Understanding Statistics As A Language. Robert Andrews
numbers. For example, IQ is an index number of the rate of general mental growth to chronological age.
Nominal Scale
The nominal scale of measurement is the most limited type of measurement and involves the process of naming or labeling. Nominal measurement can be described as classifying observations. All
observations placed in the same class are treated as equal. This is the property of identity. The categories must be mutually exclusive; that is, each observation is placed in one and only one category. The categories must also be
exhaustive; that is, a place must be provided for each observation. Nominal data
cannot be graded, ranked, or scaled for qualities such as better or worse, higher or lower, more or less. Observations in each category can be counted, which provides a frequency. Demographic data such as groups of ages, gender, nationality, or denominational affiliation are nominal scale data.
There are statistical methods designed to analyze nominal or categorical data. Chi-Square is a primary method of analyzing nominal data.
Ordinal Scale
The ordinal scale of measurement is used when observations can be placed in order, based upon a characteristic. While an order can be observed, the exact measurement between two observations cannot be determined. Thus, the distance between A and B may be much greater than the distance between B and C. The ordinal scale allows for the characteristics of more or less of a characteristic, but not how much more or less. An ordinal scale does not have an exact zero nor equal units of measurement. Likert scale items can produce ordinal scale data when the responses are ranked, for example, from one to seven and labels such as ‘Agree’ and ‘Disagree’ are attached to each end of the scale. The distance between a response of 3 is only considered to be greater than a response of 2 and smaller than a response of 4. The response of 3 is not equidistant from 2 and 4.
Because the categories of observations have rank, additional statistical methods of analysis are available including correlation and tests for significant difference. In the survey research example given above a Likert scale item might ask the respondent to circle a number from 1 to 7 with 1 indicating little or no relief from the hiccups and 7 indicating total relief. The researcher could then analyze the differences in the responses among various demographic groups.
Interval Scale
The interval measurement scale involves the application of a scale with equal units and an arbitrary zero. With equal units of measurement, most of the useful numerical operations, such as addition, subtraction, multiplication, and division can be performed. Temperature readings on a Celsius or Fahrenheit thermometer, and individual intelligence test scores like the Stanford-Binet or the Wechsler tests are examples of interval data. The measurement of zero on interval scales does not mean the absence of the trait being measured, simply that the scale was not sensitive enough to measure levels of the variable below a certain point. Interval data analysis involves the use of powerful statistical methods such as correlation, t-Tests, and Analysis of Variance (ANOVA).
Table 1 -- Scales of Measurements Summary
[Table adapted from Elmore et al. (1997)]
| Scale of Measurement | Name or Categorize | Rank Order | Discern Equal Differences | Make a Ratio of Two Values |
| Nominal | X | |||
| Ordinal | X | X | ||
| Interval | X | X | X | |
| Ratio | X | X | X | X |
Ratio Scale
A ratio measurement scale is very much like an interval measurement scale except the observations have an absolute zero. Dollars or cents, yards or feet, minutes or seconds and cash on hand are all ratio measurements or measurements on a ratio scale. Practically speaking, ratio data are equivalent to interval data and may be analyzed by many of the same statistical methods.
Hypothesis Testing
In the behavioral sciences, research is conducted to determine the acceptability of hypotheses derived from theories of behavior. The researcher develops a hypothesis from a theory and collects empirical data to determine the acceptability of the hypothesis. The empirical data may lead to retaining, revising or rejecting the hypothesis and/or the theory from which the hypothesis was developed. (Siegel 1956)
Definition: A research hypothesis is a statement of theory, previous research results, or other information that leads to an anticipated outcome.
Definition: A null hypothesis is a hypothesis of no difference. According to the null hypothesis, any observed difference between samples is regarded as a chance occurrence resulting from sampling error alone. If the null hypothesis of no difference is rejected, the alternate hypothesis that a true population or sample difference does exist is accepted.
Definition: An alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false.
In order to make an objective decision about whether or not a particular hypothesis is confirmed by a set of data, objective procedures are needed. Following are a set of procedures [The procedures were adapted from Siegel (1956). The definitions related to hypothesis testing were adapted from Siegel (1956 ), Mann (1992) and Elmore et al. (1997)], in order of performance that can be used to determine the disposition of a null hypothesis: (1) state the null hypothesis (Ho), (2) choose a statistical test for testing Ho [See the Decision Model], (3) specify a significance level () and a sample size (n), (4) find / assume the sampling distribution of the statistical test under Ho, (5) define the region of rejection, and(6) compute the value of the statistical test, using the data obtained from the sample(s). If the computed value is in the region of rejection, reject Ho. If the computed value is outside the region of rejection, Ho cannot be rejected at the chosen level of significance.
Definition: Significance level of a test is the probability of rejecting Ho when in fact it is true. The size of the rejection region depends on the value assigned to . The commonly used values of are 0.01, 0.05, and 0.10. In social science research, the most common value of is 0.05.
Definition: The critical value or critical point is the place on a curve at which the rejection and nonrejection regions are divided.
Decision Model
The Decision Model is a chart designed to facilitate a decision about which statistical procedure is most appropriate in the analysis of a set of data. (See Figure 1) A few questions about a set of data can lead quickly to a decision about which statistical procedure is most appropriate. The questions are: (1)
What type of measurement scale produced the data -- nominal, ordinal, or interval/ratio scale? (2) Were the data related or independent? Was there an absence of association between observations or were the observations related as in a pretest/posttest