Chapter6: Understanding Frequencies and Percentages

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Complete Exercises 6, 8, and 9 in Statistics for Nursing Research: A Workbook for Evidence-Based Practice, and submit as directed by the instructor.]

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chapter 6

Understanding Frequencies

and Percentages

STATISTICAL TECHNIQUE IN REVIEW

Frequency is the number of times a score or value for a variable occurs in a set of data.

Frequency distribution is a statistical procedure that involves listing all the possible

values or scores for a variable in a study. Frequency distributions are used to organize

study data for a detailed examination to help determine the presence of errors in coding

or computer programming ( Grove, Burns, & Gray, 2013 ). In addition, frequencies and

percentages are used to describe demographic and study variables measured at the nominal

or ordinal levels. Chapter6: Understanding Frequencies and Percentages

Percentage can be defi ned as a portion or part of the whole or a named amount in

every hundred measures. For example, a sample of 100 subjects might include 40 females

and 60 males. In this example, the whole is the sample of 100 subjects, and gender is

described as including two parts, 40 females and 60 males. A percentage is calculated

by dividing the smaller number, which would be a part of the whole, by the larger

number, which represents the whole. The result of this calculation is then multiplied

by 100%. For example, if 14 nurses out of a total of 62 are working on a given day, you

can divide 14 by 62 and multiply by 100% to calculate the percentage of nurses working

that day. Calculations: (14 ÷ 62) × 100% = 0.2258 × 100% = 22.58% = 22.6%. The answer

also might be expressed as a whole percentage, which would be 23% in this example.

A cumulative percentage distribution involves the summing of percentages from the

top of a table to the bottom. Therefore the bottom category has a cumulative percentage

of 100% (Grove, Gray, & Burns, 2015). Cumulative percentages can also be used to determine

percentile ranks, especially when discussing standardized scores. For example, if 75%

of a group scored equal to or lower than a particular examinee ’ s score, then that examinee ’ s

rank is at the 75 th percentile. When reported as a percentile rank, the percentage is often

rounded to the nearest whole number. Percentile ranks can be used to analyze ordinal

data that can be assigned to categories that can be ranked. Percentile ranks and cumulative

percentages might also be used in any frequency distribution where subjects have only one

value for a variable. For example, demographic characteristics are usually reported with the

frequency ( f ) or number ( n ) of subjects and percentage (%) of subjects for each level of a

demographic variable. Income level is presented as an example for 200 subjects:

Income Level Frequency ( f ) Percentage (%) Cumulative %

1. < $40,000 20 10% 10%

2. $40,000–$59,999 50 25% 35%

3. $60,000–$79,999 80 40% 75%

4. $80,000–$100,000 40 20% 95%

5. > $100,000 10 5% 100%

EXERCISE

60 EXERCISE 6 • Understanding Frequencies and Percentages

In data analysis, percentage distributions can be used to compare fi ndings from different

studies that have different sample sizes, and these distributions are usually arranged in

tables in order either from greatest to least or least to greatest percentages ( Plichta &

Kelvin, 2013 ).

RESEARCH ARTICLE

Source

Eckerblad, J., Tödt, K., Jakobsson, P., Unosson, M., Skargren, E., Kentsson, M., & Theander,

K. (2014). Symptom burden in stable COPD patients with moderate to severe airfl ow

limitation. Heart & Lung, 43 (4), 351–357.

Introduction

Eckerblad and colleagues (2014 , p. 351) conducted a comparative descriptive study to

examine the symptoms of “patients with stable chronic obstructive pulmonary disease

(COPD) and determine whether symptom experience differed between patients with moderate

or severe airfl ow limitations.” The Memorial Symptom Assessment Scale (MSAS)

was used to measure the symptoms of 42 outpatients with moderate airfl ow limitations

and 49 patients with severe airfl ow limitations. The results indicated that the mean

number of symptoms was 7.9 ( } 4.3) for both groups combined, with no signifi cant differences

found in symptoms between the patients with moderate and severe airfl ow limitations.

For patients with the highest MSAS symptom burden scores in both the moderate

and the severe limitations groups, the symptoms most frequently experienced included

shortness of breath, dry mouth, cough, sleep problems, and lack of energy. The researchers

concluded that patients with moderate or severe airfl ow limitations experienced multiple

severe symptoms that caused high levels of distress. Quality assessment of COPD

patients ’ physical and psychological symptoms is needed to improve the management of

their symptoms.

Relevant Study Results

Eckerblad et al. (2014 , p. 353) noted in their research report that “In total, 91 patients

assessed with MSAS met the criteria for moderate ( n = 42) or severe airfl ow limitations

( n = 49). Of those 91 patients, 47% were men, and 53% were women, with a mean age of

68 ( } 7) years for men and 67 ( } 8) years for women. The majority (70%) of patients were

married or cohabitating. In addition, 61% were retired, and 15% were on sick leave.

Twenty-eight percent of the patients still smoked, and 69% had stopped smoking. The

mean BMI (kg/m 2 ) was 26.8 ( } 5.7). Chapter6: Understanding Frequencies and Percentages

There were no signifi cant differences in demographic characteristics, smoking history,

or BMI between patients with moderate and severe airfl ow limitations ( Table 1 ). A lower

proportion of patients with moderate airfl ow limitation used inhalation treatment with

glucocorticosteroids, long-acting β 2 -agonists and short-acting β 2 -agonists, but a higher

proportion used analgesics compared with patients with severe airfl ow limitation.

Symptom prevalence and symptom experience

The patients reported multiple symptoms with a mean number of 7.9 ( } 4.3) symptoms

(median = 7, range 0–32) for the total sample, 8.1 ( } 4.4) for moderate airfl ow limitation

and 7.7 ( } 4.3) for severe airfl ow limitation ( p = 0.36) . . . . Highly prevalent physical symptoms

( ≥ 50% of the total sample) were shortness of breath (90%), cough (65%), dry mouth

(65%), and lack of energy (55%). Five additional physical symptoms, feeling drowsy, pain,

Understanding Frequencies and Percentages • EXERCISE 6 61

TABLE 1 BACKGROUND CHARACTERISTICS AND USE OF MEDICATION FOR PATIENTS WITH

STABLE CHRONIC OBSTRUCTIVE LUNG DISEASE CLASSIFIED IN PATIENTS WITH

MODERATE AND SEVERE AIRFLOW LIMITATION

Moderate

n = 42

Severe

n = 49 p Value

Sex, n (%) 0.607

Women 19 (45) 29 (59)

Men 23 (55) 20 (41)

Age (yrs), mean ( SD ) 66.5 (8.6) 67.9 (6.8) 0.396

Married/cohabitant n (%) 29 (69) 34 (71) 0.854

Employed, n (%) 7 (17) 7 (14) 0.754

Smoking, n % 0.789

Smoking 13 (31) 12 (24)

Former smokers 28 (67) 35 (71)

Never smokers 1 (2) 2 (4)

Pack years smoking, mean ( SD ) 29.1 (13.5) 34.0 (19.5) 0.177

BMI (kg/m 2 ), mean ( SD ) 27.2 (5.2) 26.5 (6.1) 0.555

FEV 1 % of predicted, mean ( SD ) 61.6 (8.4) 42.2 (5.8) < 0.001

SpO 2 % mean ( SD ) 95.8 (2.4) 94.5 (3.0) 0.009

Physical health, mean ( SD ) 3.2 (0.8) 3.0 (0.8) 0.120

Mental health, mean ( SD ) 3.7 (0.9) 3.6 (1.0) 0.628

Exacerbation previous 6 months, n (%) 14 (33) 15 (31) 0.781

Admitted to hospital previous year, n (%) 10 (24) 14 (29) 0.607

Medication use, n (%)

Inhaled glucocorticosteroids 30 (71) 44 (90) 0.025

Systemic glucocorticosteroids 3 (6.3) 0 (0) 0.094

Anticholinergic 32 (76) 42 (86) 0.245

Long-acting β 2 -agonists 30 (71) 45 (92) 0.011

Short-acting β 2 -agonists 13 (31) 32 (65) 0.001

Analgesics 11 (26) 5 (10) 0.046

Statins 8 (19) 11 (23) 0.691

Eckerblad, J., Tödt, K., Jakobsson, P., Unosson, M., Skargren, E., Kentsson, M., & Theander, K. (2014). Symptom burden in stable

COPD patients with moderate to severe airfl ow limitation. Heart & Lung, 43 (4), p. 353.

numbness/tingling in hands/feet, feeling irritable, and dizziness, were reported by between

25% and 50% of the patients. The most commonly reported psychological symptom was

diffi culty sleeping (52%), followed by worrying (33%), feeling irritable (28%) and feeling

sad (22%). There were no signifi cant differences in the occurrence of physical and psychological

symptoms between patients with moderate and severe airfl ow limitations”

( Eckerblad et al., 2014 , p. 353).

62 EXERCISE 6 • Understanding Frequencies and Percentages

STUDY QUESTIONS

1. What are the frequency and percentage of women in the moderate airfl ow limitation group?

2. What were the frequencies and percentages of the moderate and the severe airfl ow limitation

groups who experienced an exacerbation in the previous 6 months?

3. What is the total sample size of COPD patients included in this study? What number or frequency

of the subjects is married/cohabitating? What percentage of the total sample is married

or cohabitating?

4. Were the moderate and severe airfl ow limitation groups signifi cantly different regarding married/

cohabitating status? Provide a rationale for your answer.

5. List at least three other relevant demographic variables the researchers might have gathered data

on to describe this study sample.

6. For the total sample, what physical symptoms were experienced by ≥ 50% of the subjects? Identify

the physical symptoms and the percentages of the total sample experiencing each symptom.

Understanding Frequencies and Percentages • EXERCISE 6 63

7. Were the physical symptoms identifi ed in the study what you might expect for patients with

moderate to severe COPD? Provide a rationale for your answer with documentation.

8. What frequency and percentage of the total sample used inhaled glucocorticosteroids? Show

your calculations and round to the nearest tenth of a percent.

9. Is there a signifi cant difference between the moderate and severe airfl ow limitation groups

regarding the use of inhaled glucocorticosteriods? Provide a rationale for your answer.

10. Was the percentage of COPD patients with moderate and severe airfl ow limitations using

inhaled glucocorticosteriods what you expected? Provide a rationale for your answer with

documentation. Chapter6: Understanding Frequencies and Percentages

Answers to Study Questions

1. The moderate airfl ow limitation group included 19 women, which means 45% of this group

was female (see Table 1 ).

2. A frequency of 14 (33%) of the moderate airfl ow limitation group and a frequency of 15 (31%)

of the severe airfl ow limitation group experienced an exacerbation in the previous 6 months

(see Table 1 ).

3. The total sample was N = 91 patients with COPD in the Eckerblad et al. (2014) study (see the

narrative of study results). The number or frequency of subjects ’ who were married/cohabitating

is calculated by adding the frequencies from the two groups in Table 1 .

Calculation: Frequency married/cohabitating = 29 moderate group + 34 severe group = 63.

The percentage of the sample married/cohabitating is 70% (see narrative of study results) or

can be calculated by (frequency married/cohabitating ÷ sample size) × 100% = (63 ÷ 91) ×

100% = 69.23% = 69%. The researchers might have rounded to next higher whole percent of

70%, but 69% is a more accurate percentage of the married/cohabitating for the sample.

4. No, the moderate and severe airfl ow limitation groups were not signifi cantly different regarding

married/cohabitating status as indicated by p = 0.854 (see Table 1 ). The level of signifi –

cance or alpha ( α ) in most nursing studies is set at α = 0.05 ( Grove et al., 2015 ). Since the

p value is > 0.05, the two groups were not signifi cantly different in this study.

5. Additional demographic variables that might have been described in this study include race/

ethnicity, socioeconomic status or income level, years diagnosed with COPD, and other comorbid

medical diagnoses of these study participants. You might have identifi ed other relevant

demographic variables to be included in this study.

6. “Highly prevalent physical symptoms ( ≥ 50% of the total sample) were shortness of breath

(90%), cough (65%), dry mouth (65%), and lack of energy (55%)” ( Eckerblad et al., 2014 ,

p. 353; see study narrative of results).

7. Yes, the physical symptoms of shortness of breath, cough, dry mouth, and lack of energy or

fatigue are extremely common in patients with COPD who have moderate to severe airfl ow

limitations. Evidence-based guidelines for many chronic diseases can be found on the Agency

for Healthcare Research and Quality (AHRQ) website at www.guidelines.gov . Specifi c evidence-

based guidelines for the assessment, diagnosis, and management of COPD can be

found at the following AHRQ website: http://www.guideline.gov/content.aspx?id = 23801

&search = copd . The Global Initiative for Chronic Obstructive Lung Disease website is also an

excellent resource at http://www.goldcopd.org/Guidelines/guidelines-resources.html . You

might document with other websites, research articles, or textbooks.

Understanding Frequencies and Percentages • EXERCISE 6 65

8. Frequency = 74 and percent = 81.3%. In this study, 30 of the moderate airfl ow limitation group

and 44 of the severe group used inhaled glucocorticosteroids. Calculations: Frequency = 30

+ 44 = 74. Percentage total sample = (74 ÷ 91) × 100% = 0.8132 × 100% = 81.32% = 81.3%,

rounded to the nearest tenth of a percent.

9. Yes, the moderate and severe airfl ow limitation groups were signifi cantly different regarding

the use of inhaled glucocorticosteroids as indicated by p = 0.025 (see Table 1 ). The level of

signifi cance or alpha ( α ) in most nursing studies is set at 0.05. Since the p value is < 0.05, the

two groups were signifi cantly different for the use of inhaled glucocorticosteroids in this

study ( Grove et al., 2013 ; Shadish, Cook, & Campbell, 2002 ).

10. In this study, 30 (71%) of the patients with moderate airfl ow limitation and 44 (90%) of the

patients with severe airfl ow limitation were treated with glucocorticosteroids. The mean percentage

for the total sample who used glucocorticosteroids is (71% + 90%) ÷ 2 = 161 ÷ 2 =

80.5%, or 81%. The use of inhaled glucocorticosteroids is very common for patients with

moderate to severe COPD, in fact, recommended by national evidence-based guidelines, particularly

for those with severe airfl ow limitation. Thus, you might expect that a large number

of COPD patients in this study were using inhaled glucocorticosteroids. The Gold Standard

for the management of COPD can be found at the AHRQ (2015) website at: http://www

.guideline.gov/content.aspx?id = 23801&search = copd or at the Global Initiative for Chronic

Obstructive Lung Disease website at: http://www.goldcopd.org/Guidelines/guidelinesresources.

html . Chapter6: Understanding Frequencies and Percentages

EXERCISE

Questions to Be Graded

Follow your instructor ’ s directions to submit your answers to the following questions for grading.

Your instructor may ask you to write your answers below and submit them as a hard copy for

grading. Alternatively, your instructor may ask you to use the space below for notes and submit your

answers online at http://evolve.elsevier.com/Grove/statistics/ under “Questions to Be Graded.”

1. What are the frequency and percentage of the COPD patients in the severe airfl ow limitation

group who are employed in the Eckerblad et al. (2014) study?

2. What percentage of the total sample is retired? What percentage of the total sample is on sick

leave?

3. What is the total sample size of this study? What frequency and percentage of the total sample

were still employed? Show your calculations and round your answer to the nearest whole percent.

4. What is the total percentage of the sample with a smoking history—either still smoking or former

smokers? Is the smoking history for study participants clinically important? Provide a rationale

for your answer.

Name: _______________________________________________________ Class: _____________________

Date: ___________________________________________________________________________________

68 EXERCISE 6 • Understanding Frequencies and Percentages

5. What are pack years of smoking? Is there a signifi cant difference between the moderate and severe

airfl ow limitation groups regarding pack years of smoking? Provide a rationale for your answer.

6. What were the four most common psychological symptoms reported by this sample of patients

with COPD? What percentage of these subjects experienced these symptoms? Was there a signifi

cant difference between the moderate and severe airfl ow limitation groups for psychological

symptoms?

7. What frequency and percentage of the total sample used short-acting β2 -agonists? Show your

calculations and round to the nearest whole percent.

8. Is there a signifi cant difference between the moderate and severe airfl ow limitation groups

regarding the use of short-acting β 2 -agonists? Provide a rationale for your answer.

9. Was the percentage of COPD patients with moderate and severe airfl ow limitation using shortacting

β 2 -agonists what you expected? Provide a rationale with documentation for your answer.

10. Are these fi ndings ready for use in practice? Provide a rationale for your answer

chapter 8

Measures of Central Tendency :

Mean, Median, and Mode

EXERCISE

STATISTICAL TECHNIQUE IN REVIEW

Mean, median, and mode are the three measures of central tendency used to describe

study variables. These statistical techniques are calculated to determine the center of a

distribution of data, and the central tendency that is calculated is determined by the level

of measurement of the data (nominal, ordinal, interval, or ratio; see Exercise 1 ). The mode

is a category or score that occurs with the greatest frequency in a distribution of scores

in a data set. The mode is the only acceptable measure of central tendency for analyzing

nominal-level data, which are not continuous and cannot be ranked, compared, or subjected

to mathematical operations. If a distribution has two scores that occur more frequently

than others (two modes), the distribution is called bimodal . A distribution with

more than two modes is multimodal ( Grove, Burns, & Gray, 2013 ).

The median ( MD ) is a score that lies in the middle of a rank-ordered list of values of

a distribution. If a distribution consists of an odd number of scores, the MD is the middle

score that divides the rest of the distribution into two equal parts, with half of the values

falling above the middle score and half of the values falling below this score. In a distribution

with an even number of scores, the MD is half of the sum of the two middle numbers

of that distribution. If several scores in a distribution are of the same value, then the MD

will be the value of the middle score. The MD is the most precise measure of central tendency

for ordinal-level data and for nonnormally distributed or skewed interval- or ratiolevel

data. The following formula can be used to calculate a median in a distribution of

scores.

Median(MD) (N 1) 2

N is the number of scores

Example: N Median th score 31

31 1

32 2 16

Example: N Median . th score 40

40 1

41 2 20 5

Thus in the second example, the median is halfway between the 20 th and the 21 st scores.

The mean ( X ) is the arithmetic average of all scores of a sample, that is, the sum of its

individual scores divided by the total number of scores. The mean is the most accurate

measure of central tendency for normally distributed data measured at the interval and

ratio levels and is only appropriate for these levels of data (Grove, Gray, & Burns, 2015).

In a normal distribution, the mean, median, and mode are essentially equal (see Exercise

26 for determining the normality of a distribution). The mean is sensitive to extreme

80 EXERCISE 8 • Measures of Central Tendency: Mean, Median, and Mode

scores such as outliers. An outlier is a value in a sample data set that is unusually low or

unusually high in the context of the rest of the sample data. If a study has outliers, the

mean is most affected by these, so the median might be the measure of central tendency

included in the research report ( Plichta & Kelvin, 2013 ). The formula for the mean is:

MeanX

Σ X is the sum of the raw scores in a study

N is the sample size or number of scores in the study

Example:Raw scores 8, 9, 9,10,11,11 N 6 Mean 58 6 9.666 9.67

RESEARCH ARTICLE

Source

Winkler, C., Funk, M., Schindler, D. M., Hemsey, J. Z., Lampert, R., & Drew, B. J. (2013).

Arrhythmias in patients with acute coronary syndrome in the fi rst 24 hours of hospitalization.

Heart & Lung, 42 (6), 422–427. Chapter6: Understanding Frequencies and Percentages

Introduction

Winkler and colleagues (2013) conducted their study to describe the arrhythmias of a

population of patients with acute coronary syndrome (ACS) during their fi rst 24 hours

of hospitalization and to explore the link between arrhythmias and patients ’ outcomes.

The patients with ACS were admitted through the emergency department (ED), where a

Holter recorder was attached for continuous 12-lead electrocardiographic (ECG) monitoring.

The ECG data from the Holter recordings of 278 patients with ACS were analyzed.

The researchers found that “approximately 22% of patients had more than 50 premature

ventricular contractions (PVCs) per hour. Non-sustained ventricular tachycardia (VT)

occurred in 15% of the patients . . . . Only more than 50 PVCs/hour independently predicted

an increased length of stay ( p < 0.0001). No arrhythmias predicted mortality. Age

greater than 65 years and a fi nal diagnosis of acute myocardial infarction (AMI) independently

predicted more than 50 PVCs per hour ( p = 0.0004)” ( Winkler et al., 2013 , p. 422).

Winkler and colleagues (2013 , p. 426) concluded: “Life-threatening arrhythmias are

rare in patients with ACS, but almost one quarter of the sample experienced isolated

PVCs. There was a signifi cant independent association between PVCs and a longer length

of stay (LOS), but PVCs were not related to other adverse outcomes. Rapid treatment of

the underlying ACS should remain the focus, rather than extended monitoring for

arrhythmias we no longer treat.”

Relevant Study Results

The demographic and clinical characteristics of the sample and the patient outcomes for

this study are presented in this exercise. “The majority of the patients ( n = 229; 83%) had

a near complete Holter recording of at least 20 h and 171 (62%) had a full 24 h recorded.

We included recordings of all patients in the analysis. The mean duration of continuous

12-lead Holter recording was 21 } 6 (median 24) h. Chapter6: Understanding Frequencies and Percentages

The mean patient age was 66 years and half of the patients identifi ed White as

their race ( Table 1 ). There were more males than females and most patients (92%) experienced

chest pain as one of the presenting symptoms to the ED. Over half of the patients

Measures of Central Tendency: Mean, Median, and Mode • EXERCISE 8 81

TABLE 1 DEMOGRAPHIC AND CLINICAL CHARACTERISTICS OF THE SAMPLE ( N = 278)

Characteristic N %

Gender

Male 158 57

Female 120 43

Race

White 143 51

Asian 60 22

Black 50 18

American Indian 23 8

Pacifi c Islander 2 < 1

Presenting Symptoms to the ED (May Have > 1)

Chest pain 255 92

Shortness of breath 189 68

Jaw, neck, arm, or back pain 152 55

Diaphoresis 116 42

Nausea and vomiting 96 35

Syncope 11 4

Cardiovascular Risk Factors (May Have > 1)

Hypertension 211 76

Hypercholesterolemia 175 63

Family history of CAD 148 53

Diabetes 81 29

Smoking (current) 56 20

Cardiovascular Medical History (May Have > 1)

Personal history of CAD 176 63

History of unstable angina 124 45

Previous acute myocardial infarction 114 41

Previous percutaneous coronary intervention 85 31

Previous CABG surgery 54 19

History of arrhythmias 53 19

Final Diagnosis

Unstable angina 180 65

Non-ST elevation myocardial infarction 74 27

ST elevation myocardial infarction 24 9

Interventions during 24-h Holter Recording

PCI ≤ 90 min of ED admission 14 5

PCI > 90 min of ED admission 3 1

Thrombolytic medication 3 1

Interventions Any Time during Hospitalization

PCI 76 27

Treated with anti-arrhythmic medication 16 6

CABG surgery 22 8

Mean ( SD ) Median Range

Age (years) 66 (14) 66 30–102

ECG recording time (hours) 21 (6) 24 2–25

ED, emergency department; CAD, coronary artery disease; CABG, coronary artery bypass graft; PCI, percutaneous coronary

intervention; SD , standard deviation; ECG, electrocardiogram.

Winkler, C., Funk, M., Schindler, D. M., Hemsey, J. Z., Lampert, R., & Drew, B. J. (2013). Arrhythmias in patients with acute

coronary syndrome in the fi rst 24 hours of hospitalization. Heart & Lung, 42 (6), p. 424.

82 EXERCISE 8 • Measures of Central Tendency: Mean, Median, and Mode

Winkler, C., Funk, M., Schindler, D. M., Hemsey, J. Z., Lampert, R., & Drew, B. J. (2013). Arrhythmias in patients with acute

coronary syndrome in the fi rst 24 hours of hospitalization. Heart & Lung , 42 (6), p. 424.

TABLE 2 OUTCOMES DURING INPATIENT STAY, AND WITHIN 30 DAYS AND 1 YEAR OF

HOSPITALIZATION ( N = 278)

Outcomes N %

Inpatient complications (may have > 1)

AMI post admission for patients admitted with UA 21 8

Transfer to intensive care unit 17 6

Cardiac arrest 7 3

AMI extension (detected by 2nd rise in CK-MB) 6 2

Cardiogenic shock 5 2

New severe heart failure/pulmonary edema 2 1

Readmission *

30-day

To ED for a cardiovascular reason 42 15

To hospital for ACS 13 5

1-year ( N = 246)

To ED for a cardiovascular reason 108 44

To hospital for ACS 24 10

All-cause mortality †

Inpatient 10 4

30-day 13 5

1-year ( N = 246) 27 11

Mean ( SD ) Median Range

Length of stay (days) 5.37 (7.02) 4 1–93

AMI, acute myocardial infarction; UA, unstable angina; CK-MB, creatinine kinase-myocardial band; ED, emergency department;

ACS, acute coronary syndrome; SD , standard deviation.

* Readmission: 1-year data include 30-day data.

† All-cause mortality: 30-day data include inpatient data; 1-year data include both 30-day and inpatient data.

experienced shortness of breath (68%) and jaw, neck, arm, or back pain (55%). Hypertension

was the most frequently occurring cardiovascular risk factor (76%), followed by

hypercholesterolemia (63%) and family history of coronary artery disease (53%). A majority

had a personal history of coronary artery disease (63%) and 19% had a history of

arrhythmias” ( Winkler et al., 2013 , pp. 423–424). Chapter6: Understanding Frequencies and Percentages

Winkler et al. (2013 , p. 424) also reported: “We categorized patient outcomes into four

groups: 1) inpatient complications (of which some patients may have experienced more

than one); 2) inpatient length of stay; 3) readmission to either the ED or the hospital

within 30-days and 1-year of initial hospitalization; and 4) death during hospitalization,

within 30-days, and 1-year after discharge ( Table 2 ). These are outcomes that are reported

in many contemporary studies of patients with ACS. Thirty-two patients (11.5%) were lost

to 1-year follow-up, resulting in a sample size for the analysis of 1-year outcomes of 246

patients” ( Winkler et al., 2013 , p. 424).

Measures of Central Tendency: Mean, Median, and Mode • EXERCISE 8 83

STUDY QUESTIONS

1. In Table 1 , what is the mode for cardiovascular risk factors? Provide a rationale for your answer.

What percentage of the patients experienced this risk factor?

2. Which measure of central tendency always represents an actual score of the distribution?

a. Mean

b. Median

c. Mode

d. Range

3. What is the mode for the variable presenting symptoms to the ED? What percentage of the

patients had this symptom? Do the presenting symptoms have a single mode or is this distribution

bimodal or multimodal? Provide a rationale for your answer.

4. What are the three most common presenting symptoms to the ED, and why is this clinically

important?

5. For this study, what are the mean and median ages in years for the study participants?

6. Are the mean and median ages similar or different? What does this indicate about the distribution

of the sample?

84 EXERCISE 8 • Measures of Central Tendency: Mean, Median, and Mode

7. What are the mean and median ECG recording times in hours? What is the range for ECG

recordings? Does this distribution of data include an outlier? Provide a rationale for your answer.

8. What is the effect of outliers on the mean? If the study data have extreme outliers (either high

or low scores) in the data, what measure(s) of central tendency might be reported in a study?

Provide a rationale for your answer. Chapter6: Understanding Frequencies and Percentages

9. In the following example, 10 ACS patients were asked to rate their pain in their jaw and neck

on a 0–10 scale: 3, 4, 7, 7, 8, 5, 6, 8, 7, 9. What are the range and median for the pain scores?

10. Calculate the mean ( X ) for the pain scores in Question 9. Does this distribution of scores appear

to be normal? Provide a rationale for your answer.

Answers to Study Questions

1. Hypertension (HTN) is the mode for the cardiovascular risk factors since it is the most frequent

risk factor experienced by 211 of the study participants. A total of 76% of the study

participants had HTN.

2. Answer: c. Mode. The mode is the most frequently occurring score in a distribution; thus, it

will always be an actual score of the distribution. The mean is the average of all scores, so it

may not be an actual score of the distribution. Median is the middle score of the distribution,

which, with an even number of items, may not be an actual score in the distribution. The

range is a measure of dispersion, not a measure of central tendency.

3. Chest pain was the mode for the variable presenting symptoms to the ED, with 255 or 92%

of the participants experiencing it (see Table 1 ). The variable presenting symptoms to the ED

has one mode, chest pain, which was the most reported symptom.

4. Chest pain (92%); shortness of breath (68%); and jaw, neck, arm, or back pain (55%) are the

three most commonly reported presenting symptoms to the ED by study participants. This

is clinically important because nurses and other healthcare providers need to assess for these

symptoms, diagnose the problem, and appropriately manage patients presenting with ACS

at the ED. Since 92% of the participants had chest pain, it is clinically important to note this

symptom is common for both males and females in this study. Chapter6: Understanding Frequencies and Percentages

5. Both the mean ( X ) and median ( MD ) values were equal to 66 years.

6. In this study, the X age = MD age = 66 years, so they are the same value. In a normal distribution

of scores, the mode = MD = X ( Grove et al., 2013 ). Since the MD = X = 66 years, age seems

to be normally distributed in this sample.

7. ECG recording time has X = 21 hours and MD = 24 hours, with a range of 2–25 hours (see

Table 1 ). The 2 hours of ECG Holter monitoring seems to be an outlier, which resulted in

the difference between the mean and median ( X = 21 hours and MD = 24 hours) numbers of

monitoring hours. Winkler et al. (2013) reported that 83% of the study participants had a

near complete Holter recording of at least 20 hours, and 62% of the participants had a full

24 hours recorded, which supports the 2 hours as an outlier. You would need to examine the

study data to determine more about possible outliers. All ECG data were analyzed regardless

of the monitoring time, and more explanation is needed about outliers and the reasons for

including all recordings in the study analyses. Chapter6: Understanding Frequencies and Percentages

8. An unusually low score or outlier decreases the value of the mean as in this study (see the

answer to Question 7), and an unusually high score increases the mean value. The mean in a

study is most affected by outliers ( Grove et al., 2013 ). If the outliers cause the data to be

skewed or not normally distributed, it is best to report the median. If the data are normally

distributed, then the mean is the best measure of central tendency to report.

86 EXERCISE 8 • Measures of Central Tendency: Mean, Median, and Mode

9. Place the pain scores in order from the least to the greatest score = 3, 4, 5, 6, 7, 7, 7, 8,

8, 9. In this example, the range of pain scores = 3–9. The mode = 7 and the MD or middle

score = 7.

10. X = (3 + 4 + 5 + 6 + 7 + 7 + 7 + 8 + 8 + 9) ÷ 10 = 64 ÷ 10 = 6.4. The mode = median = approximately

the mean, so this is a normal distribution of scores. Exercise 26 provides the steps for

determining the normality of a distribution of scores.

Questions to Be Graded EXERCISE

Follow your instructor ’ s directions to submit your answers to the following questions for grading.

Your instructor may ask you to write your answers below and submit them as a hard copy for

grading. Alternatively, your instructor may ask you to use the space below for notes and submit your

answers online at http://evolve.elsevier.com/Grove/statistics/ under “Questions to Be Graded.”

1. The number of nursing students enrolled in a particular nursing program between the years of

2010 and 2016, respectively, were 563, 593, 606, 520, 563, 610, and 577. Determine the mean

( X ), median ( MD ), and mode of the number of the nursing students enrolled in this program.

Show your calculations. Chapter6: Understanding Frequencies and Percentages

2. What is the mode for the variable inpatient complications in Table 2 of the Winkler et al. (2014)

study? What percentage of the study participants had this complication?

3. Does the distribution of inpatient complications have a single mode, or is this distribution

bimodal or multimodal? Provide a rationale for your answer.

4. As reported in Table 1 , what are the three most common cardiovascular medical history events

in this study, and why is it clinically important to know the frequency of these events?

Name: _______________________________________________________ Class: _____________________

Date: ___________________________________________________________________________________

88 EXERCISE 8 • Measures of Central Tendency: Mean, Median, and Mode

5. What are the mean and median lengths of stay (LOS) for the study participants?

6. Are the mean and median for LOS similar or different? What might this indicate about the

distribution of the sample? Provide a rationale for your answer.

7. Examine the study results and determine the mode for arrhythmias experienced by the participants.

What was the second most common arrhythmia in this sample?

8. Was the most common arrhythmia in Question 7 related to LOS? Was this result statistically

signifi cant? Provide a rationale for your answer.

9. What study variables were independently predictive of the 50 premature ventricular contractions

(PVCs) per hour in this study?

10. In Table 1 , what race is the mode for this sample? Should these study fi ndings be generalized to

American Indians with ACS? Provide a rationale for your answer.

chapter 9

Measures of Dispersion :

Range and Standard Deviation

STATISTICAL TECHNIQUE IN REVIEW

Measures of dispersion , or measures of variability, are descriptive statistical techniques

conducted to identify individual differences of the scores in a sample. These techniques

give some indication of how scores in a sample are dispersed, or spread, around the mean.

The measures of dispersion indicate how different the scores are or the extent that individual

scores deviate from one another. If the individual scores are similar, dispersion or

variability values are small and the sample is relatively homogeneous , or similar, in terms

of these scores. A heterogeneous sample has a wide variation in the scores, resulting in

increased values for the measures of dispersion. Range and standard deviation are the

most common measures of dispersion included in research reports. Chapter6: Understanding Frequencies and Percentages

The simplest measure of dispersion is the range . In published studies, range is presented

in two ways: (1) the range includes the lowest and highest scores obtained for a

variable, or (2) the range is calculated by subtracting the lowest score from the highest

score. For example, the range for the following scores, 8, 9, 9, 10, 11, 11, might be reported

as 8 to 11 (8–11), which identifi es outliers or extreme values for a variable. The range can

also be calculated as follows: 11 − 8 = 3. In this form, the range is a difference score that

uses only the two extreme scores for the comparison. The range is generally reported in

published studies but is not used in further analyses ( Grove, Burns, & Gray, 2013 ).

The standard deviation ( SD ) is a measure of dispersion and is the average number of

points by which the scores of a distribution vary from the mean. The SD is an important

statistic, both for understanding dispersion within a distribution and for interpreting the

relationship of a particular value to the distribution. When the scores of a distribution

deviate from the mean considerably, the SD or spread of scores is large. When the degree

of deviation of scores from the mean is small, the SD or spread of the scores is small. SD

is a measure of dispersion that is the square root of the variance. The equation and steps

for calculating the standard deviation are presented in Exercise 27 , which is focused on

calculating descriptive statistics. Chapter6: Understanding Frequencies and Percentages

RESEARCH ARTICLE

Source

Roch, G., Dubois, C. A., & Clarke, S. P. (2014). Organizational climate and hospital nurses ’

caring practices: A mixed-methods study. Research in Nursing & Health, 37 (3), 229–240.

EXERCISE

90 EXERCISE 9 • Measures of Dispersion: Range and Standard Deviation

Introduction

Roch and colleagues (2014) conducted a two-phase mixed methods study ( Creswell, 2014 )

to describe the elements of the organizational climate of hospitals that directly affect

nursing practice. The fi rst phase of the study was quantitative and involved surveying

nurses ( N = 292), who described their hospital organizational climate and their caring

practices. The second phase was qualitative and involved a study of 15 direct-care registered

nurses (RNs), nursing personnel, and managers. The researchers found the following:

“Workload intensity and role ambiguity led RNs to leave many caring practices to

practical nurses and assistive personnel. Systemic interventions are needed to improve

organizational climate and to support RNs ’ involvement in a full range of caring practices”

( Roch et al., 2014 , p. 229). Chapter6: Understanding Frequencies and Percentages

Relevant Study Results

The survey data were collected using the Psychological Climate Questionnaire (PCQ) and

the Caring Nurse-Patient Interaction Short Scale (CNPISS). The PCQ included a fi vepoint

Likert-type scale that ranged from strongly disagree to strongly agree , with the high

scores corresponding to positive perceptions of the organizational climate. The CNPISS

included a fi ve-point Likert scale ranging from almost never to almost always, with the

higher scores indicating higher frequency of performing caring practices. The return rate

for the surveys was 45%. The survey results indicated that “[n]urses generally assessed

overall organizational climate as moderately positive ( Table 2 ). The job dimension relating

to autonomy, respondents ’ perceptions of the importance of their work, and the

feeling of being challenged at work was rated positively. Role perceptions (personal workload,

role clarity, and role-related confl ict), ratings of manager leadership, and work

groups were signifi cantly more negative, hovering around the midpoint of the scale, with

organization ratings slightly below this midpoint of 2.5.

Caring practices were regularly performed; mean scores were either slightly above or

well above the 2.5 midpoint of a 5-point scale. The subscale scores clearly indicated,

however, that although relational care elements were often carried out, they were less

frequent than clinical or comfort care” ( Roch et al., 2014 , p. 233). Chapter6: Understanding Frequencies and Percentages

TABLE 2 NURSES ’ RESPONSES TO ORGANIZATIONAL CLIMATE SCALE AND SELF-RATED

FREQUENCY OF PERFORMANCE OF CARING PRACTICES ( N = 292)

Scale and Subscales

(Possible Range) M SD

Observed

Range

Organizational Climate

Overall rating (1–5) 3.13 0.56 1.75–4.67

Job (1–5) 4.01 0.49 1.94–5.00

Role (1–5) 2.99 0.66 1.17–4.67

Leadership (1–5) 2.93 0.89 1.00–5.00

Work group (1–5) 3.36 0.88 1.08–5.00

Organization (1–5) 2.36 0.74 1.00–4.67

Caring Practices

Overall rating (1–5) 3.62 0.66 1.95–5.00

Clinical care (1–5) 4.02 0.57 2.44–5.00

Relational care (1–5) 2.90 1.01 1.00–5.00

Comforting care (1–5) 4.08 0.72 1.67–5.00

Roch, G., Dubois, C., & Clarke, S. P. (2014). Research in Nursing & Health, 37 (3), p. 234.

Measures of Dispersion: Range and Standard Deviation • EXERCISE 9 91

STUDY QUESTIONS

1. Organizational Climate was measured with which type of scale? What level of measurement was

achieved with this scale? Provide a rationale for your answer.

2. The mean ( X ) is a measure of __________________ ___________________ of a distribution,

while the standard deviation ( SD ) is a measure of _______________________ of its scores. Both

X a nd S D are __________________________ statistics.

3. What is the purpose of the range, and how is it determined in a distribution of scores?

4. What subscales were included in the description of Organizational Climate? Do these seem

relevant? Provide a rationale for your answer with documentation.

5. Which Organizational Climate subscale had the lowest mean? What does this result probably

mean?

6. What were the dispersion results for the Organization subscale in Table 2 ? What do these results

indicate?

92 EXERCISE 9 • Measures of Dispersion: Range and Standard Deviation

7. Which aspect or subscale of Organizational Climate has the lowest dispersion or variation of

scores? Provide a rationale for your answer.

8. Is the dispersion or variation of the ratings on Jobs more homogeneous or heterogeneous than

the other subscales? Provide a rationale for your answer.

9. Which subscale of Organization Climate had the greatest dispersion of scores? Provide a rationale

for your answer. Chapter6: Understanding Frequencies and Percentages

10. What additional research is needed in this area?

Answers to Study Questions

1. Organizational Climate was measured with the Psychological Climate Questionnaire (PCQ),

which is a 5-point Likert scale. This scale has multiple items, and the participants mark their

responses to each item using a scale of 1 = strongly disagree to 5 = strongly agree . The data

obtained from multiple-item Likert scales are combined and usually analyzed as though they

are interval-level data as in this study ( Grove et al., 2013 ). Some sources might describe Likert

scale data as ordinal because the 5-point rating scale used in a Likert scale lacks continuous

values. However, most nursing and healthcare researchers analyze data from multiple-item

Likert scales as interval-level data.

2. The X i s a measure of central tendency, and the S D is a measure of dispersion. Both X and SD

are descriptive or summary statistics.

3. Range is the simplest measure of dispersion, obtained by identifying the lowest and highest

scores in a distribution or by subtracting the lowest score from the highest score in the distribution

of scores.

4. The subscales included in Organizational Climate were Job, Role, Leadership, Work Group,

and Organization (see Table 2 ). Yes, these subscales seem relevant because the items used to

measure Job were related to perceived autonomy, importance of work, and being challenged.

The Role subscale included personal workload, role clarity, and role-related confl ict (see narrative

of results). Thus, the items of these fi ve subscales are important in understanding the

organizational climate in a hospital. The American Hospital Association (AHA) promotes

research to improve the climates in hospitals. For more information on AHA, review their

website at http://www.aha.org/research/index.shtml . A subsidiary of AHA is the American

Organization of Nurses Executives, which is focused on improving nursing leadership in the

current healthcare system (AONE; http://www.aone.org/ ). You might document with other

research articles, texts, and websites. Chapter6: Understanding Frequencies and Percentages

5. Organization had the lowest mean at 2.36, indicating this is the most negatively perceived of

the subscales covered by the PCQ scale. The lower the mean the more negative the nurses ’

perception of their organization.

6. The dispersion results for the Organization subscale included range = 1.00–4.67 and SD =

0.74. The score for each item on the Organization subscale could range from 1.00–5.00 based

on the Likert scale used in the PCQ. Both the range and SD seemed similar to the other subscales,

indicating the dispersion of scores was similar for the Organization subscale.

7. The Job subscale had the lowest dispersion with range = 1.94–5.00 or, when calculating the

range by subtracting the lowest score from the highest score, 5.00 − 1.94 = 3.06. The SD =

0.49 was also the lowest for Organizational Climate, indicating the scores for Job had the

lowest variation of the subscales. Focusing on the subscales ’ results rather than just on the

overall Organizational Climate rating provides readers with a richer understanding of the

nurses ’ perceptions of their organization.

94 EXERCISE 9 • Measures of Dispersion: Range and Standard Deviation

8. Job scores were the most homogeneous or had the least variation of the Organization Climate

subscales as indicated by the lowest range and SD results discussed in Question 7.

9. When compared with the other subscales, Leadership scores had the greatest dispersion or

variation among the subscales as indicated by the largest SD ( SD = 0.89) and range (1.00–5.00

or 5.00 − 1.00 = 4).

10. Additional studies in this area might include a larger sample size of RNs obtained from more

diverse hospitals. The response rate of 45% might be increased with an online survey format

and additional reminders sent to study participants reminding them to complete the survey.

An increased sample size might provide a stronger description of the hospitals ’ organizational

climate and the RNs ’ caring practices. Roch et al. (2014) indicated that interventions need to

be developed and tested to improve organizational climate and to support RNs ’ implementation

of caring practices. Chapter6: Understanding Frequencies and Percentages

Questions to Be Graded EXERCISE

Follow your instructor ’ s directions to submit your answers to the following questions for grading.

Your instructor may ask you to write your answers below and submit them as a hard copy for

grading. Alternatively, your instructor may ask you to use the space below for notes and submit your

answers online at http://evolve.elsevier.com/Grove/statistics/ under “Questions to Be Graded.” Chapter6: Understanding Frequencies and Percentages

1. What were the name and type of measurement method used to measure Caring Practices in the

Roch, Dubois, and Clarke (2014) study?

2. The data collected with the scale identifi ed in Questions 1 were at what level of measurement?

Provide a rationale for your answer.

3. What were the subscales included in the CNPISS used to measure RNs ’ perceptions of their

Caring Practices? Do these subscales seem relevant? Document your answer.

4. Which subscale for Caring Practices had the lowest mean? What does this result indicate?

Name: _______________________________________________________ Class: _____________________

Date: ___________________________________________________________________________________

96 EXERCISE 9 • Measures of Dispersion: Range and Standard Deviation

5. What were the dispersion results for the Relational Care subscale of the Caring Practices in

Table 2 ? What do these results indicate?

6. Which subscale of Caring Practices has the lowest dispersion or variation of scores? Provide a

rationale for your answer.

7. Which subscale of Caring Practices had the highest mean? What do these results indicate?

8. Compare the Overall rating for Organizational Climate with the Overall rating of Caring

Practices. What do these results indicate?

9. The response rate for the survey in this study was 45%. Is this a study strength or limitation?

Provide a rationale for your answer.

10. What conclusions did the researchers make regarding the caring practices of the nurses in this

study? How might these results affect your practice?

Description of a Study Sample

STATISTICAL TECHNIQUE IN REVIEW

Most research reports describe the subjects or participants who comprise the study

sample. This

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