval variables are the same.**
**Make sure to code variables in the most logical order.** For example, if you are trying to measure the frequency of smoking, use the highest numbers for the most amount of smoking and the lowest numbers for the least amount of smoking. For example:
Question: How often do you smoke?
0 = Never
1 = Less than once per week
2 = At least once per week
3 = 2 or 3 times per week
4 = Daily
III. Numerical Measures:
A. Measures of Central Tendency
1. Mode-the most prevalent variable (for example, in a data set of
pets: 25 cats, 13 dogs, 12 hamsters, 47 fish. Fish is the modal category with 47 being the mode). The only measure of central tendency for nominal-level data is the mode. The mode can also be used with ordinal and interval-level data.
2. Median-the mid point in a value. An equal number of the sample
will be above the median as will be below it. In a sample of
n =101: 50 of the scores will be above the median, and 50 will be
below the median). The median is most often used for interval-level data. When ranked from lowest to highest, the 51st score will be the median. If your n is an even number, take the average between the two middle scores.
3. Mean-the average of a value. The mean is usually not computed on
ordinal and nominal-level data because it has little explanatory value. However, many statistical tests require that the mean be computed for the formula. The mean is most appropriate for interval-level data.
B. Measures of Variability
1. Range—the difference between the largest and smallest values.
2. Variance—the sum of the squared deviations of n measurements
from their mean divided by (n – 1).
3. Standard Deviation—the positive square root of the variance. One
must compute the variance first to get the standard deviation because the sum of deviations from the mean always equals zero.
4. Typically, standard deviations are used in the form of standard
errors from the mean in a normal curve. In a normal distribution, the interval from one standard error below the mean to one standard error above the mean contains approximately 68% of the measurements. The interval between two standard errors below and above the mean contains approximately 95% of the measurements.
5. In a perfect normal curve, the mode, median, and mean are all the
same number.
IV. Probability:
A. Classic theory of probability—the chance of a particular outcome
occurring is determined by the ratio of the number of favorable outcomes (successes) to the total number of outcomes. This theory only pertains to outcomes that are mutually exclusive (disjoint).
B. Relative Frequency Theory—is that if an experiment is repeated an
extremely large number of times and a particular outcome occurs a percentage of the time, then that particular percentage is close to the probability of that outcome.
C. Independent Events—outcomes not affected by other outcomes.
D. Dependent Events—outcomes affected by other outcomes.
E. Multiplication Rule
1. Joint Occurrence—to compute the probability of two or more
independent events all occurring, multiply their probabilities.
F. Addition Rule—to determine the probability of at least one successful
event occurring, add their probabilities.
G. Remember to account for probabilities with and with replacement. For
example, when picking cards out of a deck, the probability of choosing on the first try the Queen of Hearts is 1/52. However, if you don’t put back the first card before choosing for the second time, your probability increases to 1/51 because there are only 51 cards left to choose from.
H. Probability Distribution is nothing more than a visual representation of the
probabilities of success for given outcomes.
I. Discrete variables are those that do not have possible outcomes in
between values. Thus, a coin toss can only result in either a heads or tails outcome. Continuous variables are those that have possible outcomes between values. For example, it is absolute possible to be 31.26 years old.
V. Sampling:
A. Independent Random Sampling: Most statistical tests are based on the
premise that samples are independently and randomly selected. If in fact a sample was purposive, statistical analysis cannot be generalized outside the sample. Random sampling is done so that inferential statistics can be interpreted outside the sample. Statistical tests can still be employed in samples that have been drawn through non-random techniques, however, their interpretation must be confined to the sample at hand, and limitations of the sampling design must be addressed in the results and methods discussions of your research.
B. Central Limit Theorem—the larger the number of observations (the bigger
your n), the more likely the sample distribution will approximate a normal curve.
VI. Principles of Testing:
A. Research hypothesis—can be stated in a certain direction or without
direction. Directional hypotheses are tested with one-tailed tests of significance. Non-directional hypotheses are tested with two-tailed tests of significance.
B. Null hypothesis—is essentially the statement that two values are not
statistically related. If a test of a research hypothesis is not significant, then the null hypothesis cannot be rejected.
C. Type I error: (alpha) is when a researcher rejects the null hypothesis when
in fact it is true. That is, stating that two variables are significantly related when in fact they are not.
D. Type II error: (beta) is when a researcher fails to reject the null hypothesis when in fact the null hypothesis is false. That is, stating that two variables are not significantly related when in fact they are.
VII. Univariate Inferential Tests: (See also Appendix A for quick reference guide)Essentially, univariate tests are only looking at one variable and how the scores for that variable differ between sample and population groups, two samples, and within groups at different points in time. Many univariate tests can be employed on proportions when that is all that is available. This is not a comprehensive listing of all univariate tests but can the most commonly used are briefly discussed below.
A. Steps in testing:
1. State the assumptions
2. State the Null and Research Hypotheses.
3. Decide on a significance level for the test, Determine the test to be used.
4. Compute the value of a test statistic.
5. Compare the test statistic with the critical value to determine
whether or not the test statistic falls in the region of acceptance or the region of rejection.
B. One-sample z-test:
1. Requirements: normally distributed population. Population
variance is known.
2. Test for population mean
3. One-tailed or two-tailed.
4. This test essentially tells you if your sample mean is statistically
different from the larger population mean.
C. One-sample t-test:
1. Requirements: normally distributed population. Population
variance is known.
2. Test for population mean
3. One-tailed or two-tailed.
4. This test essentially tells you if your sample mean is statistically
different from the larger population mean.
D. Two-sample t and z-tests for comparing two means:
1. Requirements: two normally distributed but independent
populations. Population variance can be known or unknown. Different formulas depending on whether the variance is known.
E. Paired Difference t-test:
1. Requirements: a set of paired observations from a normal
population. This test is usually employed to compare “before” and “after” scores. Also employed in twin studies.
F. Chi-Square for population distributions:
1. Requirements: assumption of normal distribution
2. This test basically compares frequencies from a known sample
with frequencies from an assumed population. There are many different types of chi-square tests which test different things—consistency, distribution, goodness of fit, independence…make sure you are employing the right chi-square for your needs.
VIII. Bivariate Relationships: Bivariate simply means analysis between two
variables. Before beginning this section it is important to note that there is a difference between measures of association and tests of significance. A Pearson’s r (a measure of association) will give you are statistic between -1 and +1. -1 is a perfect negative correlation, +1 is a perfect positive correlation. 0 is no correlation. When SPSS provides output, it has to run a separate test to tell you if the relationship is significant. Thus, if doing stats by hand, one has to complete two different formulas—one to get the correlation and one to get the significance.
A. Measures of Association: (See also Appendix A for quick reference guide)