Table Of ContentPearson New International Edition
Statistics and Data Analysis
for Nursing Research
Denise F. Polit
Second Edition
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ISBN 13: 978-1-292-02781-4
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P E A R S O N C U S T O M L I B R AR Y
Table of Contents
1. Frequency Distributions: Tabulating and Displaying Data
Denise F. Polit 1
2. Central Tendency, Variability, and Relative Standing
Denise F. Polit 25
3. Bivariate Description: Crosstabulation, Risk Indexes, and Correlation
Denise F. Polit 49
4. Statistical Inference
Denise F. Polit 77
5. t Tests: Testing Two Mean Differences
Denise F. Polit 113
6. Analysis of Variance
Denise F. Polit 139
7. Chi-Square and Nonparametric Tests
Denise F. Polit 175
8. Correlation and Simple Regression
Denise F. Polit 205
9. Multiple Regression
Denise F. Polit 233
10. Analysis of Covariance, Multivariate ANOVA, and Related Multivariate Analyses
Denise F. Polit 277
11. Logistic Regression
Denise F. Polit 323
12. Factor Analysis and Internal Consistency Reliability Analysis
Denise F. Polit 355
13. Missing Values
Denise F. Polit 391
I
444233139
Appendix: Theoretical Sampling Distribution Tables
Denise F. Polit 421
Appendix: Tables for Power Analyses
Denise F. Polit 433
Index 439
II
Frequency Distributions:
Tabulating and
Displaying Data
Frequency Distributions Kurtosis
Ungrouped Frequency Distributions The Normal Distribution
forQuantitative Variables Research Applications of Frequency
Grouped Frequency Distributions Distributions
Frequency Distributions for Categorical Variables The Uses of Frequency Distributions
Graphic Display of Frequency Distributions The Presentation of Frequency Information
Bar Graphs and Pie Charts inResearch Reports
Histograms Tips on Preparing Tables and
Frequency Polygons Graphs forFrequency
General Issues in Graphic Displays Distributions
Research Example
Shapes of Distributions
Modality Summary Points
Symmetry and Skewness Exercises
Researchers use a variety of descriptive statistics to describe data from a research sample. This chapter presents
methods of calculating, organizing, and displaying some descriptive statistics.
FREQUENCY DISTRIBUTIONS
Researchers begin their data analyses by imposing some order on their data. A simple listing of the raw data for
a variable rarely conveys much information, unless the sample is very small. Take, for example, the data shown
in Table 1, which represent fictitious resting heart rate values (in beats per minute or bpm) for 100 patients. It is
difficult to understand these data simply by looking at the numbers: We cannot readily see what the highest and
lowest values are, nor can we see where the heart rate values tend to cluster.
Ungrouped Frequency Distributions for Quantitative Variables
One of the first things that researchers typically do with data is to construct frequency distributions. A frequency
distributionis a systematic arrangement of data values—from lowest to highest or vice versa—together with a
count of how many times each value was observed in the dataset. Table 2presents a frequency distribution for the
From Chapter 2 of Statistics and Data Analysis for Nursing Research, Second Edition. Denise F. Polit. Copyright ©2010 by
Pearson Education, Inc. All rights reserved.
1
Frequency Distributions: Tabulating and Displaying Data
TABLE 1 Fictitious Data on Heart Rate for 100 Patients, in Beats Per Minute
60 65 63 57 64 65 56 64 71 67
70 72 68 64 62 66 59 67 61 66
56 69 67 73 68 63 69 70 72 68
60 66 61 60 65 67 74 66 65 66
65 72 66 58 62 60 73 64 59 72
65 68 61 59 68 71 67 65 63 70
67 59 66 69 61 70 58 62 66 63
74 69 68 57 63 65 71 67 62 66
55 70 69 62 66 67 62 72 64 68
64 58 64 66 63 69 71 64 67 57
heart rate data. Now we can tell at a glance that the lowest value is 55, the highest
value is 74, and the value with the highest frequency (11 people) is 66.
TIP: Statistical software provides options for ordering variables in
ascending or descending order. We show ascending order in our examples,
but researchers preparing tables for journal articles may use reverse
ordering for conceptual or theoretical reasons.
TABLE 2 Frequency Distribution of Heart Rate Values
Score (X) Tallies Frequency (f) Percent (%)
55 | 1 1.0
56 || 2 2.0
57 ||| 3 3.0
58 ||| 3 3.0
59 |||| 4 4.0
60 |||| 4 4.0
61 |||| 4 4.0
62 |||| | 6 6.0
63 |||| | 6 6.0
64 |||| ||| 8 8.0
65 |||| ||| 8 8.0
66 |||| |||| | 11 11.0
67 |||| |||| 9 9.0
68 |||| || 7 7.0
69 |||| | 6 6.0
70 |||| 5 5.0
71 |||| 4 4.0
72 |||| 5 5.0
73 || 2 2.0
74 || 2 2.0
N(cid:2)100 (cid:2)Σf 100.0 (cid:2)Σ%
2
Frequency Distributions: Tabulating and Displaying Data
Researchers constructing a frequency distribution manually list the data values
(the Xs) in a column in the desired order, and then keep a tally next to each value for
each occurrence of that value. In Table 2, the tallies are shown in the second column,
using the familiar system of four vertical bars and then a slash for the fifth case. The
tallies can then be totaled, yielding the frequency (f) or count of the number of cases
for each data value.
In constructing a frequency distribution, researchers must make sure that the
list of data values is mutually exclusive and collectively exhaustive. The sum of the
frequencies must equal the number of cases in the sample.
Σf(cid:2)N
where Σ(cid:2)the sum of
f(cid:2)the frequencies
N(cid:2)the sample size
This equation simply states that the sum of (symbolized by the Greek letter sigma, Σ)
all the frequencies of score values (f) equals the total number of study participants (N).
A frequency count of data values usually communicates little information in and
of itself. In Table 2, the fact that five patients had a heart rate of 70 bpm is not very in-
formative without knowing how many patients there were in total, or how many pa-
tients had lower or higher heart rates. Because of this fact, frequency distributions al-
most always show not only absolute frequencies (i.e., the count of cases), but also
relative frequencies, which indicate the percentage of times a given value occurs. The
far right column of Table 2 indicates that 5% of the sample had a heart rate of 70.
Percentages are useful descriptive statistics that appear in the majority of research
reports. A percentage can be calculated easily, using the following simple formula:
% (cid:2)(f(cid:3)N) (cid:4)100
That is, the percentage for a given value or score is the frequency for that
value, divided by the number of people, times 100. The sum of all percentages must
equal 100% (i.e., Σ% (cid:2) 100%). You will probably recall that a proportion is the
same as a percentage, before multiplying by 100 (i.e., proportion (cid:2)f(cid:3)N).
Of course, researchers rarely use a tally system or manually compute percentages
with their dataset. In SPSS and other statistical software packages, once the data have
been entered and variable information has been input, you can proceed to run analyses
by using pull-down menus that allow you to select which type of analysis you want to
run. For the analyses described in this chapter, you would click on Analyze in the top
toolbar, then select Descriptive Statistics from the pull-down menu, then Frequencies.
Another commonly used descriptive statistic is cumulative relative frequency,
which combines the percentage for the given score value with percentages for all val-
ues that preceded it in the distribution. To illustrate, the heart rate data have been an-
alyzed on a computer using SPSS, and the resulting computer printout is presentedin
Figure 1. (The SPSS commands that produced the printout in Figure 1are Analyze
➞Descriptive Statistics➞Frequencies ➞hartrate.)
3
Frequency Distributions: Tabulating and Displaying Data
Frequencies
Heart Rate in Beats per Minute
Valid Cumulative
Frequency Percent Percent Percent
Valid 55 1 1.0 1.0 1.0
56 2 2.0 2.0 3.0
57 3 3.0 3.0 6.0
58 3 3.0 3.0 9.0
59 4 4.0 4.0 13.0
60 4 4.0 4.0 17.0
61 4 4.0 4.0 21.0
62 6 6.0 6.0 27.0
63 6 6.0 6.0 33.0
64 8 8.0 8.0 41.0
65 8 8.0 8.0 49.0
66 11 11.0 11.0 60.0
67 9 9.0 9.0 69.0
68 7 7.0 7.0 76.0
69 6 6.0 6.0 82.0
70 5 5.0 5.0 87.0
71 4 4.0 4.0 91.0
72 5 5.0 5.0 96.0
73 2 2.0 2.0 98.0
74 2 2.0 2.0 100.0
Total 100 100.0 100.0
FIGURE 1 SPSS printout of a frequency distribution.
In Figure 1, cumulative relative frequencies are shown in the last column,
labeled Cumulative Percent. The advantage of these statistics is that they allow you to
see at a glance the percentage of cases that are equal to or less than a specified value.
For example, we can see that 87.0% of the patients had heart rates of 70 bpm or
lower. This figure also has a column labeled Valid Percent. In this example, the values
in this column are identical to the values in the preceding column (Percent) because
there are no missing data—heart rate information is available for all 100 patients. It
is common, however, to have missing data in actual studies. The percentages in the
column Valid Percentare computed after removing any missing cases. Thus, if heart
rate data were missing for ten sample members, the valid percent for the value of 55
would be 1.1% ([1 (cid:3)90] (cid:4)100) rather than 1.0%.
Grouped Frequency Distributions
The values in the heart rate example ranged from a low of 55 to a high of 74, for a
total of 20 different values. For some variables, the range of values is much greater.
For example, in a sample of 100 infants, it would be possible to obtain 100 different
values for the variable birth weight measured in grams. An ordinary frequency table
to examine the birth weight data would not be very informative, because each value
would have a frequency of 1. When a variable has many possible values, researchers
4
Frequency Distributions: Tabulating and Displaying Data
sometimes construct a grouped frequency distribution. Such a distribution
involves grouping together values into sets, called class intervals, and then tabulating
the frequency of cases within the class intervals. For example, for infants’ birth
weights, we might establish the following class intervals:
• 1,500 or fewer grams
• 1,501–2,000 grams
• 2,001–2,500 grams
• 2,501–3,000 grams
• 3,001 or more grams
In grouping together data values, it is useful to strike a balance between insuffi-
cient detail when too few groups are used, and lack of clarity when too many groups
are created. For example, if infants’ birth weight was grouped in clusters of 10 grams
(e.g., 1,001 to 1,010; 1,011 to 1,020, and so on), there would be dozens of groups. On
the other hand, for some purposes it might be inadequate to cluster the birth weight
data into only two groups (e.g., (cid:6)2,000 grams and (cid:7)2,000 grams). As a rule of thumb,
a good balance can usually be achieved using between four and 10 class intervals.
Once you have a general idea about the desired number of intervals, you can
determine the size of the interval. By subtracting the lowest data value in the dataset
from the highest data value and then dividing by the desired number of groups, an
approximate interval size can be determined. However, you should also strive for in-
tervals that are psychologically appealing. Interval sizes of two and multiples of five
(e.g., 10, 100, 500) often work best. All class interval sizes in a grouped frequency
distribution should be the same.
Given that the heart rate data resulted in a total of 20 different values, it might be
useful to construct a grouped frequency distribution. Clustering five values in a class
interval, for example, we would have four intervals. The printout for this grouped fre-
quency distribution is shown in Figure 2.1In this distribution, we can readily see that,
for example, there were relatively few cases at either the low end or high end of the
distribution, and that there is a substantial clustering of values in the 65 to 69 interval.
On the other hand, there is also an information loss: For example, we cannot deter-
mine from this distribution what percentage of cases is 70 or below, as we could with
the original ungrouped distribution. Decisions on whether to use an ungrouped or
grouped distribution depend, in part, on the reason for constructing the distribution.
Frequency Distributions for Categorical Variables
When a variable is categorical or qualitative (i.e., measured on the nominal scale),
you can also construct a frequency distribution. As with quantitative variables, the
variable categories are listed in the first column, followed by frequencies and/or
relative frequencies in succeeding columns. A fictitious example of a frequency
distribution for the nominal variable marital status is shown in Table 3.
With categorical variables, it is usually not meaningful to display cumulative
relative frequencies because there is no natural ordering of categories along any di-
mension. In Table 3, for example, the ordering of the categories could be changed
without affecting the information (e.g., the category “Single, never married” could
come first). Several strategies can be used to order the categories in tables prepared
1For producing the frequency distribution in Figure 2, we created a new variable (we called it grouphr) by
➞
using the Transform Compute commands. For example, we instructed the computer to set grouphrto
(cid:5) (cid:6)
1 if hartrate 54 and hartrate 60. A procedure in SPSS called “Visual Binning” (within the
“Transform” set of commands) can also be used.
5
