Frequency
distributions

Analyzing
single
variables

Presenting univariate
data

Measures of
central
tendency

Measures of
variability

Standard
deviation and
the normal distribution

Computer
analysis
reminder

Aids

Key terms

Main points

1. Univariate analysis is the analysis of data for a single variable. Univariate analysis provides descriptions of variables. Univariate analysis begins with examination of the frequency distribution of each variable. This tells how many times each attribute occurred.
   
2. A proportion is a fraction of something. Mathematically, it is expressed as the frequency (f) for an attribute of interest over the total number of cases, referred to as the N (for the number). Proportions vary from 0 to 1.00. The formula for a proportion is f/N. A proportion of 0.5, for example, means that one attribute of a variable makes up half of all the frequencies for that variable.
   
3. Percentages are proportions converted to a base of 100. To calculate a percentage, find the proportion and then multiply by 100. The formula for percentage is f/N(100).
   
4. A ratio is a proportion of one frequency divided by another and multiplied by a standardizing index such as 100.
   
5. A rate is a measure of occurrence of some event. It is similar to a ratio except that a larger standardizing base such as 1,000, 10,000, or even larger is used to express the rate of occurrence of an event, such as births in a population.
   
6. Proportions, percentages, ratios and rates are useful for comparing results from samples of different sizes. By reducing results to a base of 100 or some other standardizing base, differences in sizes of samples or population are eliminated.
   
7. A cumulative frequency distribution shows the sub-totals for each attribute and all attributes above or below it. A cumulative percentage distribution expresses the same property for a percentage distribution.
   
8. Measures of central tendency describe the average or typical occurrence of something. The three measures of central tendency are the mean, median, and the mode. The mean is the arithmetical average – the sum of scores divided by the number of scores; the median is the middle score in an ordered distribution (half the scores are above the median and half below it); the mode is the score that occurs most frequently.
   
9. Selection of a measure of central tendency depends upon properties of a variable. For variables measured at the nominal level, the mode is the only measure of central tendency. Medians can be used for ordinal data, but the mode often is the best choice for measuring central tendency.
   
10. The mean is used as measure of central tendency for most interval or ratio variables (age, intelligence, etc.)
    
11. Scores at the extreme ends of a distribution affect the mean, but not the median, which is affected only by the relative position of scores, not their values. When a distribution has a number of extreme scores at the high or low end, it is said to be skewed. With skewed distribution, the median is generally a more appropriate measure of central tendency. It is not pushed in the direction of the extreme scores, whereas the mean is.
    
12. Another objective of univariate analysis is to learn about the degree of variation or dispersion among scores making up a distribution. Three measures of variability are used: the range (R), the variance (s2), and the standard deviation (s).
    
13. The range is measured by the difference between the highest minus the lowest score plus 1. Variance is based on the deviation of each score from the mean of a distribution. Standard deviation is the square root of the variance.
    
14. A normal distribution or normal curve has precise mathematical properties. It is symmetrical in shape: If folded at the middle, the left and right halves of the curve would match perfectly. In a normal distribution, therefore, the mean, median, and mode are identical. But most important, a fixed proportion of observations lies between the mean and specific units of the standard deviation.
    
15. In a normal distribution, approximately 68% of the scores lie within ±1s (standard deviation) of the mean; approximately 95% lie within ±2s of the mean; and over 99% lie within ±3s of the mean. Although most distributions for variables do not meet the exact criteria for a normal distribution, standard deviation is frequently used to describe variations among scores or other values.

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